Appendix A
Solutions of Linear Differential Equations
A.l Linear Differential Equations with Constant Coefficients
Linear diflFerential equations with constant coefficients are usually writ ten as 2/("> + ai2/("-i) + ... + a„_i2/(i) + anV = g, (A.l) where a^, fc = 1,..., n, are numbers, y^^^ = ^ , and g = g{t) is a known function of t. We shall denote hy D = ^ the derivative operator^ so that the differential equation now becomes
p{D)y = (D^ + aiD^-i + ... + a^_iD + an)y = g. (A.2)
If g(t) = 0, the equation is said to be homogeneous. If g{t) ^ 0, then the homogeneous or reduced equation is obtained from (A.2) by replacing g byO. If y and y* are two different solutions of (A.2), then it is easy to show that y — y* solves the reduced equation of (A.2). Hence, if y is any solution to (A.2), it can be written as
y = y*+y\ (A.3) where y* is any other particular solution to (A.2) and y^ is a suitable solution to the homogeneous equation. Therefore, solving (A.2) involves (a) finding all the solutions to the homogeneous equation, caUed the gen eral solution, and (b) finding a particular solution to the given equation. 364 A. Solutions of Linear Differential Equations
The rest of these notes indicate how to solve these two problems. Given (A.l) the auxiliary equation is p{m) = mP + aim^'^ + ... + an-im + an = 0, (A.4) In other words, p{m) is obtained from p{D) by replacing D by m. The auxiliary equation is an ordinary polynomial of nth degree and has n real or complex roots, counting multiple roots according to their multiplicity. We will see that, given these roots, we can write the general solution forms of homogeneous Unear differential equations.
A.2 Homogeneous Equations of Order One
Here the equation is (D - a)y = y'-ay = 0, which has y = Ce^^ as its general solution form.
A.3 Homogeneous Equations of Order Two
Here the differential equation can be factored (using the quadratic for mula) as (D-mi)(Z)-m2)2/-0, where m\ and m^ can be real or complex. Examples are given in Table A.l and the solution forms are given in Table A.2.
Differential Equation General Solution Form
1. y"-Ay' + Ay = Q y{t) = e2*(Ci + Cit)
2. y" - %' + 3y = 0 y{t) = e2*(Di sinh t) + D2 sinh t)
3. y" - 4y' + 5y = 0 y{t) = e^\Disva. t + Di cos t)
Table A.l: Examples of Homogeneous Equations of Order Two A.4. Homogeneous Equations of Order n 365
Root General Solution Form
^1 7^ ^2? real y{t) = Cie^i* + C2e^2t
= e^*(Cie^^ + C2e-^^)
or
mi = a + 6, 1712 = a — b y(t) = e"*(Ci sinh 6t + C2 cosh bt)
mi = 7722 = ^^ y{t) = iCi+C2t)e^'
^1 7^ ^2 J complex y(t) = Cie^i* + C2e^2t
- e"^(Cie^^* + C2e-'^^)
or
mi = a + bi^ m2 = a — bi y(^) 3.: e"^(Di sin 6t + D2 cos bt)
or
y(^) = e^'^lEi sm{bt + ^2)]
or
y{t) = e''^[Ficos{bt + F2)] |
Table A.2: General Solution Forms for Second-Order Linear Homogeneous Equations, Constant Coefficients
A.4 Homogeneous Equations of Order n
When (A.2) is of order n, the auxiliary equation p(m) = 0 has n roots, when multiple roots are coimted according to their multiplicity. Also, complex roots occur in conjugate pairs. The general solutions of the homogeneous equations is the sum of the solutions associated with each multiple root. They can be foimd in Table A.4 for each root and should be added together to form the general solution. First, we give some examples in Table A.3. 366 A. Solutions of Linear Differential Equations
Differential Equation General Solution Form
1. D2(2)2 _4D + 4)2/-0 y(t) = Ci + C2i + e2*(C3 + C^t)
2. (D-3)2(D + 5)3(D2-4D + 5)2y=0 2/(t)-e3*(Ci+C2t)
+e-5t(C3 + C4t + C5t2)
+e2*[(C6+C7)sin t
^(Cs + CgOoos t]
3. (D2 -2D-\- 2)3(D2 - 2D -- 3)^2/ = 0 2/(i) = e*[(Ci -f C2* + C3t2) sin t
+(C4-f-C5t-f C6t2)cos i]
+(C7 + Cst)e^^ + (Cg + Ciot)e-^
Table A.3: Examples of Homogeneous Equations of Order n
Root Multiplicity General Solution Form
r,=l yj{t) = Ce'^J^
rrij, real rj >1 yj{t) = (Ci 4- C2* + ... + Crjf-J -i)e"^i*
Complex Conjugate r,=l e'^J*(Ci sin 6jt + C2 cos 6jt)
aj lb bj2 0>1 e^i*[Ci + C2t + ... + Cr^-Ti -^)sin 6jt]
+(Cr^- + l + Cr^+2t + .. . + C2r,-r:'' -^) COS bjt]
Table A.4: General Solution Forms for Multiple Roots of Auxiliary Equation
A.5 Particular Solutions of Linear D.E. with Constant Coefficients
The particular solution to the inhomogeneous equation (A.2) can usually be found by guessing the form of the answer and then verifying the guess by substitution. Table A.5 shows the correct forms for guessing for various kinds of forcing fimctions g(t). Note that the form of the guess depends on whether certain nimibers are roots of the auxiliary equation. Table A.6 gives examples of differential equations along with their particular integrals. A,5. Particular Solutions of Linear D,E, — Constant Coefficients 367
Forcing Function, g{t) K Particular Integral, y{t)
(i)c 0 A
(2) h{t) 0 X
(3) csin qt or ccos qt 9 A sin qt — B cos qt
(4) ce"* 9 Ae (5) ce^* sin qt or ce^* cos qt p + iq AeP* sin ^t - Be^* cos ^i (6) /i(i)e«* Q Xe«* (7) /i(t) sin ^t or /i(f) cos qt iq X sin gi + y cos qfi (8) /i(t)eP*sin gt or /i(i)eP*cos gt p + iq XeP*sin ^i + FeP^cos qt Notation. (a) In the forcing function column, p, q, and c are given constants and h{t) is a given polynomial of degree s. (b) In the p>articular integral column, A and B are coefficients to be determined and X = ^0 + Alt + ... + Ast\ Y = Bo-\-Bit+ ... + Bst^ are s degree polynomials whose coefficients are to be determined. Rules. (a) If the number in the K column is not a root of the auxiliary equation p(m) = 0, then the proper guess for the particular integral is as shown. (b) If the number in the K colimin is a root of the auxiliary equation of degree r, then multiply the guess in the last column by t^. Table A.5: Particular Solution Forms for Various Forcing Functions If the forcing function g{t) is the sum of several functions, 9^=91 + g2 + *"+9ky each having one of the forms in the table, then solve for each Qi separately and add the results together to get the complete solution. In the next table, we wiU apply the formulas and the rules in Table A.6 to obtain particular integrals in specific examples. 368 A. Solutions of Linear Differential Equations Differential Equation Particular Integral 1.2/'"-32/" = 5 At^ 2. y'" - 3y" = l + 3t + 5t^ t^Ao + Ait + A2t'^) 3. 2/"-V + 42/ = 3-i2 Ao + Ait + A2t^ 4. y" -4y' + 4y = 2sm t Asm t + Bcos t 5. 2/"-43/'+ 42/= 5sin 2t Asm 2t + Bcos 2t 6. 2/" - 42/' + 4y = lOe^* Ae'^ 7. 2/" - 4y' + 42/ = lOe^* i2(Ae2«) 8. 2/" - 42/' + 52/ = e^* cos t t{Ae^^ sin t + Be^* cos t) 9. 2/"-42/' + 52/ = t^sin f X^r_o(^r^'^ sin i + B^V cos i) 10. y" - 42/' + 5y = i^e^* cos t * Er=o(^r*''e2* sin « + B^fe^* cos *)J Table A.6: Particular Integrals in Specific Examples Consider the first-order linear equation 2/' + ay = /(i). (A.5) If we multiply both sides of the equation by the integrating factor e"*, we get d ^(ye«')=j/'e"*+aye«* = e"V(i). (A.6) Integrating from 0 to t we have y{t) = y{0)e--' + T e'^(--^)/(T)rfr, (A.7) Jo which is the complete solution (homogeneous solution plus particular solution) to the equation. A, 7. Reduction of Higher-Order to First-Order Linear Equations 369 A.7 Reduction of Higher-Order Linear Equations to Systems of First-Order Linear Equations Another way of solving equation (A.l) is to convert it into a system of first-order linear equations. We use the transformations zi = y, Z2 = y^^\...,zn = y^'' ^\ (A.8) so that (A.l) can be written as 0 1 0 .. 0 zi 0 4 0 0 1 .. 0 ^2 0 + 0 0 0 .. 1 Zn-1 0 Z^ —dn -an- -1 -CLn- -2 • ai Zn 9 J (A.9) In vector form this equation reads z' = Az + b (A.10) with the obvious definitions obtained by comparing (A.9) and (A. 10). We will present two ways of solving the first-order system (A. 10). The first method involves the matrix exponential function e*"* defined by the power series /2 42 (A.11) E A;! It can be shown that this series converges (component by component) for all values of t. Also it is differentiable (component by component) for all values of t and satisfies ^(e*^) = Ae'^ = {e'^)A. (A.12) By analogy with Section A.6, we try e^* as the integrating factor for (A. 10) to obtain 370 A. Solutions of Linear Differential Equations (Note that the order of matrix multiphcation here is important.) Using the product rule for matrix multiphcation of fimctions, which can be shown to be vahd, the above equation becomes dV ' Integrating from 0 to i gives TA b{r)dT. Jo Evaluating and solving, we have z{t) = e'^z{0) + e'^ r .-TA b{T)dr, (A.13) Jo The analogy between this equation and (A.6) is clear. Although (A.13) represents a formal expression for the solution of (A. 10), it does not provide a computationally convenient way of getting explicit solutions. In order to demonstrate such a method we assimie that the matrix A is diagonalizable, i.e., that there exists a nonsingtilar square matrix P such that P-^AP = A. (A.14) Here A is the diagonal matrix Ai 0 ••• 0 0 A2 ••• 0 A = (A.15) 0 0 An where the diagonal elements, Ai,..., A^i, are eigenvalues of A, The ith colunm of P is the column eigenvector associated with the eigenvalue A^ (to see this multiply both sides of (A.14) by P on the left). By looking at (A.ll) it is easy to see that p-l^tAp^^tA^ (A.16) Suppose we make the following definitions: z = Pw, z(0) = Pw{Q), z' = Pw\ (A.17) A. 7. Reduction of Higher-Order to First-Order Linear Equations 371 These in turn imply w = P-^Z, w{0) = P-h{0), w' = p-^z\ (A.18) Substituting (A. 17) into (A. 10) gives Pw' = APw + b, vJ = p-^APw + p-^b, which by using (A. 14) gives w' = Aw + p-^b. (A.19) Since A is a diagonal matrix, it is easy to solve the homogeneous part of (A.19), which is w' = Aw. (A.20) The solution is Wi = Wi{0)e~^^^ for i = 1,..., n. We solve (A.19) completely by multiplying through by the integrating factor e~*^: ^(e-'^w) = e'^w' - e-^^Aw = e'^^p-^b. at Integrating this equation from 0 to ^ gives w{t) = e^^w(0) + e^^ f e-^^p-^b{T)dT. (A.21) Jo Using the substitutions (A.18) yields z{t) = {Pe^^p-^)z{0) + Pe^^ f e-^^p-^b{r)dT, (A.22) Jo which is the formal solution to (A. 10). Since well-known algorithms are available for finding eigenvalues and eigenvectors of a matrix, the solution to (A.22) can be found in a straightforward manner. 372 A. Solutions of Linear Differential Equations A.8 Solution of Linear Two-Point Boundary Value Problems In linear-quadratic control problems with linear salvage values (e.g., the production-inventory problem in Section 6.1) we require the solution of linear two-point boundary value problems of the form X 11 An X hi + (A.23) 21 A22 A 62 J with boundary conditions a;(0) = XQ and A(T) = Ay. (A.24) The solution of this system will be of the form (A.22), which can be restated as x{t) Quit) Quit) rr(O) Rx{t) + (A.25) m Q2l{t) Q22{t) . ^(°^ . Mt) where the A(0) is a vector of unknowns. They can be determined by setting AT = Q2i{T)x{0) + Q22(T)A(0) + R^iT), (A.26) which is a system of linear equations for the variables A(0). A.9 Homogeneous Partial Differential Equations A homogeneous partial differential equation is an equation containing one or more partial derivatives of an unknown function with respect to its independent variables. If the highest partial derivative appearing ex plicitly in the equation has order n, then the partial differential equation is said to be of order n. As we saw in the previous sections, the general solutions of ordinary differential equations involve expressions containing arbitrary constants. Similarly, the solutions of partial differential equations are expressions containing arbitrary (differentiable) functions. Conversely, when arbi trary fimctions can be eliminated algebraically from a given expression, A.9, Homogeneous Partial Differential Equations 373 after suitable partial derivatives have been taken, then the result is a partial differentiable equation. Example A,l Eliminate the arbitrary function / from the expression u = f{ax — by), where a and b are non-zero constants. Solution. Taking partial derivatives, we have Ux = af and Uy = —bf so that fru^ + auy = abf - abf = 0. (A.27) Here u = f{ax — by) is a, solution for the equation bux + aUy = 0. To show that any solution u = g{x, y) can be written in this form, we set s = ax — by, and define \s + by G{s,y) = g .y = 9{^^y)' Then, gx = Gs% = aGs and gy = G,^ + Gy = -bGs + Gy. Since we assume g solves the equation bux + aUy = 0, we have Q = bgx- agy = abGg - abGs + aGy = 0, (A.28) but this implies Gy = 0 so that G is a function oi s = ax — by only, and hence g{x, y) = G{s) = G{ax — by) is of the required form. We conclude that u = f(ax — by) is a general solution form for bux +auy = 0. Example A.2 Eliminate the arbitrary fimctions /i and /2 from the expression u = fi{x)f2{y). Solution. Taking partial derivatives, we have "^x = /i/2, Uy = /1/2, and Uxy = /I/2 so that uuxy - UxUy = /i/2/{/2 - /i/2/1/2 = 0- As in Example A.l we conclude that u = h{^)f2{y) is the general solu tion form of the equation uUxy — UxUy = 0. The subject of partial differential equations is too vast to even sur vey here. However, Table A.7 gives general solution forms for all the homogeneous partial differential equations we will consider in this book, as well as others. 374 A. Solutions of Linear Differential Equations Partial Differential General Solution Form Ekjuation (1) bux + auy = 0 u = f(ax — by) (2) XUx + yUy =0, X y/^0 u = f{y/x) (3) xux — yUy == 0 u = fi^y) (4) Ux -\-Uy =^ au u = h{x- y)e^ + f2{x - y)e''y (5) Uz-\-Uy = au^, k ^ 1 u^[{k-\){h{x-y)-ax)Y/^^-'^ ^[k-l)h{x-y)-ay)Y^^^-') (6) ^xx ^ yy — ^ u=:fi{y + ax) + hiy - ax) (7) Uxx + CL^Uyy = 0 u = fi{y + iax) -\- f2(y- iax), i = y/^ (8) Uxy =0 u = fi(x) - f2(y) (9) UUxy —UxUy =0 u = fi(x)f2(y) (10) UUxy +UxUy=0 u = Mx)/f2{y) (11) bcux + acuy + abuz = 0 u =: fi{ax - by) ^ f2(by - cz) + fsicz - ax) (12) xUx + yuy 4- zuz = 0 u = fi(x/y)-\-f2{y/z) (13) xUx — yUy -1- xuz = 0 u = fi(xy)-\-f2{yz) (15) U^Uxyz — UxUyUz = 0 u = Mx)f2(y)f3(z) (16) ^txy^ = 0 u=-fi{x)-\-f2iy)-\-f3{z) (17) Uxx — 0^{Uyy 4- Uzz) = 0 u = fi(y + ax) + f2{y - ax)-\- h{z + ax) +f4{z-ax) (18) Ux -\- Uy + Uz = au u = h{x- y)e^ + h{y- ^)e"^ + h{z - ^)e"' [ Note. The function fi are arbitrary differentiable functions of a single variable; a, 6, c,... stand for arbitrary (non-zero) constants. Table A.7: General Solution Forms for Some Homogeneous Partial Differential Equations A. 10 Inhomogeneous Partial Differential Equations As in the ordinary case, an inhomogeneous partial differential equation is obtained from a homogeneous one by adding one or more forcing A.iJ. Solutions of Finite Difference Equations 375 functions. The general case of this problem is too difficult to treat here. We consider only the case in which the forcing functions are separable, i.e., can be written as a sum of functions each involving only one of the independent variables. In solving such problems we can make use of the solutions to ordinary differential equations considered earlier. Example A.3 Solve the partial differential equation Ux + Uy = 3x + e^. Solution. We know from the previous section that the general solution to the homogeneous equation is of the form f{x—y). To get the particular solutions we solve separately the ordinary equations Ux = Sx^ and Uy = e^^ obtaining solutions x^ and e^. Therefore, the general solution to the original equation is u = f{x-y)+x^ + e^. Generally speaking, the above philosophy of finding particular solu tions to separable partial differential equations (when it works) follows the same method of "dividing and conquering." Other methods involve the use of series. We will not go further here for lack of space. A. 11 Solutions of Finite Difference Equations In this book we will have uses for finite difference equations only in Chapters 8 and 9. For that reason we will give only a brief introduction to solution techniques for them. Readers who wish more details can consult one of several texts on difference equations; see, e.g., Goldberg (1986) or Spiegel (1971). If f{k) is a real function of time, then the difference operator applied to / is defined as A/(fc) = /(fc + l)-/(fc). (A.29) The factorial power of k is defined as A;(") = k{k - l)ik -2)...{k-{n- 1)). (A.30) It is easy to show that Aifc("> = nfc("-i). (A.31) 376 A. Solutions of Linear Differential Equations Because this formula is similar to the corresponding formula for the derivative d{k'^)/dk^ the factorial powers of k play an analogous role for finite differences that the ordinary powers of k play for differential calculus. K /(fc) is a real function of time, then the anti-difference operator A~-^ applied to / is defined as another fimction g — A~^f{k) with the property that A^ = f(k). (A.32) One can easily show that A-ifcW - (l/(n + l))fc(^+i) + c, (A.33) where c is an arbitrary constant. Equation (A.33) corresponds to the integration formula for powers of k in calculus. Note that formulas (A.31) and (A.33) are similar to, respectively, differentiation and integration of the power function k"^ in calculus. By analogy with calculus, therefore, we can solve difference equations in volving polynomials in ordinary powers of k by first rewriting them as polynomials involving factorial powers of k so that (A.31) and (A.33) can be used. We show next how to do this. A. 11.1 Changing Polynomials in Powers of k into Facto rial Powers of k We first give an abbreviated list of formulas that show how to change powers of k into factorial powers of k: fcO = fc(o) = 1 (by definition), k'=k('\ fc4 = fc(i)+7/c(2)+6fc(3)+fcW, k^ = fc(i) + 15fc(2) + 25fc(^) + lOfcW + fc(^). The coefficients of the factorial powers on the right-hand sides of these equations are called Stirling numbers of the second kind, after the person who first derived them. This list can be extended by using a A,ll. Solutions of Finite Difference Equations 377 more complete table of these nimibers, which can be found in books on difference equations cited earlier. Example A.4 Express A;^ — 3fc + 4 in terms of factorial powers. Solution. Using the equations above we have k^ = fed) + 7fc(2) + 6fc(3) + fcW, -3k = -3fc(i), 4 = 4, SO that Example A.5 Solve the following difference equation in Example 8.7: AA^ = -k + 6,X^ = 0. Solution. We first change the right-hand side into factorial powers so that it becomes AA^ = -fc(^) + 5. Applying (A.33), we obtain A^ = -(l/2)fc(2) + 5fcW + c, where c is a constant. Applying the condition A^ = 0, we find that c= —15, so that the solution is A^ = -(l/2)fc(2) + 5fc(^) - 15. (A.34) However, we would like the answer to be in ordinary powers of fc. The way to do that is discussed in the next section. A. 11.2 Changing Factorial Powers of k into Ordinary Powers of k In order to change factorial powers of k into ordinary powers of fc, we make use of the following formulas: fc(i) = k, ki'^) = ^k + k\ fc(3) = 2k- 3A;2 + k^, k^^) = -6k + llfc2 _ 6^3 ^ ^4^ 378 A. Solutions of Linear Difkrential Equations fc(5) = 24k - 50A;2 + 35k^ - lOJt^ + k^. The coefficients of the factorial powers on the right-hand sides of these equations are called Stirling numbers of the first kind. This list can also be extended by using a more complete table of these nimibers, which can be found in books on difference equations. Solution of Example A.5 Continued. By substituting the first two of the above formulas into (A.34), we see that the desired answer is A'^ = -(1/2)A;2 + (n/2)k - 15, (A.35) which is the solution needed for Example 8.7. EXERCISES FOR APPENDIX A 3 2 5 0 1 1 A.l If ^ = show that A andF = 2 3 0 2 1 -1 Use (A.22) to solve (A. 10) for this data, given that ^(O) = 3 3 6 0 1 3 A.2 If A , show that A = andP = 2 4 0 1 1 -2 0 Use (A.22) to solve (A. 10) for this data, given that z{0) = 5 A.3 After you have read Section 6.1, re-solve the production-inventory example stated in equations (6.1) and (6.2), (ignoring the control constraint (P > 0) by the method of Section A.8. The linear two-point boundary value problem is stated in equations (6.6) and (6.7). Appendix B Calculus of Variations and Optimal Control Theory Here we introduce the subject of the calculus oi variations by analogy with the classical topic of maximization and miniiir:zation in calculus; see Gelfand and Fomin (1963), Young (1969), and Leitmann (1981) for rigorous treatments of the subject. The problem of the calculus of varia tions is that of determining a function that maximizes a given functional, the objective fimction. An analogous problem in calculus is that of de termining a point at which a specific function, the objective function, is maximum. This, of course, is done by taking the first derivative of the function and equating it to zero. This is what is called the first-order condition for a maximum. A similar procedure will be employed to de rive the first-order condition for the variational problem. The analogy with classical optimization extends also to the second-order maximiza.- tion condition of calculus. Finally, we will show the relationship between the maximum principle of optimal control theory and the necessary con ditions of the calculus of variations. It is noted that this relationship is similar to the one between the Kuhn-Tucker conditions in mathematical programming and the first-order conditions in classical optimization. We start with the "simplest" variational problem in the next section. B.l The Simplest Variational Problem Assume a function x : C-^[0^t] —> E^, where C^[O^T] is a class of func tions defined over the interval [0,T] with continuous first derivatives. (For simplicity in exposition, assimie x to be a scalar fimction. The 380 B. Calculus of Variations and Optimal Control Theory extension to a vector function is straightforward.) Assume further that a function in this class is termed admissible if it satisfies the terminal conditions x{0) = xo and x{T) = XT- (B.l) We are thus dealing with a fixed-end-point problem. Examples of admis sible functions for the problem are shown in Figure B.l; see Section 6 and Chapters 2 and 3 of Gelfand and Fomin (1963) for problems other than the simplest problem, i.e., the problems with other kinds of conditions for the end points. 0 T Figure B.l: Examples of Admissible Functions for the Problem The problem under consideration is to obtain the admissible function X* for which the fimctional fT J{x) = / g{x^x^t)dt (B.2) Jo has a relative maximum. We will assume that all first and second partial derivatives of the function g : E^ x E^ x E^ —^ E^ are continuous. B.2 The Euler Equation The necessary first-order conditions in classical optimization were ob tained by considering small changes about the solution point. For the B.2, The Euler Equation 381 variational problem, we consider small variations about the solution func tion. Let x{t) be the solution and let y{t)=x{t) + sri{t), where T]{t) : C^[0,T] —» E^ is an arbitrary continuously differentiable function satisfying r){0) = r,{T) = 0, (B.3) and £ > 0 is a small number. A sketch of these functions is shown in Figure B.2. Figure B.2: Variation about the Solution Function The value of the objective functional associated with y{t) can be considered a fimction of s, i.e., fT V{e) = J{y) = g{x + erj, x + ef],, t)dt Jo However, x(t) is a solution and therefore V{e) must have a maximum at £ = 0. This means A dV\ -0, de e=Q where 8J is known as the variation 8J in J. Differentiating V{e) with respect to e and setting e — Q yields 8J = = / {9xV + 9xf])dt = 0, de =0 ^0 382 B. Calculus of Variations and Optimal Control Theory which after integrating the second term by parts provides «5^ = -^1 = j^ 9xVdt + {9iv)\o - I Jt^9i)vdt = 0. (B.4) Because of the end conditions on 77, the expression simplifies to = [9x- •:^9x]vdt = 0. de zzo Jo dt We now use the fundamental lemma of the calculus of variations which states that if /i is a continuous fimction and /Q h(t)rj{t)dt = 0 for every continuous function r]{t)^ then h{t) — 0 for all t G [0, T]. The reason that this lemma holds, without going into details of a rigorous proof which is available in Gelfand and Fomin (1963), is as follows. Suppose that h(t) ^ 0 for some t G [0,T]. Since h{t) is continuous, there is, therefore, an interval (^1, ^2) C [0, T] over which h is nonzero and has the same sign. Now selecting r/(t) such >0, te{ti,t2) r]{t) is < 0, otherwise, it is possible to make the integral /Q h(t)r){t)dt 7^ 0. Thus, by contrar- diction, h{t) must be identically zero over the entire interval [0,T]. By using the fimdamental lemma, we have the necessary condition 9x - -Tjgx = 0 (B.5) known as the Euler equation, which must be satisfied by a maximal solution X*, We note that the Euler equation is a second-order ordinary differen tial equation. This can be seen by taking the total time derivative of QX and collecting terms: ^9xx + i^9xx + {9tx - 9x) = 0. The boimdary conditions for this equation are obviously the end-point conditions x{0) = XQ and x{T) = x^- B.3. The Shortest Distance Between Two Points on the Plane 383 Special Case (i) When g does not depend explicitly on x. In this case, the Enler equation (B.5) reduces to which is nothing but the first-order condition of classical optimization. In this case, the dynamic problem is a succession of static classical optimization problems. Special Case (ii) When g does not depend explicitly on x. The Euler equation reduces to j^9^ = 0, (B.6) which we can integrate as gx = constant. (B.7) Special Case (iii) When g does not depend explicitly on t, Finally, we have the important special case in which g is explicitly independent of t. In this case, we write the Euler equation (B.5) as j^{9-i9x)-9t = 0. (B.8) But gt = 0 and therefore we can solve the above equation as g-xgx = C, (B.9) where C is the constant of the integration. B.3 The Shortest Distance Between Two Points on the Plane The problem is to show that the straight line passing through two points on a plane is the shortest distance between the two points. The problem 384 B. Calculus of Variations and Optimal Control Theory can be stated as follows: imn Jo subject to x(0) = xo and x{T) = XT- Here t refers to distance rather than time. Since g = — vT+^ does not depend explicitly on x^ we are in the second special case and the first integral (B.7) of the Euler equation is This implies that i is a constant, which results in the solution x{t) = Cit + C2, where Ci and C2 are constants. These can be evaluated by imposing boimdary conditions which give Ci — {XT — xo)/T and C2 = XQ. Thus, XT-XQ x{t) = t + Xo, T which is the straight line passing through XQ and XT- B.4 The Brachistochrone Problem The problem arises from the search for the shape of a wire along which a bead will slide in the least time from a given point to another, imder the influence of gravity; see Figure 1.1. The Brachistochrone problem has a long history. It was first studied (incorrectly) by Galileo in 1630. The problem was correctly posed by Johann Bernoulli in 1696 and later solved by Johann Bernoulli, Jacob Bernoulli, Newton, and L'Hospital. Note that Euler deduced the Euler's equation in 1744, and we will solve the Brachistochrone problem using Euler's equation. But first we must formulate the problem. Assimie the bead slides with no friction. Let m denote the mass of the bead, s denote the arc length, t denote the horizontal axis, x denote the vertical axis (measured vertically down), and r denote the time. Assimie ^0 = 0, rr(io) = 0, T-l, x{T) = l. B.4. The Brachistochrone Problem 385 We wish to minimize ^^ ds Jo Jo V where v represents velocity, and ST is the final displacement measured on the curve. We can write ds = \/l + x'^dt and, from elementary physics, it is known that if v{to = 0) — 0 and a denotes the gravitational acceleration constant, then v = V2^ax, Therefore, the variational problem can be stated as , 1 f'/* ! h 1+x^ min^TT- / \l—- dt \2aJo V X where x = dx/dt (note that t does not denote time), and x(0) = 0 and x{l) = 1. Since a is a constant, we can rewrite the problem as mm J{x) = / g(x^x,t)dt= / \ dt> . Since g does not depend explicitly on f, the problem belongs to the third special case. Using the first integral (B.9) of the Euler equation for this case, we have 1/2 i;2[x(l+i2)]-i/2 1+x^ = Ci (a constant). X We can reduce this to dx dt y xCf To solve this equation, we separate the variables as •^xdx dt ^l ""x 386 B. Calculus of Variations and Optimal Control Theory and substitute X = (sin^ e)/Cl = (1 - cos 26)/Cl (B.IO) The resulting expression can be integrated to yield t=[e- (1/2) sm2e]/Cf + C2. (B.ll) The condition ^ = 0 at ^ = 0 implies C2 = 0 providing Cf > 0. The value of Cl can be obtained in terms of the value of 6 at t = 1; let this be 9i. Then, since a; = 1 at ^ = 1, we have Ci = sin 6, where Oi satisfies 26>i --I = sin20i-cos26>i. This equation must be solved nimierically. An iterative numerical pro cedure yields 0i = 1.206 and therefore Cf = 0.873. Defining 0 = 26, we can write (B.IO) and (B.ll) as X = 0.573(1-cos 20), t = 0.573(0-sin0), which are equations of a cycloid in the parametric form. The shape of the curve is shown in Figure 1.1 in Chapter 1. B.5 The Weierstrass-Erdmann Corner Conditions So far we have only considered functionals defined for smooth curves. This is, however, a restricted class of curves which qualify as solutions, since it is easy to give examples of variational problems which have no solution in this class. Consider, for example, the objective functional min h{x) = I x'^il - xfdt\ , x(-l) = 0, x{l) = 1. The greatest lower bound for J(x) for smooth x = x{t) satisfying the boimdary conditions is obviously zero. Yet there is no x € C-^[—1,1] with x(—l) = 0 and x{l) = 1, which achieves this value of J(x). In fact, the minimum is achieved for the curve 0, -1 < < < 0, x(t) t, 0 which has a comer (i.e., a discontinuous first derivative) at t = 0, Such a piecewise smooth extremal with corners is called a broken extremal We now enlarge the class of admissible functions by relaxing the requirement that they be smooth everywhere. The larger class is the class of piecewise continuous functions which are continuously differentiable almost everywhere in [0,r], i.e., except at some points in [0, T]. Let x{t) be an extremal with a corner at r G [0, T]. Let us decompose J{x) as rT PT PT J{x) = / g{x,x^t)dt= / g{x,x,t)dt+ / g{x^x,t)dt Jo Jo JT = Ji{x)+J2{x), It is clear that on each of the intervals [0, r) and {r^T], the Euler equation must hold. To compute variations SJi and SJ2, we must recognize that the two 'pieces' oi x are not fixed-end-point problems. We must require that the two pieces of x join continuously at ^ = r; the point t = T can, however, move freely as shown in Figure B.3. Figure B.3: A Broken Extremal with Corner at r This will require a slightly modified version of formula (B.4) for writ ing out the variations; see pp. 55-56 in Gelfand and Fomin (1963). Equat ing the sum of variations SJ = SJi +SJ2 = 0 388 B. Calculus of Variations and Optimal Control Theory for x{t) to be an extremal and using the fact that x{t) must be continuous at t = r imphes g^l^- = g^l^^ , (B.12) [9 - igxlr- = [9- i9x]T+' (B.13) These conditions are called Weierstrass-Erdmann corner conditions, which must hold at the point r where the extremal has a corner. In each of the interval [0, r) and (r, t], the extremal x must satisfy the Euler equation (B.5). Solving these two equations will provide us with four constants of integration since the Euler equations are second-order differential equations. These constants can be foiind from the end-point conditions (B.l) and Weierstrass-Erdmann conditions (B.12) and (B.13). B.6 Legendre's Conditions: The Second Variation The Euler equation is a necessary conditions analogous to the first-order condition for a maximum (or minimimi) in the classical optimization problems of calculus. The condition analogous to the second-order nec essary condition for a maximum is the Legendre condition 9xx < 0. (B.14) To obtain this condition, we use the second-order condition of classical optimization on function V(e) to be a maximum at e = 0, i.e., d^V{e) PT / {9xxrj^ + '^gxxm + 9xxf]^)dt < 0. (B.15) de'^ e=0 JO Integrating the middle term by parts and using (B.3), we can transform (B.15) into a more convenient form / {Qri^ + Pfi^)dt While it is possible to rigorously obtain (B.14) from (B.16), we will only provide a qualitative argument for this. If we consider the quadratic functional (B.16) for functions r]{t) satisfying 77(0) = 0, then r]{t) will be B.7, Necessary Condition for a Strong Maximum 389 small in [0, T] if f]{t) is small in [0, T], The converse is not true, however, since it is easy to construct r]{t) which is small but has a large derivative 7]{t) in [0,r]. Thus, Pfj^ plays the dominant role in (B.16); i.e., Pif can be much larger than Qrf but it cannot be much smaller (provided P 7^ 0). Therefore, it might be expected that the sign of the fimctional in (B.6) is determined by the sign of the coefficient P(t), i.e., (B.16) implies (B.14). For a rigorous proof, see Gelfand and Fomin (1963). We note that the strengthened Legendre condition (i.e., with a strict inequahty in (B.14)), the Euler equation, and one other condition called strengthened Jacobi condition are sufficient for a maximum. The reader can consult Chapter 5 of Gelfand and Fomin (1963) for details. B.7 Necessary Condition for a Strong Maximum So far we have discussed necessary conditions for a weak maximum. By weak maximum we mean that the candidate extremals are smooth or piecewise smooth functions. The concept of a strong m>axim.um, on the other hand requires that the candidate extremal need only be continuous functions. Without going into details, which are available in Gelfand and Fomin (1963), we state a necessary condition for a strong maximum. This is called the Weierstrass necessary condition. The condition is analogous to the one in the static case that the objective function be concave. It states that if the fimctional (B.2) has a strong maximum for the extremal 7 satisfying (B.l), then E{x,x,t,v) <0 (B.17) along 7 for every finite v, where E is the Weierstrass Excess Function defined as E{x^ i, t, v) = g{x^ v^ t) — g{x^ i, t) — gxi^i ^, t){'^ — x), (B.18) Note that this condition is always met if g{x^x^t) is concave in x. The proof of (B.17) is by contradiction. Suppose there exists a r G [0, T] and a vector q such that E{T,x{T),x{T),q) >0, where x = x(t) is the equation of the extremal 7. It is then possible to suitably modify 7 to /? which is close to 7 in C^[0,T] such that ^J — I g{x^x^t)dt— / g{x^x^t)dt > 0, JB J-i 390 B. Calculus of Variations and Optimal Control Theory contradicting the hypothesis that J{x) has a strong maximum for 7. B.8 Relation to the Optimal Control Theory It is possible to derive the necessary conditions of the calculus of varia tions from the maximum principle. This is strongly reminiscent of the relationship between the first-order conditions of classical optimization and the Kuhn-Tucker conditions of mathematical progranmiing. First, we note that the calculus of variations problem can be stated as an optimal control problem as follows: max J — j g{x,u^t)dt > subject to X = n, x(0) = xo, x(T) = XT^ The Hamiltonian is H(x^ u^ A, t) = g(x^ u, t) + Xu (B.19) with the adjoint variable A satisfying A = —Hx = —Qx- (B.20) Maximizing the Hamiltonian with respect to u yields Hu = 9x + ^=> ^ = -Qx- (B.21) Differentiating with respect to time, we have d A dt 9x' This equation with (B.20) implies 9x - ^9x = 0, which is the Euler equation of the calculus of variations. B.8, Relation to the Optimal Control Theory 391 The second-order condition for the maximization of the Hamiltonian, i.e., Huu < 0 => pi;i < 0, which is the Legendre condition. Again, by the maximum principle, if u is an optimal control, then H{x,t,X,t) >H{x,v,\t), where v is any other control. By the definition of the Hamiltonian (B.19) and equation (B.21), we have g{x, i, t) - QxX > g(x, v, t) - g±v, which by transposition of terms yields the Weierstrass necessary condi tion E(x^ ir, i, v) = g{x^ v^ t) — g(x^ i, t) — gx{v — i) < 0. We have just proved the equivalence of the maximum principle and the Weierstrass necessary condition in the case where Q is open. In cases when O is closed and when the optimal control is on the boundary of f2, the Weierstrass necessary condition is no longer valid, in general. The maximum principle still applies, however. Finally, according to the maximum principle, both A and H are con tinuous functions of time. However, A == -gx and H = g - g±x, which means that the right-hand sides must be continuous with respect to time, i.e., even across corners. These are precisely Weierstrass-Erdmann corner conditions. Appendix C An Alternative Derivation of the Maximum Principle Recall that in the derivation of the maximum principle in Chapter 2, we assumed the twice differentiability of the return function V, Looking at (2.32), we can observe that the smoothness assumptions on the return function do not arise in the statement of the maximum principle. Also since it is not an exogenously given fimction, there is no a priori reason to assume the twice differentiabihty. In many important cases as a matter of fact, V has no derivatives at individual points, e.g., at points on switching manifolds. In what foUows, we wiU give an alternate derivation. This proof fol lows the course pointed out by Pontryagin et al. (1962) but with certain simplifications. It appears in Fel'dbaum (1965) and, in our opinion, it is one of the simplest proofs for the maximiim principle which is not related to dynamic programming and thus permits the elimination of assumptions about the differentiability of the return fimction V(t, x), We select the Mayer form of the problem (2.5) for deriving the max imum principle in this section. It will be convenient to reproduce (2.5) here as (C.l): max {J = cx(T)} u{t)en{t) subject to (C.l) X = f{x,U,t), x{0) =Xo, 394 C. An Alternative Derivation of the Maximum Principle C.l Needle-Shaped Variation Let u*{t) be an optimal control with corresponding state trajectory x*{t). We sketch u*{t) in Figure C.l and x*{t) in Figure C.2 in a scalar case. Note that the kink in x*{t) aX t = 9 corresponds to the discontinuity in u*{t) 8itt = e. •-> t 6 T-8 T T Figure C.l: Needle-Shaped Variation Let r denote any time in the open interval (0,r). We select a suffi ciently small e to insure that r — e > 0 and concentrate our attention on this small interval (r — e^r]. We vary the control on this interval while keeping the control on the remaining intervals [0, r — s] and (r, T] fixed. Specifically, the modified control is ^-> t 0 T-s T r Figure C.2: Trajectories x*{t) and x(t) in a One-Dimensional Case. C.l. Needle-Shaped Variation 395 V eft, t e (r — s,r], u{t) = I (C.2) u*{t), otherwise. This is called a needle-shaped YSuYiaXion as shown in Figure C.l. It is a jump function and is different from variations in the calculus of variations (see Appendix B). Also the difference v —u* is finite and need not be small. However, since the variation is on a small time interval, its influence on the subsequent state trajectory can be proved to be 'small'. This is done in the following. Let the subsequent motion be denoted by x{t) ^ x*{t) for t > r — s, In Figure 0.2, we have sketched x{t) corresponding to u{t). Let 6x{t) = x{t) - a;*(i), t>T-e, denote the change in the state variables. Obviously 6X{T — S) = 0. Clearly, Sx{T)^e[x{s)-x\s)], (C.3) where s denotes some intermediate time in the interval (r — e^r]. In particular, we can write (C.3) as 6x{r) = 4H^) - ^*('^)] + ^(^) = ^[/(x(r),^,r) -/(x*(r),u*(r),r] + o(s). (C.4) But Sx{r) is small since / is assumed to be boimded. Furthermore, since / is continuous and the difference 6X{T) = X(T) — X*{T) is small, we can rewrite (C.4) as 6x{t) « e[f{x*{T),v,T)-f{x*{r),u*iT),T)]. (C.5) Since the initial difference SX(T) is small and since U*{T) does not change from t > T on, we may conclude that Sx{t) will be small for aU t > r. Being small, the law of variation of Sx{t) can be foimd from linear equa tions for small changes in the state variables. These are called variational equations. From the state equation in (C.l), we have ^^El^M. = f(a:* + Sx,u*,t) (C.6) or, ^ + ^^fix\u*,t)+fjx (C.7) 396 C. An Alternative Derivation of the Maximum Principle or using (C.l), -r{Sx) ^ U{x\ u*, t)6x, for t > r, (C.8) tit/ with the initial condition 6X{T) given by (C.5). The basic idea in deriving the maximum principle is that equations (C.8) are linear variational equations and result in an extraordinary sim plification. We next obtain the adjoint equations. C.2 Derivation of the Adjoint Equation and the Maximum Principle For this derivation, we employ two methods. The direct method, similar to that of Hartberger (1973), is the consequence of directly integrating (C.8). The indirect method avoids this integration by a trick which is instructive. Direct method. Integrating (C.8) we get 6x{T) = 6x{r)+ f f^[x\t),u\t),t\8x{t)dt, (C.9) where the initial condition 6X{T) is given in (C.5). Since 8x{T) is the change in the terminal state from the optimal state x*(r), the change in the objective function 6J must be negative. Thus, 8J = c8x{T) = C8X{T) + f cfx[x%t),u*{t),t\8x{t)dt < 0. (CIO) Furthermore, since (C.8) is a linear homogeneous differential equation, we can write its general solution as 8x{t) = ^t,T)8x{T), (C.ll) where the fundamental solution matrix or the transition matrix $(t, r) G ^$(^,r) = Ux%t),u*m^t,r), ^T,T) ^ I, (C.12) where / is an n x n identity matrix; see Appendix A. C.2. Derivation of Adjoint Equation and the Maximum Principle 397 Substituting for 6x{t) from (C.ll) into (C.IO), we have 6J = C6X{T) + / cfa,[x*{t), u\t), t]^{t, T)8x{T)dt < 0. (C.13) This induces the definition X*(t) = J cf4x'{t),u*{t),t]^t,r)dt + c, (C.14) which when substituted into (C.13), yields SJ=^X*{T)SX{T)<0, (C.15) But Sx{r) is supphed in (C.5). Noting that £ > 0, we can rewrite (C.15) as X*{T)f[x*(T),v,T]-X*(r)f[x*iT),u*iT),T]<0. (C.16) Defining the Hamiltonian for the Mayer form as H[x, u, A, t] = A/(x, u, t), (C.17) we can rewrite (C.16) as if[:r*(r),u*(r),A(r),r] >ff[ar*(r),t;,A(r),r]. (C.18) Since this can be done for almost every r, we have the required Hamil tonian maximizing condition. The differential equation form of the adjoint equation (C.14) can be obtained by taking its derivative with respect to r. Thus, -C/X[X'(T),M'(T),T]. (0.19) It is also known that the transition matrix has the property: ^^^ = -$(i,r)/,[x*(r),«*(r),T], which can be used in (C.19) to obtain rfA(r) rT ^^ -j cU[x*{t),u*{t),t]^{t,T)U[x*{Tlu*{r),T]dt -cU[x*{T)X{r),T]. (C.20) 398 C. An Alternative Derivation of the Maximum Principle Using the definition (C.14) of A(r) in (C.20), we have ^ = -A(r)/,[x*(r),«*(r),r] with A(T) = c, or using (C.17) and noting that r is arbitrary, we have A= -A/^[x*,n*,t] = -H^[x*,u\\t), X(T) = c. (C.21) This completes the derivation of the maximum principle along with the adjoint equation using the direct method. Indirect method. The indirect method employs a trick which simplifies considerably the derivation. Instead of integrating (C.8) explicitly, we now assume that the result of this integration yields cSx{T) as the change in the state at the terminal time. As in (C.IO), we have 6J = c6x{T) < 0. (C.22) First, we define A(r) = c, (C.23) which makes it possible to write (C.22) as SJ - c6x{T) = X{T)6x(T) < 0. (C.24) Note parenthetically that if the objective fimction J = S{x{T)), we must define A(r) = dS[x{T)]/dx{T) giving us *^ = ^i^^^(^) = A(r)6^(T). Now, X(T)Sx{T) is the change in the objective fimction due to a change Sx{T) at the terminal time T. That is, A(T) is the marginal return or the marginal change in the objective function per unit change in the state at time T. But Sx{T) cannot be known without integrating (C.8). We do know, however, the value of the change SX{T) at time r which caused the terminal change Sx(T) via (C.8). We would therefore like to pose the problem of obtaining the change 6J in the objective function in terms of the known value SX{T); see FeFdbaimi (1965). Simply stated, we would like to obtain the marginal return A(r) per unit change in state at time r. Thus, X{T)SX{T) = 6J = X{T)SX{T) < 0. (C.25) C,2. Derivation of Adjoint Equation and the Maximum Principle 399 Obviously, knowing A(r) will make it possible to make an inference about SJ which is directly related to the needle-shaped variation applied in the small interval (r — e^r], However, since r is arbitrary, our problem of finding A(r) can be translated to one of finding A(t), t G [0,r], such that X{t)6x{t) = \{T)6x{T), t e [0,T], (C.26) or in other words, X(t)Sx{t) = constant, A(r) = c. (C.27) It turns out that the differential equation which X{t) must satisfy can be easily foimd. From (C.27), ^^[X(t)Sx{t)] = A^ + XSx = 0, (C.28) which after substituting for dSx/dt from (C.8) becomes Xfjx + X6x = (A/^ + X)Sx = 0. (C.29) Since (C.29) is true for arbitrary 6x, we have A = -XU = -Ha: (C.30) using the definition (C.17) for the Hamiltonian. The Hamiltonian maximizing condition can be obtained by substi tuting for 6x{r) from (C.5) into (C.25). This is the same as what we did in (C.15) through (C.18). The purpose of the alternative proof was to demonstrate the valid ity of the maximum principle for a simple problem without knowledge of any return function. For more complex problems, one needs compli cated mathematical analysis to rigorously prove the maximum principle without making use of return fimctions. A part of mathematical rigor is in proving the existence of an optimal solution without which necessary conditions are meaningless; see Young (1969). Appendix D Special Topics in Optimal Control In this appendix we will discuss three specialized topics. These are linear- quadratic problems, second-order variations, and singular control. These topics are referred to but not discussed in the main body of the text because of their advanced nature. While we shall not be able to go into a great detail, we will provide an adequate description of these topics and list relevant references. D.l Linear-Quadratic Problems An important problem in systems theory, especially engineering sciences, is to synthesize feedback controllers. These controllers provide optimal control as a function of the state of the system. A usual method of ob taining these controllers is to solve the Hamilton-Jacobi-Bellman partial differential equation (2.19). This equation is nonlinear in general, which makes it very difficult to solve in closed form. Thus, it is not possible in most cases to obtain optimal feedback control schemes explicitly. It is, however, feasible in many cases to obtain perturbation feedback control, which refers to control in the vicinity of an optimal path; see Bryson and Ho (1969). These perturbation schemes reqiiire the approx imation of the problem by a linear-quadratic problem in the vicinity of an optimal path (see Section D.2), and feedback control for the approx imating problem is easy to obtain. A linear-quadratic control problem is a problem with Unear dynamics 402 D, Special Topics in Optimal Control and quadratic objective function. More specifically, it is: max I J = -x'^Gx + - I {x^Cx + u^Du)dt I (D.l) u subject to x = Ax + Bu, x{Q) =- XQ. (D.2) The matrices G, C, D, A, and B are in general time dependent. Fur thermore, the matrices G, (7, and D are assumed to be negative definite and the superscript ^ denotes the transpose operation. Note that this problem is a special case of Row (c) of Table 3.3. To solve this problem for the explicit feedback controller, we write the Hamilton-Jacobi-Bellman equation (2.19) as 0 = max[if + Vt] = max I--(a:^Cx + u^Du) + V^\Ax + Bu\ -\-v\ (D.3) with the terminal boundary condition V{x,T) = \x^Gx. (D.4) The maximization of the maximand in (D.3) can be carried out by taking its derivative with respect to u and setting it to zero. Thus, ^M^ = ^ = (^Duf + V,B = 0=^u' = -V,B{D^)-\ (D.5) au ou Note that (D.5) is the same as the Hamiltonian maximizing condition. Substituting (D.5) in (D.3) and simplifying, we obtain 0 = ^x^Cx + V^Ax - ^V^BD-^B^V^. (D.6) This is a nonlinear partial differential equation of first order and it has a solution of the form V{x,t) = lx'^S{t)x. (D.7) Substitution of (D.7) into (D.6) yields 0 = \x^\S + 5A + ^S - SBD-^B^S + C\x. (D.8) D.l, Linear-Quadratic Problems 403 Since (D.8) must hold for all x, it implies the following matrix differential equation S + SA + A^S - SBD'^B^S + C = 0, (D.9) called a matrix Riccati equation, with the boundary condition S(T) = G. (D.IO) A solution procedure for Riccati equations appears in Bryson and Ho (1969). Once we have the solution S(t) of (D.9) and (D.IO), the optimal feedback control can be written as u\t) = D{t)-^B' {t)S{t)x{t), (D.ll) A generaUzation of (D.l), which would be useful in the next section on the second variation, is to set 1 T C N X J = -x^ Gx + jf'^^"^) dt. (D.12) N^ D u The state equation is given by (D.2). It is possible to derive the optimal control for this problem as u*{t) = Dity^iN'^it) + B'^{t)S{t)]x{t), (D.13) where S+SA+A^S-{SB+N)D-^(B^S+N^)+C = 0, S{T) = G. (D.14) For other variations and extensions of the linear-quadratic problem (D.l) and (D.2), for which explicit feedback controllers can be developed, the reader is referred to Bryson and Ho (1969). D.1.1 Certainty Equivalence or Separation Principle Suppose equation (D.2) is changed by the presence of a Gaussian white noise w{t) and becomes X = Ax + Bu + w, where E[w{t)] = 0, E[w{t)w{Tf] = Q{t)6{t - r), 404 D. Special Topics in Optimal Control and x{0) is a normal random variable with E[x{0)] = 0, E[x{0)x{Of] = Po. Because of the presence of uncertainty in the system equation, we must modify the objective fimction in (D.12) as follows: ( \ C N ' X max < J = E ^x'^Gx + ^{x^,u^) dt\ iV^ D ^ [u] Assume further that x cannot be directly measured and the measure ment process is given by (13.21), i.e., y{t) = H{t)x{t)+v{t), where v{t) is a white noise as defined in (12.72). The optimal control u* (t) for this linear-quadratic stochastic optimal control problem can be shown to be given by (D.13) with x{t) replaced by its estimate x{t); see Bryson and Ho (1969). Thus, n*(t) - D{t)-^[N^{t) + B^it)S{t)]x{t), where S is given in (D.14) and x is given by the Kalman-Bucy filter. 'x = Ax + Bu*+w + K[y-Hx], :r(0) = 0, K - PH^R-\ P = AP + PA^ -KHP + Q, P{0)^Po, The above procedure has received two different names in the liter ature. In economics it is called the certainty equivalence principle; see Simon (1956). In engineering and mathematics literature it is called the separation principle, Joseph and Tou (1961). When we call it the cer tainty equivalence principle, we are emphasizing the fact that x(t) can be used for the purposes of optimal feedback control as if it were the certain value of the state variable x{t). Whereas the term separation principle emphasizes the fact that the process of determining the optimal control can be broken down into two steps: first, estimate x by using the optimal filter; second, use that estimate in the optimal feedback control formula for the deterministic problem. D,2. Second-Order Variations 405 D.2 Second-Order Variations Second-order variations in optimal control theory are analogous to the second-order conditions in the classical optimization problem of calculus. To discuss the second-order variational condition is difficult when the control variable u is constrained to be in the control set fl. So we make the simplifying assimiption that Q = R^, and thus the control u is unconstrained. As a result, we are now dealing with the problem: max \j=f F{x, u, t)dt + ^x{T)] \ (D.15) subject to X = f{x^ u, t), x{0) = xo> (D.16) From Chapter 2, we know that the first-order necessary conditions for this problem are given by A = -Ha:, A(T) - 0, (D.17) Hu = 0, (D.18) where the Hamiltonian H is given by H = F + Xf, (D.19) Since u is unconstrained, these conditions may be easily derived by the method of calculus of variations. To see this, we write the augmented objective fimctional as J = ^x(T)] + I [Hix, u, A, t) - Xx]dt (D.20) Jo Consider small perturbation from the extremal path given by (D.16) - (D.19) as a result of small perturbations 6x{0) in the initial state. Define the resulting perturbations in state, adjoint, and control variables by Sx{t), <5A(i), and 6u(t), respectively. These, of course, will be obtained by linearizing (D.16 - D.18) around the external path: dux —rr = fx^x + fu^'^i 6x{0) specified, (D.21) do ^ = -{HxJxf - SXf - (H^uSu), (D.22) 406 D. Special Topics in Optimal Control SHu = {HuxSxf + SX{HuXf + (HuuSuf = {HuuSxf + 6Xfu + {HuuSuf = 0. (D.23) Alternatively, we may consider an expansion of the objective function and the state equation to second order since the first-order terms vanish about a trajectory which satisfies (D.15 - D.18), Prom Bryson and Ho (1969), this may be accomplished by expanding (D.20) to second order and all the constraints to first order. Thus, we have ^x dt ^ 2 Jo 8u (D.24) subject to d8x IT = fxSx + fuSu^ 6x{0) specified. (D.25) Since we are interested in a neighboring extremal path, we must deter mine 6u{t) so as to maximize 6^J subject to (D.25). This problem is a linear-quadratic problem discussed in the previous section. For this problem, the optimal control 6u*{t) is given by the formula (D.14), pro vided Huu{i) is nonsingular for 0 < t < T. The case when Huuify is singular for a finite time interval is treated in Section D.3. Thus, rec ognizing that G = ^xx, C = Hxx, N = H^u, D = Huu, A = fx, and B = fu, we have 6u%t) = H-^[Hux + f^S{t)]8x{t), (D.26) where S + Sfx + f^S-{Sfu+Hxu)H-^{f^S + Hux) + Hxx = 0, S{T) = $^^. (D.27) While a ntmiber of second-order conditions can be obtained by pro ceeding further from this manner, we shall be interested only in the concavity condition (or strengthened Legendre-Clebsch condition). It is possible to show that neighboring stationary paths exist (in a weak sense; i.e., Sx and Su are small) if Huu{t) < 0 for 0 represent sufficient conditions for a trajectory to be a local maximum. We are not being specific here because in this book we would be relying mostly on the sufficiency conditions developed in Chapters 2-4, which are based on certain concavity requirements. We are stating (D.28) because of its similarity to the second-order condition for a local maximum in the classical maximization problem. We must note that Hu = 0 and Huu < 0 (D.29) form necessary conditions for a trajectory to be a local maximimi. D.3 Singular Control In some optimization problems including some problems treated in this text, extremal arcs satisfying Hu = 0 occur on which the matrix Huu is singular. Such arcs are called singular arcs. Note that these arcs sat isfy (D.29) but not the strengthened condition (D.28). While no general sufficiency conditions are available for singular arcs, some additional nec essary conditions known as the generalized Legendre-Clebsch conditions have been developed. A good reference on singular control is Bell and Jacobson (1975). We shall only discuss the case in which the Hamiltonian is linear in one or more of the control variables. For these systems, Hu = 0 implies that the coefficient of the linear control term in the Hamiltonian vanishes identically along a singular arc. Thus, the control is not determined in terms of x and A by the Hamiltonian maximizing condition Hu = 0. Instead, the control is determined by the requirement that the coefficient of these linear terms remain zero on the singular arc. That is, the time derivatives of Hu must be zero. Having obtained the control by setting dHu/dt = 0 (or by setting higher time derivatives to equal zero) along the singular arc, we must check additional necessary conditions analogous to the second-order condition (D.28). For a maximization problem with a single control variable, these conditions turn out to be (P^Hu (-')'!; <0, it = 0,1,2,.... (D.30) dfi^ The conditions (D.30) are called the generalized Legendre-Clebsch conditions. 408 D. Special Topics in Optimal Control Example D.l We present an example treated by Johnson and Gibson (1963): max \j=-{—ir^- I ?x\dt \ (D.31) subject to xi = X2+u^ xi{Q) = a^ (D.32) i2 = -u, a:(0) = fe, (D.33) xi{T) - X2{T) = 0. (D.34) Solution. We form the Hamiltonian H = --x\ + Xi{x2 + u)+ Mi-u), (D.35) where the adjoint equations are Ai=xi, A2 = -Ai. (D.36) The optimal control is bang-bang plus singular. Singular arcs must sat isfy if = Ai - A2 = 0 (D.37) for a finite time interval. The optimal control can, therefore, be obtained by ^ = Ai-A2 = :ci+Ai = 0. (D.38) Differentiating once more with respect to time t, we obtain 2 = xi + \i = X2 + u + xi = Q, which implies u = -{xi+X2) (D.39) along the singular arc. We now verify for the example, the generalized Legendre-Clebsch condition (D.30) for fc = 1: Appendix E Answers to Selected Exercises Completely worked solutions to all exercises in this book are contained in a forthcoming Teachers' Manual, which wiU be made available to instructors by the publisher when it is ready. Chapter 1 1.1 (a) Feasible. J = -333,333. 1.2 J = 36. 1.3 (a) C = $157,861/year. (b) J = 103.41 utils. (c) $15,000/year. 1.4 (b) W{20) = 985,648; J = 104.34. 1.12 imp(Gi,G2;i) = (Gi - G2)e-^*. Chapter 2 2.2 The optimal control is 2 if 0 u* (t) = { undefined if i = 2 - In 2.5, 0 ift>2-ln2.5. 410 E. Answers to Selected Exercises 2.8 u* = bang(0,1; Ai - A2), where X{t) = (8e-2(*-i8), 4e-2(*-i8)). 2.10 (a) u* = bang[0,1; (giKi + 52^2)(Ai - A2)]. (c) i = T- (1/52) Hio^bi - 9ih2)/{g2 - giM- 2.12 (a) a;(100) = 30 - 20e-i° « 30. (b) w* = 3fori€ [0,100]. 3 for is [0, 100-101n2], (c) u*{t) = 0 otherwise. 2.14 (a) C*{t) = pWoe^'-P^yil - e-P^). (b) C*it) = Kir-p). 2.17 (a) X = x + 3Aa;2, A(l) = 0, and x =-x^ + A, x{0) = 1. 2.18 X = f{x) + b{x)u, x{0) = xo, x{T) = 0. u = [b{xfg'{x) - 2<^u{b(x)f'{x) - V{x)f{x)y\l\l(?h{x)\. Chapter 3 3.1 X — Ml > 0, «1 — M2 > 0, Ml > 0, 1 + M2 > 0- 3.2 X = [-1, 5]. 3.7 L = F(x, v) + A/(x, M, i) + iig{x, u, t), A = -(a/a)X - 1^, /x > 0, /x^ = 0. 3.11 A(i) = t - 1. 3.12 (a) A(i) = 10 I _g0.1(t-100) 0 if ii: - 300, -10 2 _ gO.iCic/s-ioo) if ii: < 300, M*(t) = bang[0,3;A + /Lt]. The problem is infeasible for K > 300. E. Answers to Selected Exercises 411 (b) r* = niin[0, 100-K/3], 0 for t < t**, u*{t) = I 3 iort>t**. V 3.18 11.87 minutes. 3.19 u* = -1, T* = 5. 3.20 tx* = -2, T* = 5/2. 3.29 (a) {I, P,X} = {h - p{S - Pi), S, 2(5 - Pi)}. (b) I = h. Chapter 4 4.1 u*{t) = -l, iJ,i = -X = 1/2-t, H2 = 7] = 0. 4.2 One solution appears in Figure 3.1. Another solution is u(t) = 1/2 for t G [0,2]. There are many others. 4.4 (a) u* = 0. 1, 0 (e) J = -(1/8 + 1/8K). (f) J =-1/8. Chapter 5 5, t< l + 61n0.99«0.94, 5.1 (a) u*{t) 0, t> 0.094. -5, 0 < ^ < 0.28, 0, 0.28 < t < 0.4, (b) A2(t)/Ai(i) = e3(*'-«+i)/i2,^t*(f) = J 5, 0.4 < i < 0.93, 0, 0.93 < t < 1.0. 412 E. Answers to Selected Exercises 5.5 u* = v* = 0 for all t. 5.7 u* = 0, V* = 4/5 for t € [0,49], u* = 0, ^;* = 0 for ^€[49,60], J* = 34,420. 5.10 (b) /(**) = t* - 101n(l - 0.3e°-"*). (c) t* = 1.969327, J{t*) = 19.037. Chapter 6 6.5 Q(t) = t^ - 160*3 ^ 1740^2 _ 7350^ + 9639. 6.7 V* = sat[-F2,14; (A2 - Aip)2/?Ai]. 6.9 J* = 10.56653. 6.10 y*{t) « 3e-3*, y*(t) w 1 - Se'^*. 0, 0 < i < 7/3, 2, 7/3 0, 13/3 -fi+|, t€[0,l], 6.13 /xi = < 0, tG{0,3]. 0, ie [0,1.8), f^2= \ -ii + f, ie [1.8,3]. 0, tG[0,l)U(1.8,3], 77= < -fi+|, «€ [1,1.8). E. Answers to Selected Exercises 413 -1 for tG [0,1.8), 6.14 (a) v*{t) = { 1 for ie (1.8,3]. (b) t;*(t) = 1 for iG [0,10]. -1 for tG [0,1/2], 0 for f G (1/2, ti], where ti = 23/12, 6.15 V* = < +1 foriG(ii,ii + l/2], 0 for tG (ii + 1/2,4]. 0, for0 where ti=T- ^2BC/h. Chapter 7 7.1 p* = 102.5 + 0.2G. 7.7 {u)/{pS) = {Sp)/{rj{p + S)). 7.10 G + SG = bang[0, oo; A + 1], -X + (p + S)X = 7r'{G). 7.12 The equations corresponding to (6.28) and (6.29) can be obtained by replacing p by p+r/r. The form of (6.30) remains imchanged. 7.17 (b) 1 1 Xo _, 1 . X-X^ h rQ + 6:ln— x^ , t2 = rQ„ + S. In •X — XT 7.18 T > ^ In ^<^(1 ~ ^o) ~ ^^0 + i In fi ~ rQ + 6 rQ(l — x^) — 6x^ 6 XT ' 414 E. Answers to Selected Exercises 7.19 The reachable set is [xoe'^^, {xo - x)e-(*+'''5)^ + x], where x = rQ/{W + rQ). 7.25 1-A imp{A^B;t) = - r 1-B 7.26 (b) J = 0.6325. Chapter 8 8.1 (a) y = l^z = 3. (b) y - 2, z - 10. 8.2 (a) (1,3) is a relative maximum. (b) (2,10) is a relative maximum. 8.3 x = 50;x = 80. 8.6 (a) a: = 4 is a local maximum. (b) a: = 8 is a local maximimn and a: = 20 is a local and a global maximum. 8.7 (a) (0, 0) is the nearest point. (b) (1/2,1/2) is the nearest point. 8.8 (1/V^, 2/\/5) is the closest point. 8.9 (a) (2^,0). (b) (0,2). (c) (0,2). 8.10 Xj = dF/dxJ for i = 1,2,..., n; A^+i = 1. Note that here T denotes the terminal time, and not the transpose operation. +1 ifA'=+i6>l, k* 8.14 u -1 ifA'^+iJx-l ,whereA*= = (7 + A)^-'=A^ 0 if iX^+^b] < 1. E. Answers to Selected Exercises 415 Chapter 9 9.1 t' = 5.25, T = 11. 9.3 T = i* = 2.47. 9.4 t^ = 0, T = 30. l2 9.5 u*{t) = sat[0,1;u^{t)], where u^{t) = \2 - e005(t-34.8)|'/(i + i), ti^S;t2-T = 34.8. Chapter 10 10.3 X = 0.734. 10.4 (a) X Sep X = (-^i)W('-^i) +prX (b) For p = 0, 5 = 220,000. Fovp = 0.1,x = 86,000. For p = oo, X = 40,000. 10.5 [g'{x) - p]\p - c{x)] - c^{x)g{x) = 0. 10.7 [g'{x) -p\\p- c{x)] - c'{x)g{x) + p = 0. Chapter 11 11.1 A(i) = \oe^P-^^\ where [Koe^T + 5(1 _ g/3T)/^ _ ^^j(2p - 0) Xo = e/?T _ e2(/3-p)T e^'). p p Zp:(-' Chapter 12 12.5 0 < T + ;^ ha(l - ^) = i. 12.8 ti = r/2. Chapter 13 13.5 q*{x) = j^^^,c*ix) = j^{p-r0-i^)x, Bibliography [1] Abad, P.L. (1982a). An optimal control approach to marketing- production planning, Optimal Control Applications & Methods^ 3,1,1-13. [2] Abad, P.L. (1982b). Approach to decentralized marketing-production planning, International Journal of Systems Science, 13, 3, 227-235. [3] Abad, P.L. (1987). A hierarchical optimal control model for coordina tion of functional decisions in a firm, European Journal of Operational Research, 32, 62-75. [4] Abad, P.L. (1989). Multi-product multi-market model for coordination of marketing-production decisions. International Journal of Systems Sci ence, 20, 2011-2027. [5] Abad, P.L. and Sweeney, D.J. (1982). Decentralized planning with an interdependent marketing-production system. Omega, 10, 353-359. [6] Agnew, C.E. (1976). Dynamic modeling and control of congestion-prone systems, Operations Research, 24, 400-419. [7] Alam, M., Lynn, J.W. and Sarma, V.V.S. (1976). Optimal maintenance policy for equipment subject to random deterioration and random failure, International Journal of Systems Science, 7, 1071-1080. [8] Alam, M. and Sarma, V.V.S. (1974). Optimal maintenance policy for equipment subject to deterioration and random failure, IEEE Transac tions on System, Man, and Cybernetics SMC-4, 172-175. [9] Alam, M. and Sarma, V.V.S. (1977). An application of optimal control theory to repairman problem with machine interference, IEEE Transac tions on Reliability, R-26, 121-124. [10] Allen, K.R. (1973). Analysis of the stock-recruitment relation in Antarc tic fin whales, in Fish Stock and Recruitment, J. du Conseil International pour VExploration de la Mer, Rapports et Proces-Verbaux de Reunions, B. Parrish (Ed.), 164, 132-137. 418 Bibliography [11] Amit, R. (1986). Petroleum reservoir exploitation: switching from pri mary to secondary recovery, Operations Research, 34, 4, 534-549. [12] Amit, R. and Ilan, Y. (1990). The choice of manufacturing technology in the presence of dynamic demand and experience effects, IIE Transactions, 22, 2, 100-111. [13] Anderson, R.M. and May, R. M. (1992). Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford. [14] Aoki, M. (1976). Dynamic Economics: A System Theoretic Approach to Theory and Control, Elsevier, New York. [15] Arnold, L. (1974). Stochastic Differential Equations: Theory and Appli cations, Wiley, New York. [16] Aronson, J.E. and Thompson, G.L. (1984). A survey on forward methods in mathematical programming. Journal of Large Scale Systems, 7, 1-16. [17] Arora, S.R. and Lele, P.T. (1970). A note on optimal maintenance policy and sale date of a machine. Management Science, 17, 170-173. [18] Arrow, K.J. (1968). Applications of control theory to economic growth, in Mathematics of the Decision Sciences, G.B. Dantzig and A.F. Veinott (Eds.), Vol. 12 in the Series Lectures in Applied Mathematics, American Mathematical Society, Providence, RI, 85-119. [19] Arrow, K.J. and Chang, S. (1980). Optimal pricing, use, and exploration of uncertain natural resource stocks, in Dynamic Optimization in Math ematical Economics, P.-T. Liu (Ed.), Plenum Press, New York. [20] Arrow, K.J. and Kurz, M. (1970). Public Investment, The Rate of Return, and Optimal Fiscal Policy, The John Hopkins Press, Baltimore. [21] Arthur, W.B. and McNicoll, G. (1977). Optimal time paths with age dependence: a theory of population policy. Review of Economic Studies, 44, 111-123. [22] Arutyunov, A.V. and Aseev, S.M. (1977). Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM Journal on Control and Optimization, 35, 3, 930-952. [23] Aubin, J.-P. and Cellina, A. (1984). Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin. [24] Axsater, S. (1985). Control theory concepts in production and inventory control. International Journal of Systems Science, 16, 161-169, Bibliography 419 [25] Bagchi, A. (1984). Stackelberg Differential Games in Economic Models, Lecture Notes in Control and Information Sciences, Vol. 64, Springer- Verlag, Berlin. [26] Basar, T. (1986). A tutorial on dynamic and differential games, in Dy namic Games and Applications in Economics, T. Basar (Ed.), Springer- Verlag, Berlin, 1-25. [27] Basar, T. and Olsder, G.J. (1982). Dynamic Noncooperative Game The ory, Academic Press, London. [28] Bass, F.M. (1969). A new product growth model for consumer durables, Management Science, 15, 5, 215-227. [29] Bass, F.M. and Bultez, A.V. (1969). A note on optimal strategic pricing of technological innovations. Marketing Science, 1, 371-378. [30] Bean, J.C. and Smith, R.L. (1984). Conditions for the existence of plan ning horizons. Mathematics of Operations Research, 9, 3, 391-401. [31] Bell, D.J. and Jacobson, D.H. (1975). Singular Optimal Control, Aca demic Press, New York. [32] Bellman, R.E. (1957). Dynamic Programming, Princeton University Press, Princeton, NJ. [33] Bellman, R.E. and Kalaba, R.E. (1965a). Quasi linearization and Bound ary Value Problems, Elsevier, New York. [34] Bellman, R.E. and Kalaba, R.E. (1965b). Dynamic Programming and Modern Control Theory, Academic Press, New York. [35] Bensoussan, A., Bultez, A.V. and Naert, P.A. (1978). Leader's dynamic marketing behavior in oligopoly, in TIMS Studies in the Management Sciences, A. Bensoussan et al., (Eds.), Vol. 9, North-Holland, Amster dam, 123-145. [36] Bensoussan, A., Crouhy, M. and Proth, J.-M. (1983). Mathematical The ory of Production Planning, North-Holland, Amsterdam. [37] Bensoussan, A., Hurst, Jr., E.G. and Naslund, B. (1974). Management Applications of Modern Control Theory, Elsevier, New York. [38] Bensoussan, A. and Lesourne, J. (1980). Optimal growth of a self- financing firm in an uncertain environment, in Applied Stochastic Control in Econometrics and Management Science, A. Bensoussan et al. (Eds.), North-Holland, Amsterdam, 235-269. 420 Bibliography [39] Bensoussaii, A. and Lesourne, J. (1981). Growth of firms: a stochastic control theory approach, in Unternehmensplanung, K. BrockhofFand W. Krelle (Eds.), Springer-Verlag, Berlin, 101-116. [40] Bensoussan, A. and Lions, J.L. (1975). Nouvelles methodes en controle impulsionnel. Applied Mathematics & Optimization, 1, 289-312. [41] Bensoussan, A. and Lions, J.L. (1982). Application of Variational In equalities in Stochastic Control, North-Holland, Amsterdam, The Nether lands. [42] Bensoussan, A. and Lions, J.L. (1984). Impulse Control and Quasi- Variational Inequalities, Bordas, Paris. [43] Bensoussan, A., Nissen, G. and Tapiero, C.S. (1975). Optimum inventory and product quality control with deterministic and stochastic deteriora tion - an application of distributed parameter control systems, IEEE Transactions on Automatic Control, AC-20, 407-412. [44] Bensoussan, A., Sethi, S.P., Vickson, R.G. and Derzko, N. A. (1984). Stochastic production planning with production constraints, SI AM Jour nal on Control and Optimization, 22, 6, 920-935. [45] Berkovitz, L.D. (1961). Variational methods in problems of control and programming, Journal of Mathematical Analysis and Applications, 3, 145-169. [46] Berkovitz, L.D. (1994). A theory of diflferential games, in Advances in Dynamic Games and Applications, T. Basar and A. Haurie (Eds.), Birkhauser, Boston, 3-22. [47] Berkovitz, L.D. and Dreyfus, S.E. (1965). The equivalence of some nec essary conditions for optimal control in problems with bounded state variables. Journal of Mathematical Analysis and Applications, 10, 275- 283. [48] Bertsekas, D.P. and Shreve, S.E. (1996). Stochastic Optimal Control: The Discrete-Time Case, Athena Scientific, New York. [49] Bes, C. and Sethi, S.P. (1988). Concepts of forecast and decision horizons: applications to dynamic stochastic optimization problems. Mathematics of Operations Research, 13, 2, 295-310. [50] Bes, C. and Sethi, S.P. (1989). Solution of a class of stochastic linear- convex control problems using deterministic equivalents. Journal of Op timization Theory and Applications, 62, 1, 17-27. Bibliography 421 [51] Beyer, D. and Sethi, S.R (1998). A proof of the EOQ formula using quasi- variational inequalities. The International Journal of Systems Science^ 29, 11, 1295-99. [52] Bhaskaran, S. and Sethi, S.P. (1981). Planning horizons for the wheat trading model. Proceedings of AMS SI Conference^ 5: Life, Men, and Societies, 197-201. [53] Bhaskaran, S. and Sethi, S.P. (1983). Planning horizon research - the dynamic programming/control theory interface, Proceedings of Interna tional AMSE Winter Symposium, Bermuda, 155-160. [54] Bhaskaran, S. and Sethi, S.P. (1987). Decision and forecast horizons in a stochastic environment: a survey. Optimal Control Applications & Meth ods, 8, 201-217. [55] Bhaskaran, S. and Sethi, S.P. (1988). The dynamic lot size model with stochastic demands: a planning horizon study. Information Systems and Operational Research, 26, 3, 213-224. [56] Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637-659. [57] Blaquiere, A. (1979). Necessary and sufficiency conditions for optimal strategies in impulsive control, in Differential Games and Control Theory III, Part A, P.-T. Liu and E.O. Roxin (Eds.), M. Dekker, New York, 1-28. [58] Blaquiere, A. (1985). Impulsive optimal control with finite or infinite time horizon. Journal of Optimization Theory and Applications, 46, 431-439. [59] Boiteux, M. (1955). Reflexions sur la concurrence du rail et de la route, le declassement des lignes non rentables et le deficit du chemin de fer, L'Economie Electrique, 2. [60] Boltyanskii, V.G. (1971). Mathematical Methods of Optimal Control, Holt, Rinehard & Winston, New York. [61] Bookbinder, J.H. and Sethi, S.P. (1980). The dynamic transportation problem: a survey. Naval Research Logistics Quarterly, 27, 65-87. [62] Bourguignon, F. and Sethi, S.P. (1981). Dynamic optimal pricing and (possibly) advertising in the face of various kinds of potential entrants, Journal of Economic Dynamics and Control, 3, 119-140. [63] Breakwell, J.V. (1968). Stochastic optimization problems in space guid ance, in Stochastic Optimization and Control, H.F. Karreman (Ed.), Wi ley, New York, 91-100. 422 Bibliography [64] Brekke, K.A. and 0ksendal, B.K. (1994). Optimal switching in an eco nomic activity under uncertainty, SIAM Journal on Control and Opti mization, 32, 4, 1021-1036. [65] Brito, D.L. and Oakland, W.H. (1977). Some properties of the optimal income tax. International Economic Review, 18, 407-423. [66] Brown, R.G. (1959). Statistical Forecasting for Inventory Control, McGraw-Hill Book Co., New York. [67] Brown, R.G. (1963). Smoothing, Forecasting and Prediction, Prentice- Hall, Inc., Englewood Cliffs, NJ. [68] Bryant, G.F. and Mayne, D.Q. (1974). The maximum principle. Inter national Journal of Control, 20, 1021-1054. [69] Bryson, Jr., A.E. (1998). Dynamic Optimization, Addison-Wesley, Read ing, Mass. [70] Bryson, Jr., A.E. and Ho, Y.-C. (1969). Applied Optimal Control, Blais- dell, Waltham, MA. [71] Buchanan, L.F. and Norton, F.E. (1971). Optimal control applications in economic systems, in Advances in Control Systems, C.T. Leondes (Ed.), Vol. 8 in the series. Academic Press, New York, 141-187. [72] Bulirsch, R. and Kraft, D. (Eds.) (1994). Computational Optimal Control, Birkhauser-Verlag, Boston. [73] Bulirsch, R., Oetth, W. and Stoer, J. (Eds.) (1975). Optimization and Op timal Control, Lecture Notes in Mathematics, Vol. 477, Springer-Verlag, Berlin. [74] Bultez, A.V. and Naert, P.A. (1979). Does lag structure really matter in optimizing advertising spending, Management Science, 25, 5, 454-465. [75] Bultez, A.V. and Naert, P.A. (1988). When does lag structure really matter...indeed?. Management Science, 34, 7, 909-916. [76] Burdet, C.A. and Sethi, S.P. (1976). On the maximum principle for a class of discrete dynamical system with lags, Journal Of Optimization Theory & Applications, 19, 445-454. [77] Burmeister, E. and Dobell, A.R. (1970). Mathematical Theories of Eco nomic Growth, MacMillan, London. [78] Butkowskiy, A.G. (1969). Distributed Control Systems, Elsevier, New York. Bibliography 423 [79] Bylka, S. and Sethi, S.R (1992). Existence of solution and forecast hori zons in dynamic lot size model with nondecreasing holding costs, Pro duction and Operations Management, 1, 2, 212-224. [80] Bylka, S., Sethi, S.R and Sorger, G. (1992). Minimal forecast horizons in equipment replacement models with multiple technologies and general switching costs, Naval Research Logistics, 39, 487-507. [81] Caines, R, Sethi, S.R and Brotherton, T. (1977). Impulse response iden tification and casualty detection for the Lydia Pinkham data. Annals of Economic and Social Measurement, 6, 2, 147-163. [82] Canon, M.D., CuUum, CD. and Polak, E. (1970). Theory of Optimal Control and Mathematical Programming, McGraw-Hill, New York. [83] Carlson, D.A. (1986a). On the existence of catching up optimal solu tions for Lagrange problems defined on unbounded intervals. Journal of Optimization Theory and Applications, 49, 2. [84] Carlson, D.A. (1986b). The existence of finitely optimal solutions for in finite horizon optimal control problems, Journal of Optimization Theory and Applications, 51, 1, 41-62. [85] Carlson, D.A. (1987a). On the existence of sporadically catching up op timal solutions for infinite horizon optimal control problems. Journal of Optimization Theory and Applications, 53, 2, 219-235. [86] Carlson, D.A. (1987b). An elementary proof of the maximum principle for optimal control problems governed by a Volterra integral equation, Journal of Optimization Theory and Applications, 54, 1, 43-61. [87] Carlson, D.A. (1988). Sufficient conditions for optimality and supported trajectories for optimal control problems governed by Volterra integral equations, in Advances in Optimization and Control, Lecture. Notes in Econ. and Math. Sys., 302, Springer-Verlag. New York, 274-282. [88] Carlson, D.A. (1989). Some concepts of optimality for infinite horizon control and their interrelationships, in Modern Optimal Control; A Con ference in Honor of Solomon Lefschetz and Joseph P. LaSalle, E.O. Roxin (Ed.), Marcel Dekker, Inc., New York, 13-22. [89] Carlson, D.A. (1990a). Uniform overtaking and weakly overtaking opti mal solutions in optimal control: when optimal solutions are agreeable, Journal of Optimization Theory and Applications, 64, 1, 55-69. [90] Carlson, D.A. (1990b). The existence of catching-up optimal solutions for a class of infinite horizon optimal control problems with time delay, SIAM Journal on Control and Optimization, 28, 2, 402-422. 424 Bibliography [91] Carlson, D.A. (1990c). Infinite horizon optimal controls for problems gov erned by a Volterra integral equation with a state and control dependent discount factor, Journal of Optimization Theory and Applications, 66, 2, 311-336. [92] Carlson, D.A. (1991). Asymptotic stability for optimal trajectories of infinite horizon optimal control models with state and control dependent discounting, Proceedings of the 4th Annual Workshop in Analysis and its Applications, 281-299. [93] Carlson, D.A. (1993). Nonconvex and relaxed infinite horizon optimal control problems. Journal of Optimization Theory and Applications, 78, 3, 465-491. [94] Carlson, D.A. (1995). An existence theorem for hereditary Lagrange problems on an unbounded interval, in Boundary Value Problems for Functional Differential Equation, World Scientific Publishing, Singapore, 73-83. [95] Carlson, D.A. (1997). Overtaking optimal solutions for convex Lagrange problems with time delay, J. Math. Appl, 208, 31-48. [96] Carlson, D.A. and Haurie, A. (1987a). Infinite Horizon Optimal Control Theory and Applications, Lecture. Notes in Econ. and Math. Sys., 290, Springer-Verlag, New York. [97] Carlson, D.A. and Haurie, A. (1987b). Optimization with Unbounded Time Interval for a Class of Non Linear Systems, Springer-Verlag, Berhn. [98] Carlson, D.A. and Haurie, A. (1992). Control theoretic models of environment-economy interactions, The Proceedings of the 31st IEEE Conference on Decision and Control, 2860-2861. [99] Carlson, D.A. and Haurie, A. (1995). A turnpike theory for infinite hori zon open-loop differential games with decoupled dynamics. New Trends in Dynamic Games and Applications^ Annals of the International Society of Dynamic Games, 3, Birkhauser, Boston, 353-376. [100] Carlson, D.A. and Haurie, A. (1996). A turnpike theory for infinite hori zon competitive processes, SI AM Journal on Optimization and Control, 34, 4. [101] Carlson, D.A., Haurie, A. and Jabrane, A. (1987). Existence of overtaking solutions to infinite dimensional control problems on unbounded time intervals, SIAM Journal on Control and Optimization, 25, 6, 1517-1541. [102] Carlson, D.A., Haurie, A. and Leizarowitz, A. (1991). Infinite Horizon Optimal Control: Deterministic and Stochastic Systems, Second Edition, Springer-Verlag, New York. Bibliography 425 [103] Carlson, D.A., Haurie, A. and Leizarowitz, A. (1994). Equilibrium points for linear-quadratic infinite horizon differential games, in Advances in Dynamic Games and Applications^ Annals of the International Society of Dynamic Games, T. Baar and A. Haurie (Eds.), Vol. 1, Birkhauser, 247-268. [104] Carraro, C. and Filar, J. (Eds.), (1995). Control and Game Theoretic Models of the Environment, Birkhauser, Boston. [105] Carrillo, J. and Gaimon, C. (2000). Improving manufacturing perfor mance through process change and knowledge creation. Management Sci ence, forthcoming. [106] Case, J.H. (1979). Economics and the Competitive Process, New York University Press, New York. [107] Cass, D. and Shell, K. (1976). The Hamiltonian Approach to Dynamic Economics, Academic Press, New York. [108] Cesari, L. (1983). Optimization - Theory and Applications: Problems with Ordinary Differential Equations, Springer-Verlag, New York. [109] Chand, S. and Sethi, S.P. (1982). Planning horizon procedures for ma chine replacement models with several possible replacement alternatives, Naval Research Logistics Quarterly, 29, 3, 483-493, [110] Chand, S. and Sethi, S.P. (1983). Finite production rate inventory models with first and second shift setups, Naval Research Logistics Quarterly, 30, 401-414. [Ill] Chand, S. and Sethi, S.P. (1990). A dynamic lot size model with learning in setups, Operations Research, 38, 4, 644-655. [112] Chand, S., Sethi, S.P. and Proth, J.-M. (1990). Existence of forecast hori zons in undiscounted discrete time lot size models, Operations Research, 38, 5, 884-892. [113] Chand, S., Sethi, S.P. and Sorger, G. (1992). Forecast horizons in the discounted dynamic lot size model. Management Science, 38, 7, 1034- 1048. [114] Charnes, A. and Kortanek, K. (1966). A note on the discrete maximum principle and distribution problems. Journal of Mathematics and Physics, 45, 121-126. [115] Chen, S.F. and Leitmann, G. (1980). Labour-management bargaining modelled as a dynamic game. Optimal Control Applications & Methods, 1, 11-25. 426 Bibliography [116] Chiarella, C, Kemp, M.C., Long, N.V. and Okuguchi, K. (1984). On the economics of international fisheries. International Economic Review^ 25, 85-92. [117] Chichilinsky, G. (1981). Existence and Characterization of optimal growth paths including models with non-convexities in utilities and tech nologies, Review of Economic Studies, 48, 51-61. [118] Chintagunta, P.K. (1993). Investigating the sensitivity of equilibrium profits to advertising dynamics and competitive effects. Management Sci ence, 39, 9, 1146-1162. [119] Chintagunta, P.K. and Jain, D. (1992). A dynamic model of channel member strategies for marketing expenditures. Marketing Science, 11, 2, 168-188. [120] Chintagunta, P.K. and Jain, D. (1994). A study of manufacturer-retailer marketing strategies: a differential game approach. Lecture Notes in Con trol and Information Sciences, Springer-Verlag, New York. [121] Chintagunta, P.K. and Jain, D. (1995). Dynamic duopoly models of ad vertising competition: estimation and a specification tests. Journal of Economics and Management Strategy, 4, 1, 109-131. [122] Chintagunta, P.K. and Vilcassim, N.J. (1992). An empirical investigation of advertising strategies in a dynamic duopoly. Management Science, 38, 9, 1230-1244. [123] Chintagunta, P.K. and Vilcassim, N.J. (1994). Marketing investment de cisions in a dynamic duopoly: a model and empirical analysis. Interna tional J. Res. Marketing, 11, 3, 287-306. [124] Chow, G.C. (1975). Analysis and Control of Dynamic Economic Systems, Wiley, New York. [125] Clark, C.W. (1973). The economics of overexploitation. Science, 181, 630-634. [126] Clark, C.W. (1976). Mathematical Bioeconomics: The Optimal Manage ment of Renewal Resources, Wiley, New York. [127] Clark, C.W. (1979). Mathematical models in the economics of renewable resources, SIAM Review, 21, 81-99. [128] Clark, C.W. (1985). Bioeconomic Modelling and Fisheries Management, Wiley, New York. Bibliography 427 [129] Clarke, F.H. (1976). Necessary conditions for a general control problem, in Calculus of Variations and Control Theory, D.L. Russell (Ed.), Aca demic Press, New York. [130] Clarke, F.H. (1983). Optimization and Nonsmooth Analysis, Wiley- Interscience, New York. [131] Clarke, F.H. (1989). Methods of Dynamic and Nonsmooth Optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA. [132] Clarke, F.H., Darrough, M.N. and Heineke, J.M. (1982). Optimal pricing policy in the presence of experience effects, Journal of Business, 55, 517- 530. [133] Clarke, F.H., and Loewen, P.D. (1987). State constraints in optimal con trol: a case study in proximal normal analysis, SI AM Journal of Control and Optimization, 25, 1440-1456. [134] Clemhout, S. and Wan, Jr., H.Y. (1985). Dynamic common property resources and environmental problems. Journal of Optimization Theory and Applications, 46, 471-481. [135] Coddington, E.A. and Levinson, N.L. (1955). Theory of Ordinary Differ ential Equations, McGraw-Hill, New York. [136] Cohen, K.J. and Cyert, R.M. (1965). Theory of the Firm: Resource Al location in a Market Economy, Prentice-Hall, Inc., Englewood Cliffs, NJ. [137] Connors, M.M. and Teichroew, D. (1967). Optimal Control of Dynamic Operations Research Models, International Textbook, Scranton, PA. [138] Conrad, K. (1982). Advertising, quality and informationally consistent prices, Zeitschrift filr ges. Staatswiss, 138, 680-694. [139] Conrad, K. (1985). Quality, advertising and the formation of goodwill un der dynamic conditions, in Optimal Control Theory and Economic Anal ysis, G. Feichtinger (Ed.), Vol. 2, North-Holland, Amsterdam, 215-234. [140] Constantinides, G.M. and Richard, S.F. (1978). Existence of optimal simple policies for discounted cost inventory and cash management in continuous time, Operations Research, 26, 620-636. [141] Dantzig, G.B. (1966). Linear control processes and mathematical pro gramming, SIAM Journal on Control, 4, 56-60. [142] Dantzig, G.B. and Sethi, S.P. (1981). Linear optimal control problems and generalized linear programs. Journal of Operational Research Society, 32, 467-476. 428 Bibliography [143] Dasgupta, P. and Heal, G.M. (1974). The optimal depletion of ex haustible resources, The Review of Economic Studies, 41, 3-28. [144] D'Autume, A. and Michel, P. (1985). Future investment constraints re duce present investment, Econometrica, 53, 203-206. [145] Davis, B.E. (1970). Investment and rate of return for the regulated firm, The Bell Journal of Economics and Management Science, 1, 245-270. [146] Davis, B.E. and Elzinga, D.J. (1972). The solution of an optimal control problem in financial modeling. Operations Research, 19, 1419-1433. [147] Davis, M.H.A. (1993). Markov Models and Optimization, Chapman & Hall, New York. [148] Dawid, H. and Feichtinger, G. (1996a). Optimal allocation of drug control efforts: a differential game analysis, Journal of Optimization Theory and Applications, 91, 279-297. [149] Dawid, H. and Feichtinger, G. (1996b). On the persistence of corruption, Journal of Economics, 64, 177-193. [150] Dawid, H., Feichtinger, G. and Hartl, R.F. (Eds.) (1999). Special Issue on Optimal control and differential games. Annals of Operations Research, 88. [151] Dawid, H., Kopel, M. and Feichtinger, G. (1997). Complex solutions of nonconcave dynamic optimization models, Economic Theory, 9, 427-439. [152] Deal, K.R. (1979). Optimizing advertising expenditures in a dynamic duopoly. Operations Research, 27, 4, 682-692. [153] Deal, K.R., Sethi, S.P. and Thompson, G.L. (1979) A bilinear-quadratic differential game in advertising, in Control Theory in Mathematical Eco nomics, P.-T. Liu and J.G. Sutinen (Eds.), Marcel Dekker, Inc., New York, 91-109. [154] Deger, S. and Sen, S.K.(1984). Optimal control and differential game models of military expenditure in less developed countries, J. of Eco nomic Dynamics and Control, 7, 153-169. [155] Deissenberg, C. (1980). Optimal control of linear econometric models with intermittent control, Economics of Planning, 16, 1, 49-56. [156] Deissenberg, C. (1981). A simple model of closed-loop optimal explo ration for oil. Policy Analysis and Information Systems, 5, 3, 167-183. Bibliography 429 [157] Deissenberg, C. and Stoppler, S. (1982). Optimal control of LQG sys tems with costly observations, in Economic Applications of Optimal Con trol Theory, G. Feichtinger (Ed.), North-Holland, Amsterdam-New York, 301-320. [158] Deissenberg, C. and Stoppler, S. (1983). Optimal information gathering and planning policies of the profit-maximizing firm. International Journal on Policy and Information, 7, 2, 49-76. [159] Derzko, N.A. and Sethi, S.P. (1981a). Optimal exploration and consump tion of a natural resource: stochastic case. International J. of Policy Analysis, 5, 3, 185-200. [160] Derzko, N.A. and Sethi, S.P. (1981b). Optimal exploration and consump tion of a natural resource: deterministic case. Optimal Control Applica tions & Methods, 2, 1, 1-21. [161] Derzko, N.A., Sethi, S.P. and Thompson, G.L. (1980). Distributed pa rameter systems approach to the optimal cattle ranching problem. Opti mal Control Applications & Methods, 1, 3-10. [162] Derzko, N.A., Sethi, S.P. and Thompson,G.L. (1984). Necessary and suf ficient conditions for optimal control of quasilinear partial differential systems. Journal of Optimization Theory and Applications, 43, 89-101. [163] Dhrymes, P.J. (1962). On optimal advertising capital and research ex penditures under dynamic conditions, Economica, 39, 275-279. [164] Dixit, A.K. and Pindyck, R.S. (1994). Investment Under Uncertainty, Princeton University Press, Princeton, NJ. [165] Dockner, E.J. (1984). Optimal pricing of a monopoly against a competi tive producer. Optimal Control Applications & Methods, 5, 345-351. [166] Dockner, E.J. and Feichtinger, G. (1991). On the optimality of limit cycles in dynamic economic systems, Journal of Economics, 53, 31-50. [167] Dockner, E.J. and Feichtinger, G. (1993). Cyclical consumption patterns and rational addiction, American Economic Review, 83, 256-263. [168] Dockner, E.J., Feichtinger, G. and J0rgensen, S. (1985). Tractable classes of nonzero-sum open-loop Nash differential games: theory and examples, Journal of Optimization Theory and Applications, 45, 179-197. [169] Dockner, E.J., Feichtinger, G. and Mehlmann, A. (1993). Dynamic R & D competition with memory. Journal of Evolutionary Economics, 3, 145-152. 430 Bibliography [170] Dockner, E.J. and J0rgensen, S., (1984). Cooperative and non- cooperative differential game solutions to an investment and pricing prob lem, Journal of Operational Research Society^ 35, 731-739. [171] Dockner, E.J. and J0rgensen, S., (1986). Dynamic advertising and pricing in an oligopoly: a Nash equilibrium approach, in Dynamic Games and Applications in Economics, T. Basar (Ed.), Springer-Verlag, Berlin, 238- 251. [172] Dockner, E.J. and J0rgensen, S., (1988). Optimal advertising policies for diffusion models of new product innovation in monopolistic situations, Management Science, 34, 1, 119-130. [173] Dockner, E.J. and J0rgensen, S., (1992). New product advertising in dynamic oligopolies, Zeitschrift fur Operations Research, 36, 5, 459-473. [174] Dockner, E.J., J0rgensen, S., Long, N.V. and Sorger, G. (2000). Differ ential Games in Economics and Management Science, Cambridge Uni versity Press, Cambridge, UK. [175] Dohrmann, C.R. and Robinett, R.D. (1999). Dynamic programming method for constrained discrete-time optimal control. Journal of Op timization Theory and Applications, 101, 2, 259-283. [176] Dolan, R.J. and Jeuland, A.P., (1981). Experience curves and dynamic demand models: implications of optimal pricing strategies, J. Marketing, 45, 52-73. [177] Dolan, R.J. and Muller, E., (1986). Models of new product diffusion: extension to competition against existing and potential firms over time, in Innovation Diffusion Models of New Product Acceptance, V. Mahajan and Y. Wind (Eds.), Ballinger, Cambridge, MA, 117-150. [178] Dorfman, R. (1969). Economic interpretation of optimal control theory, American Economic Review, 49, 817-831. [179] Dorfman, R. and Steiner, P.O. (1954). Optimal advertising and optimal quality, American Economic Review, 44, 826-836. [180] Drews, W., Hartberger, R.J. and Segers, R. (1974). On continuous math ematical programming in Optimization Methods for Resource Allocation, R.W. Cottle and J. Krarup (Eds.), The English University Press, London. [181] Dubovitskii, A.Y. and Milyutin, A.A. (1963). Extremum problems with constraints, Soviet Math. Dokl, 4, 452-455. [182] Dunn, J.C. and Bertsekas, D.P. (1989). Efficient dynamic programming implementations of Newton's method for unconstrained optimal control problems. Journal of Optimization Theory and Applications, 63,1, 23-38. Bibliography 431 [183] Durrett, R. (1996). Stochastic Calculus: A Practical Introduction, Second Edition, CRC Press. [184] El-Hodiri, M. and Takayama, A. (1981). Dynamic behavior of the firm with adjustment costs under regularity constraint, J. of Economic Dy namics and Control, 3, 29-41. [185] Ehashberg, J. and Steinberg, R. (1987). Marketing-production decisions in an industrial channel of distribution, Management Science, 33, 8, 981- 1000. [186] Elton, E. and Gruber, M. (1975). Finance as a Dynamic Process, Prentice-Hall, Englewood Cliffs, NJ. [187] Erickson, G.M. (1991). Dynamic Models of Advertising Competi tion: Open- and Closed-Loop Extensions, Kluwer Academic Publishers, Boston. [188] Erickson, G.M. (1992). Empirical analysis of closed-loop duopoly adver tising strategies. Management Science, 38, 1732-1749. [189] Fan, L.T. and Wang, C.-S. (1964). An application of the discrete maxi mum principle to a transportation problem. Journal of Mathematics and Physics, 43, 255-260. [190] Feichtinger, G. (1982a). Optimal pricing in a diffusion model with concave price-dependent market potential, O.R. Letters, 1, 6, 236-240. [191] Feichtinger, G. (1982b). Saddle-point analysis in a price-advertising model, Journal of Economic Dynamics and Control, 4, 319-340. [192] Feichtinger, G. (1982c). Optimal repair policy for a machine service prob lem, Optimal Control Applications & Methods, 3, 15-22. [193] Feichtinger, G. (1982d). The Nash solution of a maintenance-production differential game, European Journal of Operational Research, 10, 165- 172. [194] Feichtinger, G. (Ed.) (1982e). Optimal Control Theory and Economic Analysis, First Viennese Workshop on Economic Applications of Control Theory, Vienna, October 28-30, 1981, North-Holland, Amsterdam. [195] Feichtinger, G. (1982f). Anwendungen des Maximumprinzips im Opera tions Research, Teil 1 und 2, OR-Spektrum, 4, 171-190 und 195-212. [196] Feichtinger, G. (1983a). The Nash solution of an advertising differential game: generalization of a model by Leitmann and Schmitendorf, IEEE Transactions on Automatic Control, AC-28, 1044-1048. 432 Bibliography [197] Feichtinger, G. (1983b). A differential games solution to a model of com petition between a thief and the police, Management Science, 29, 686- 699. [198] Feichtinger, G. (1984a). Optimal employment strategies of profit- maximizing and labour-managed firms, Optimal Control Applications & Methods, 5, 235-253. [199] Feichtinger, G. (1984b). On the synergistic influence of two control vari ables on the state of nonlinear optimal control models. Journal of Oper ational Research Society, 35, 907-914. [200] Feichtinger, G. (Ed.) (1985a). Optimal Control Theory and Economic Analysis 2, Second Viennese Workshop on Economic Applications of Con trol Theory, Vienna, May 16 - 18, 1984, North-Holland, Amsterdam. [201] Feichtinger, G. (1985b). Optimal modification of machine reliability by maintenance and production, OR-Spektrum, 7, 43-50. [202] Feichtinger, G. (1987). Intertemporal optimization of wine consumption at a party: an unusual optimal control model, in Keynesian Theory Plan ning Models and Quantitative Economics, Essays in Memory of Vittorio Marrama, G. Gandolfo and F. Marzano (Eds.), Vol. II, Giuffre, Milano, 777-797. [203] Feichtinger, G. (Ed.) (1988). Optimal Control Theory and Economic Analysis, Vol. 3, North-Holland, Amsterdam. [204] Feichtinger, G. (1992). Optimal control of economic systems, in Auto matic Control Handbook, S.G. Tzafestas (Ed.), M. Dekker, New York, 1023-1044. [205] Feichtinger, G. and Dockner, E.J. (1984). A note to J0rgensen's logarith mic advertising differential game, Zeitschrift fur Operations Research, 28, B133-B153. [206] Feichtinger, G. and Dockner, E.J. (1985). Optimal pricing in a duopoly: a noncooperative differential games solution. Journal of Optimization Theory and Applications, 45, 199-218. [207] Feichtinger, G. and Hartl, R.F. (1985a). Optimal pricing and production in an inventory model, European Journal of Operational Research, 19, 45-56. [208] Feichtinger, G. and Hartl, R.F. (1985b). On the use of Hamiltonian and maximized Hamiltonian in non-differentiable control theory, Journal of Optimization Theory & Applications, 46, 493-504. Bibliography 433 [209] Feichtinger, G. and Hartl, R.F. (1986). Optimale Kontrolle Okonomischer Prozesse: Anwendungen des Maximumprinzips in den Wirtschaftswis- senschaften, Walter De Gruyter, Berlin. [210] Feichtinger, G. and Hartl, R.F. (Eds.) (1992). Nonlinear methods in eco nomic dynamics and optimal control, Annals of Operations Research, 37, Baltzer, Basel. [211] Feichtinger, G., Hartl, R.F., Haunschmied, J. and Kort, RM. (1998). Optimal enforcement policies (crackdowns) on a drug market. Optimal Control Applications & Methods, 19, 169-184. [212] Feichtinger, G., Hartl, R.F. and Sethi, S.P. (1994). Dynamic optimal con trol models in advertising: recent developments. Management Science, 40, 2, 195-226. [213] Feichtinger, G. and J0rgensen, S. (1983). Differential game models in management science, European Journal of Operational Research, 14, 137- 155. [214] Feichtinger, G., J0rgensen, S. and Novak, A. (1999). Petrarch's Can- zoniere: rational addiction and amorous cycles. Journal of Mathematical Sociology, 23, 3, 225-240. [215] Feichtinger, G., Kaitala, V.T. and Novak, A. (1992). Stable resource- employment limit cycles in an optimally regulated fishery, in Dynamic Economic Models and Optimal Control, G. Feichtinger (Ed.), North- Holland, Amsterdam, 163-184. [216] Feichtinger, G., Luhmer, A. and Sorger, G. (1988). Optimal price and advertising policy in convenience goods retailing. Marketing Science, 7, 187-201. [217] Feichtinger, G. and Mehlmann, A. (1986). Planning the unusual: appli cations of control theory to non-standard problems, Acta Appl. Math., 7, 79-102. [218] Feichtinger, G. and Novak, A. (1992a). Optimal consumption, training, working time and leisure over the life cycle. Journal of Optimization Theory and Applications, 75, 369-388. [219] Feichtinger, G. and Novak, A. (1992b). A note on the optimal exploitation of migratory fish stocks. Dynamics and Control, 2, 255-263. [220] Feichtinger, G. and Novak, A. (1994). Optimal pulsing in an advertising diflFusion model, Optimal Control Applications & Methods, 15, 267-276. 434 Bibliography [221] Feichtinger, G., Novak, A. and Wirl, F. (1994). Limit cycles in intertem poral adjustment models - theory and applications, Journal of Economic Dynamics and Control, 18, 353-380. [222] Feichtinger, G. and Sorger, G. (1986). Optimal oscillations in control models: how can constant demand lead to cyclical production?. Opera tions Research Letters, 5, 277-281. [223] Feichtinger, G. and Sorger, G. (1988). Periodic research and development, in Optimal Control Theory and Economic Analysis 3, Third Viennese Workshop on Optimal Control Theory and Economic Analysis, Vienna, May 20 - 22, 1987, G. Feichtinger (Ed.), North-Holland, Amsterdam, 121-141. [224] Feichtinger, G. and Wirl, F. (1993). A dynamic variant of the battle of the sexes. International Journal of Game Theory, 22, 359-380. [225] Feinberg, F.M. (1992). Pulsing policies for aggregate advertising models, Marketing Science, 11, 3, 221-234. [226] Fel'dbaum, A.A. (1965). Optimal Control Systems, Academic Press, New York. [227] Ferreira, M.M.A. and Vinter, R.B. (1994). When is the maximum prin ciple for state constrained problems nondegenerate?. Journal of Mathe matical Analysis and Applications, 187, 438-467. [228] Ferreyra, G. (1990). The optimal control problem for the Vidale-Wolfe advertising model revisited, Optimal Control Applications & Methods, 11, 363-368. [229] Filipiak, J. (1982). Optimal control of store-and-forward networks. Opti mal Control Applications & Methods, 3, 155-176. [230] Fischer, T. (1985). Hierarchical optimization methods for the coordina tion of decentralized management planning, in Optimal Control Theory and Economic Analysis 2, G. Feichtinger (Ed.), North-Holland, Amster dam. [231] Fleming, W.H. and Rishel, R.W. (1975). Deterministic and Stochastic Optimal Control, Springer-Verlag. New York. [232] Fleming, W.H. and Soner, H.M. (1992). Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York. [233] Fletcher, R. and Reeves, CM. (1964). Function minimization by conju gate gradients. Computer Journal, 7, 149-154. Bibliography 435 [234] Forster, B.A. (1973). Optimal consumption planning in a polluted envi ronment, Economic Record, 49, 534-545. [235] Forster, B.A. (1977). On a one state variable optimal control problem: consumption-pollution trade-offs, in Applications of Control Theory to Economic Analysis, J.D. Pitchford and S.J. Turnovsky (Eds.), North- Holland, Amsterdam. [236] Fourgeaud, C, Lenclud, B. and Michel, P. (1982). Technological renewal of natural resource stocks. Journal of Economic Dynamics and Control, 4, 1-36. [237] Francis, P.J. (1997). Dynamic epidemiology and the market for vaccina tions, Journal of Public Economics, 63, 3, 383-406. [238] Frankena, J.F. (1975). Optimal control problems with delay, the maxi mum principle and necessary conditions. Journal of Eng. Math., 9, 53-64. [239] Friedman, A. (1964). Optimal control for hereditary processes. Archive Rational Mechanics Analysis, 15, 396-416. [240] Friedman, A. (1971). Differential Games, Wiley, New York. [241] Friedman, A. (1977). Oligopoly and the Theory of Games, North-Holland, Amsterdam. [242] Friedman, A. (1986). Game Theory with Applications to Economics, Ox ford University Press, New York. [243] Fruchter, G.E. (1999). The many-player advertising game. Management Science, 45, 11, 1609-1611. [244] Fruchter, G.E. (1999). Oligopoly advertising strategies with market ex pansion, Optimal Control Application & Methods, 20, 199-211. [245] Fruchter, G.E. and S. Kalish (1997). Closed-loop advertising strategies in a duopoly, Management Science, 43, 54-63. [246] Fuller, D. and Vickson, R.G. (1987). The optimal construction of new plants for oil from the Alberta tar sands, Operations Research, 35, 5, 704-715. [247] Funke, U.H. (1976). Mathematical Models in Marketing: A Collection of Abstracts, Lecture Notes in Economics and Mathematical Systems, Vol. 132, Springer-Verlag, Berlin. [248] Gaimon, C. (1985a). The acquisition of automation subject to diminish ing returns, HE Transactions, 17, 147-156. 436 Bibliography [249] Gaimon, C. (1985b). The optimal acquisition of automation to enhance the productivity of labor, Management Science^ 31, 1175-1190. [250] Gaimon, C. (1986a). An impulsive control approach to deriving the op timal dynamic mix of manual and automatic output, European Journal of Operational Research, 24, 360-368. [251] Gaimon, C. (1986b). The optimal times and levels of impulsive acquisi tion of automation. Optimal Control Applications & Methods, 7, 259-270. [252] Gaimon, C. (1986c). The optimal acquisition of new technology and its impact on dynamic pricing policies. Studies in Management Science and Systems, B. Lev (Ed.), 13, 187-206. [253] Gaimon, C. (1988). Simultaneous and dynamic price, production, inven tory, and capacity decisions, European Journal of Operational Research, 35, 426-441. [254] Gaimon, C. (1989). Dynamic game results on the acquisition of new technology. Operations Research, 37, 3, 410-425. [255] Gaimon, C. (1994). Subcontracting versus capacity expansion and the impact on the pricing of services. Naval Research Logistics, 41, 7, 875- 892. [256] Gaimon, C. (1997). Planning information technology - knowledge worker systems. Management Science, 43, 9, 1308-1328. [257] Gaimon, C. and Thompson, G.L. (1984a). Optimal preventive and repair maintenance of a machine subject to failure. Optimal Control Applica tions & Methods, 5, 57-67. [258] Gaimon, C. and Thompson, G.L. (1984b). A distributed parameter co hort personnel planning model using cross-sectional data. Management Science, 30, 750-764. [259] Gaimon, C. and Thompson, G.L. (1989). Optimal preventive and repair maintenance of a machine subject to failure and a declining resale value, Optimal Control Applications & Methods, 10, 211-228. [260] Gamkrelidze, R.V. (1978). Principles of Optimal Control Theory, Plenum Press, New York. [261] Gandolfo, G. (1980). Economic Dynamics: Methods and Models, North- Holland, Amsterdam. [262] Gaskins, Jr., D.W. (1971). Dynamic limit pricing: optimal pricing under threat of entry. Journal of Economic Theory, 3, 306-322. Bibliography 437 [263] Gaugusch, J. (1984). The non-cooperative solution of a differential game: advertising versus pricing, Optimal Control Applications & Methods^ 5, 4, 353-360. [264] Gelfand, I.M. and Fomin, S.V. (1963). Calculus of Variations, Prentice- Hall, Englewood Cliffs, NJ. [265] Gerchak, Y. and Parlar, M. (1985). Optimal control analysis of a simple criminal prosecution model, Optimal Control Applications & Methods, 6, 305-312. [266] Gfrerer, H. (1984). Optimization of hydro energy storage plant problems by variational methods, Zeitschrift filr Operations Research, 28, B87- BlOl. [267] Gihman, LI. and Skorohod, A.V. (1972). Stochastic Differential Equa tions, Springer-Verlag, New York. [268] Girsanov, I.V. (1972). Lectures on Mathematical Theory of Extremum Problems, Springer-Verlag, Berlin. [269] Glad, S.T. (1979). A combination of penalty function and multiplier methods for solving optimal control problems. Journal of Optimization Theory and Applications, 28, 303-329. [270] Goh, B.-S. (1980). Management and Analysis of Biological Populations, Elsevier, Amsterdam. [271] Goh, B.-S., Leitmann, G. and Vincent, T.L. (1974). Optimal control of a prey-predator system. Mathematical Biosciences, 19, 263-286. [272] Goldberg, S. (1986). Introduction to Difference Equations, Dover Publi cations, New York. [273] Goldstine, H.H. (1980). A History of the Calculus of Variations from the 17th through the 19th Century, Springer-Verlag, New York. [274] Gopalsamy, K. (1976). Optimal control of age-dependent populations, Math. Biosci., 32, 155-163. [275] Gordon, H.S. (1954). Economic theory of a common-property resource: the fishery. Journal of Political Economy, 62, 124-142. [276] Gordon, M.J. (1962). The Investment, Financing and Valuation of the Corporation, Richard D. Irwin, Inc., Homewood, 111. [277] Gould, J.P. (1970). Diffusion processes and optimal advertising policy, in Microeconomic Foundation of Employment and Inflation Theory, E.S. Phelps et al. (Eds.), Norton, New York, 338-368. 438 Bibliography [278] Grimm, W., Well, K.H. and Oberle, H.J. (1986). Periodic control for Minimum-fuel aircraft trajectories, Journal of Guidance, 9, 169-174. [279] Gross, M. and Lieber, Z. (1984). Competitive monopolistic and efficient utilization of an exhaustible resource in the presence of habit-formation effects and stock dependent costs, Econ. Letters, 14, 383-388. [280] Hadley, G. and Kemp, M.C. (1971). Variational Methods in Economics, North-Holland, Amsterdam. [281] Hahn, M. and Hyun, J.S. (1991). Advertising cost interpretations and the optimality of pulsing. Management Science, 37, 2, 157-169. [282] Halkin, H. (1966). A maximum principle of the Pontryagin type for sys tems described by nonlinear difference equations, SI AM Journal on Con trol, 4, 90-111. [283] Halkin, H. (1967). On the necessary condition for optimal control of non linear systems, in Topics in Optimization, G. Leitmann (Ed.), Academic Press, New York. [284] Hamalainen, R.P., Haurie, A. and Kaitala, V.T. (1984). Bargaining on whales: a differential game model with Pareto optimal equilibria. Oper ations Research Letters, 3, 1, 5-11. [285] Hamalainen, R.P., Haurie, A. and Kaitala, V.T. (1985). Equilibria and threats in a fishery management game. Optimal Control Applications & Methods, 6, 315-333. [286] Hamalainen, R.P., Ruusunen, J. and Kaitala, V.T. (1986). Myopic Stack- elberg equilibria and social coordination in a share contract fishery. Ma rine Resource Economics, [287] Hamalainen, R.P., Ruusunen, J. and Kaitala, V.T. (1990). Cartels and dynamic contracts in sharefishing, Journal of Environmental Economics and Management, 19, 175-192. [288] Han, M., Feichtinger, G. and Hartl, R.F. (1994). Nonconcavity and proper optimal periodic control. Journal of Economic Dynamics and Control, 18, 976-990. [289] Hanssens, D.M., Parsons, L.J. and Schultz, R. L. (1990). Market Re sponse Models: Econometric and Time Series Analysis, Kluwer Aca demic Publishers, Boston. [290] Harris, F.W. (1913). How many parts to make at once. Factory, The Magazine of Management, 10, 135-136, 152. Bibliography 439 [291] Harris, H. (1976). Optimal planning under transaction costs: the demand for money and other assets, Journal of Economic Theory, 12, 298-314. [292] Harrison, J.M. and Pliska, S.R. (1981). Martingales, stochastic integrals, and continuous trading, Stochastic Process. Appl, 11, 215-260. [293] Hartberger, R. J. (1973). A proof of the Pontryagin maximum principle for initial-value problem, Journal of Optimization Theory and Applications, 11, 139-145. [294] Hartl, R.F. (1982a). A mixed linear/nonlinear optimization model of pro duction and maintenance for a machine, in Optimal Control Theory and Economic Analysis, G. Feichtinger (Ed.), North-Holland, Amsterdam, 43-58. [295] Hartl, R.F. (1982b). Optimal control of non-linear advertising models with replenishable budget. Optimal Control Applications & Methods, 3, 1, 53-65. [296] Hartl, R.F. (1982c). Optimal control of concave economic models with two control instruments, in Operations Research in Progress, G. Fe ichtinger and P. Kail (Eds.), D. Reidel Publishing Company, Dordrecht, Holland, 227-245. [297] Hartl, R.F. (1983a). Optimal allocation of resources in the production of human capital. Journal of the Operational Research Society, 34, 599-606. [298] Hartl, R.F. (1983b). Optimal maintenance and production rates for a machine: a nonlinear economic control problem. Journal of Economic Dynamics and Control, 6, 281-306. [299] Hartl, R.F. (1984). Optimal dynamic advertising policies for hereditary processes. Journal of Optimization Theory and Applications, 43, 1, 51-72. [300] Hartl, R.F. (1986a). A forward algorithm for a generalized wheat trading model, Zeitschrift filr Operations Research, 30, A 135-A 144. [301] Hartl, R.F. (1986b). Arrow-type sufficient optimality conditions for non- difFerentiable optimal control problems with state constraints, Applied Mathematics & Optimization, 14, 229-247. [302] Hartl, R.F. (1987). A simple proof of the monotonicity of the state tra jectories in autonomous control problems. Journal of Economic Theory, 40, 211-215. [303] Hartl, R.F. (1988a). A wheat trading model with demand and spoilage, in Optimal Control Theory and Economic Analysis, G. Feichtinger (Ed.), Vol. 3, North-Holland, Amsterdam, 235-244. 440 Bibliography [304] Hartl, R.F. (1988b). A dynamic activity analysis for a monopolistic firm, Optimal Control Applications & Methods, 9, 253-272. [305] Hartl, R.F. (1988c). The control of environmental pollution and opti mal investment and employment decisions: a comment, Optimal Control Applications & Methods, 9, 337-339. [306] Hartl, R.F. (1989a). On forward algorithms for a generalized wheat trad ing model, in Progress in Inventory Research (Proceedings of 4th Interna tional Symposium on Inventories), A. Chikan (Ed.), Hungarian Academy of Science, Budapest. [307] Hartl, R.F. (1989b). Most rapid approach paths in dynamic economic problems, in Methods of Operations Research 58 (Proceedings of SOR 12, 1987), P. Kleinschmidt et al. (Eds.), Athenaun, 397-410. [308] Hartl, R.F. (1992). Optimal acquisition of pollution control equipment under uncertainty. Management Science, 38, 609-622. [309] Hartl, R.F. (1993). On the properness of one-dimensional periodic control problems. Systems & Control Letters, 20, 3, 393-395. [310] Hartl, R.F. (1995). Production smoothing under environmental con straints, Production and Operations Management, 4, 1, 46-56. [311] Hartl, R.F. and Feichtinger, G. (1987). A new sufficient condition for most rapid approach paths. Journal of Optimization Theory and Applications, 54, 2, 403-411. [312] Hartl, R.F., Feichtinger, G. and Kirakossian, G.T. (1992). Optimal re cycling of tailings for the production of building materials, Czechoslovak Journal of Operations Research, 1,3, 181-192. [313] Hartl, R.F. and J0rgensen, S. (1985). Optimal manpower policies in a dy namic staff-maximizing bureau, Optimal Control Application & Methods, 6, 1, 57-64. [314] Hartl, R.F. and J0rgensen, S. (1988). Aspects of optimal slidesmanship, in Optimal Control Theory and Economic Analysis, G. Feichtinger (Ed.), Vol. 3, North-Holland, Amsterdam, 335-350. [315] Hartl, R.F. and J0rgensen, S. (1990). Optimal slidesmanship in confer ences with unpredictable chairmen. Optimal Control Application & Meth ods, 11, 143-155. [316] Hartl, R.F. and Kort, P.M. (1996a). Marketable permits in a stochastic dynamic model of the firm. Journal of Optimization Theory and Appli cations, 89, 1, 129-155. Bibliography 441 [317] Hartl, R.F. and Kort, P.M. (1996b). Capital accumulation of a firm facing an environmental tax scheme, Journal of Economics, 63, 1, 1-23. [318] Hartl, R.F. and Kort, P.M. (1996c). Capital accumulation of a firm facing environmental constraints. Optimal Control Applications & Methods, 17, 253-266. [319] Hartl, R.F. and Kort, P.M. (1997). Optimal input substitution of a firm facing an environmental constraint, European Journal of Operational Re search, 99, 1. [320] Hartl, R.F., Kort, P.M. and Novak, A. (1999). Optimal investment facing possible accidents. Annals of Operations Research, 88, 99-117. [321] Hartl, R.F. and Krauth, J. (1989). Optimal production mix. Journal of Optimization Theory and Applications, 66, 255-273. [322] Hartl, R.F. and Luptacik, M. (1992). Environmental constraints and choice of technology, Czechoslovak Journal of Operations Research, 1, 2, 107-125. [323] Hartl, R.F. and Mehlmann, A. (1982). The Transylvanian problem of renewable resources. Revue Prancaise d'Automatique, Informatique et de Recherche Operationelle, 16, 379-390. [324] Hartl, R.F. and Mehlmann, A. (1983). Convex-concave utility function: optimal blood-consumption for vampires. Applied Mathematical Mod elling, 7, 83-88. [325] Hartl, R.F. and Mehlmann, A. (1984). Optimal seducing policies for dy namic continuous lovers under risk of being killed by a rival. Cybernetics & Systems: An Intern. J., 15, 119-126. [326] Hartl, R.F. and Mehlmann, A. (1986). On remuneration patterns for medical services. Optimal Control Applications & Methods, 7, 185-193. [327] Hartl, R.F., Mehlmann, A. and Novak, A. (1992). Cycles of fear: opti mal periodic blood-sucking rates for vampires. Journal of Optimization Theory and Applications, 75, 3, 559-568. [328] Hartl, R.F. and Sethi, S.P. (1983). A note on the free terminal time transversality condition, Zeitschrift fur Operations Research^ Series: Theory, 27, 5, 203-208. [329] Hartl, R.F. and Sethi, S.P. (1984a). Optimal control problems with differential inclusions: sufficiency conditions and an application to a production-inventory model. Optimal Control Applications & Methods, 5, 4, 289-307. 442 Bihliography [330] Hartl, R.F. and Sethi, S.P. (1984b). Optimal control of a class of systems with continuous lags: dynamic programming approach and economic in terpretations, Journal of Optimization Theory and Applications, 43, 1, 73-88. [331] Hartl, R.F. and Sethi, S.P. (1985a). Solution of generalized linear optimal control problems using a simplex-like method in continuous-time I: the ory, in Optimal Control Theory and Economic Analysis 2, G. Feichtinger, (Ed.), North-Holland, Amsterdam, 45-62. [332] Hartl, R.F. and Sethi, S.P. (1985b). Solution of generalized linear op timal control problems using a simplex-like method in continuous-time H: examples, in Optimal Control Theory and Economic Analysis 2, G. Feichtinger (Ed.), North-Holland, Amsterdam, 63-87. [333] Hartl, R.F., Sethi, S.P. and Vickson, R.G. (1995). A survey of the max imum principles for optimal control problems with state constraints, SIAM Review, 37, 2, 181-218. [334] Harvey, A.C. (1994). Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press, New York. [335] Haurie, A. (1976). Optimal control on an infinite time horizon: the turn pike approach, J. Math. Economics, 3, 81-102. [336] Haurie, A. and Hung, N.M. (1977). Turnpike properties for the optimal use of a natural resource. Review of Economic Studies, 44, 329-336. [337] Haurie, A. and Leitmann, G. (1984). On the global asymptotic stability of equilibrium solutions for open-loop differential games. Large Scale Syst., 6, 107-122. [338] Haurie, A. and Sethi, S.P. (1984). Decision and forecast horizons, agree able plans, and the maximum principle for infinite horizon control prob lems, Operations Research Letters, 3, 5, 261-265. [339] Haurie, A., Sethi, S.P. and Hartl, R.F. (1984). Optimal control of an age- structured population model with applications to social services planning, J. of Large Scale Systems, 6, 133-158. [340] Haurie, A., Tolwinski, B. and Leitmann, G. (1983). Cooperative equilib ria in differential games. Proceedings ACC, San Francisco. [341] Haussmann, U. G. (1981). Some examples of optimal stochastic controls or: the stochastic maximum principle at work, SIAM Review, 23, 292- 307. Bibliography 443 [342] Heal, G.M. (1976). The relationship between price and extraction cost for a resource with a backstop technology, Bell Journal of Economics, 7, 371-378. [343] Heaps, T. (1984). The forestry maximum principle, Journal of Economic Dynamics and Control, 7, 131-151. [344] Heckman, J. (1976). A Hfe cycle model of earnings, learning, and con sumption, Journal of Political Economy, 84, 511-544. [345] Hestenes, M.R. (1966). Calculus of Variations and Optimal Control The ory, Wiley, New York. [346] Ho, Y.-C. (1970). Differential games, dynamic optimization and general ized control theory. Journal of Optimization Theory and Applications, 6, 179-209. [347] Hoffmann, K.H. and Krabs, W. (1984). Optimal Control of Partial Dif ferential Equations, Birkhauser, Basel. [348] Holly, S., Riistem, B. and Zarrop, M.B. (Eds.) (1979). Optimal Control for Econometric Models. An Approach to Economic Policy Formulation, Macmillan, London. [349] Holt, C.C, Modigliani, F., Muth, J.F. and Simon, H.A. (1960). Planning Production^ Inventories and Workforce, Prentice-Hall, Englewood Cliffs, NJ. [350] Holtzman, J.M. (1966). On the maximum principle for nonlinear discrete- time systems, IEEE Transactions on Automatic Control, AC-11, 273- 274. [351] Horsky, D. (1977). An empirical analysis of the optimal advertising policy, Management Science, 23, 10, 1037-1049. [352] Horsky, D. and Mate, K. (1988). Dynamic advertising strategies of com peting durable good producers. Marketing Science, 7, 4, 356-367. [353] Horsky, D. and Simon, L.S. (1983). Advertising and the diffusion of new products. Marketing Science, 2, 1, 1-17. [354] Hotelling, H. (1925). A general mathematical theory of depreciation, Journal of Amer. Statist. Assoc, 20, 340-353. [355] Hotelling, H. (1931). The economics of exhaustible resources. Journal of Political Economy, 39, 137-175. [356] Hwang, C.L., Fan, L.T. and Erickson, L.E. (1967), Optimal production planning by the maximum principle, Management Science, 13, 750-755. 444 Bibliography [357] Ijiri, Y. and Thompson, G.L. (1970). Applications of mathematical con trol theory to accounting and budgeting (the continuous wheat trading model), The Accounting Review^ 45, 246-258. [358] Ijiri, Y. and Thompson, G.L. (1972). Mathematical control theory solu tion of an interactive accounting flows model. Naval Research Logistics Quarterly, 19, 411-422. [359] Intriligator, M.D. (1971). Mathematical Optimization and Economic The ory, Prentice-Hall, Englewood Cliffs, NJ. [360] Intriligator, M.D. (1980). Applications of control theory to economics, in Analysis and Optimization of Systems, Lecture Notes in Control and Information Sciences, A. Bensoussan and J.L. Lions (Eds.), Vol. 28, Springer-Verlag, BerHn, 607-626. [361] Intriligator, M.D. and Smith, B.L.R. (1966). Some aspects of the allo cation of scientific effort between teaching and research, American Eco nomic Review, 61, 494-507. [362] loffe, A.D. and Tihomirov, V.M. (1979). Theory of Extremal Problems, North-Holland, Amsterdam. [363] Isaacs, R. (1965). Differential Games, Wiley, New York. [364] Isaacs, R. (1969). Differential games: their scope, nature, and future, Journal of Optimization Theory and Applications, 3, 283-295. [365] Jacobson, D.H., Lele, M.M. and Speyer, J.L. (1971) New necessary con ditions of optimality for control problems with state-variable inequality constraints, Journal of Mathematical Analysis and Applications, 35, 255- 284. [366] Jacquemin, A.P. (1973). Optimal control and advertising policy, Metro- Economica, 25, 200-207. [367] Jacquemin, A.R and Thisse, J. (1972). Strategy of the firm and market structure: an application of optimal control theory, in Market Structure and Corporate Behavior, K. Cowling (Ed.), Gray-Mills, London, 61-84. [368] Jagpal, S. (1999). Marketing Strategy and Uncertainty, Oxford University Press, New York. [369] Jamshidi, M. (1983). Large-Scale Systems: Modelling and Control, North- Holland, New York. [370] Jazwinski, A.H. (1970). Stochastic Processes and Filtering Theory, Aca demic Press, New York. Bibliography 445 [371] Jedidi, K., Eliashberg, J. and DeSarbo, W. (1989). Optimal advertising and pricing for a three-stage time-lagged monopolistic diffusion model incorporating income, Optimal Control Applications & Methods, 10, 313- 331. [372] Jennings, L.S., Sethi, S.P. and Teo, K.L. (1997). Computation of optimal production plans for manufacturing systems. Nonlinear Analysis^ Theory^ Methods & Applications, 30, 7, 4329-4338. [373] Jennings, L.S. and Teo, K.L. (1997). Computation of manufacturing sys tems using an enhancing control, in IPMM'97: Australiasia-Pacific Fo rum on Intelligent Processing & Manufacturing of Materials, Vol. 1, T. Chandra, S.R. Leclair, J.A. Meech, B. Verma, M. Smith, and B. Bal- achandran (Eds.), Watson Ferguson & Co., Brisbane, Australia. [374] Jeuland, A.P. and Dolan, R.J. (1982). An aspect of new product plan ning: dynamic pricing, in Marketing Planning Models, TIMS Studies in the Management Science, A.A. Zoltners (Ed.), Vol. 18, North-Holland, Amsterdam, 1-21. [375] Jiang, J. and Sethi, S.P. (1991). A state aggregation approach to man ufacturing systems having machine states with weak and strong interac tions, Operations Research, 39, 6, 970-978. [376] Johnson, CD. and Gibson, J.E. (1963). Singular solutions in problems of optimal control, IEEE Transactions on Automatic Control, AC-8, 4-15. [377] Jones, P. 1983). Analysis of a dynamic duopoly model of advertising, Mathematics of Operations Research, 8, 1, 122-134. [378] J0rgensen, S., (1982a). A survey of some differential games in advertising, Journal of Economic Dynamics and Control, 4, 341-369. [379] J0rgensen, S., (1982b). A differential games solution to a logarithmic advertising model. Operations Research Soc, 33, 5, 425-432. [380] J0rgensen, S., (1983). Optimal control of a diffusion model of new product acceptance with price-dependent total market potential. Optimal Control Application & Methods, 4, 3, 269-276. [381] J0rgensen, S., (1984). A Pareto-optimal solution of a maintenance- production differential game, European Journal of Operational Research, 18, 76-80. [382] J0rgensen, S., (1985). An exponential differential game which admits a simple Nash solution. Journal of Optimization Theory and Applications, 45, 383-396. 446 Bibliography [383] J0rgensen, S., (1986a). Optimal production, purchasing and pricing: a differential games approach, European Journal of Operational Research, 24, 64-76. [384] J0rgensen, S., (1986b). Optimal dynamic pricing in an oligopolistic mar ket: a survey, in Dynamic Games and Applications in Economics, T. Basar (Ed.), Springer-Verlag, Berhn, 179-237. [385] J0rgensen, S. (1992). The dynamics of extramarital affairs, in Dynamic Economic Models and Optimal Control, G. Feichtinger (Ed.), North- Holland, Amsterdam, 239-266. [386] J0rgensen, S. and Dockner, E.J. (1985). Optimal consumption and re plenishment policies for a renewable resource, in Optimal Control Theory and Economic Analysis 2, G. Feichtinger (Ed.), North-Holland, Amster dam, 647-664. [387] J0rgensen, S. and Kort, P.M. (1993). Optimal dynamic investment poli cies under concave-convex adjustment costs, Journal of Economic Dy namics and Control, 17, 1/2, 153-180. [388] J0rgensen, S. and Kort, P.M. (1997). Optimal investment and finance in renewable resource harvesting, Journal of Economic Dynamics and Control, 21, 603-630. [389] J0rgensen, S., Kort, P.M. and Zaccour, G., (1999). Production, inventory, and pricing under cost and demand learning effects, European Journal of Operational Research, 117, 382-395. [390] Joseph, P.D. and Tou, J.T. (1961). On hnear control theory. Trans. AIEE, Part II, 80, 193-196. [391] Journal of Marketing Research (1985). Special issue on competition in marketing, 22, 3. [392] Kaitala, V.T. (1986). Game theory models of fisheries management - a survey, in Dynamic Games and Applications in Economics, T. Basar (Ed.), Springer-Verlag, Berlin, 252-266. [393] Kalish, S. (1983). Monopolist pricing with dynamic demand and produc tion cost. Marketing Science, 2, 2, 135-159. [394] Kalish, S. (1985). A new product adoption model with price, advertising, and uncertainty. Marketing Science, 31, 12, 1569-1585. [395] Kalish, S. and Lilien, G.L. (1983). Optimal price subsidy policy for ac celerating the diffusion of innovation. Marketing Science, 2, 4, 407-420. Bibliography 447 [396] Kalish, S. and Sen, S.K. (1986). Diffusion models and the marketing mix for single products, in Series in Econometrics and Management Science: Innovation Diffusion Models of New Products Acceptance^ V. Mahajan and Y. Wind (Eds.), Ballinger, Cambridge, MA, Vol. V, 87-116. [397] Kalman, R.E. (1960a). A new approach to linear filtering and prediction problems. Trans. ASME Ser. D: J. Basic Eng., 82, 35-45. [398] Kalman, R.E. (1960b). Contributions to the theory of optimal control, Bol. de Soc. Math. Mexicana, 102-119. [399] Kalman, R.E. and Bucy, R. (1961). New results in linear filtering and prediction theory. Trans. ASME Ser. D: J. Basic Eng., 83, 95-108. [400] Kamien, M.I. and Schwartz, N.L. (1971a). Optimal maintenance and sale age for a machine subject to failure. Management Science^ 17, 427-449. [401] Kamien, M.I. and Schwartz, N.L. (1971b). Limit pricing and uncertain entry, Econometrica, 39, 441-454. [402] Kamien, M.I. and Schwartz, N.L. (1978). Optimal exhaustible resource depletion with endogenous technical change. Review of Economic Studies, 45, 179-196. [403] Kamien, M.I. and Schwartz, N.L. (1982a). Market Structure and Innova tion, Cambridge University Press, Cambridge. [404] Kamien, M.I. and Schwartz, N.L. (1982b). The role of common property resources in optimal planning models with exhaustible resources, in Ex plorations in Natural Resource Economics, V.K. Smith and J.V. Krutilla (Eds.), John Hopkins University Press, Baltimore, Maryland, 47-71. [405] Kamien, M.I. and Schwartz, N.L. (1998). Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Manage ment, Second Edition, Fourth Impression, North-Holland, New York. [406] Kaplan, W. (1958). Ordinary Differential Equations, Addison-Wesley, Reading, PA. [407] Karatzas, I., Lehoczky, J.P., Sethi, S.P. and Shreve, S.E. (1986). Explicit solution of a general consumption/investment problem. Mathematics of Operations Research, 11, 2, 261-294. [408] Karatzas, I. and Shreve, S.E. (1997). Brownian Motion and Stochastic Calculus, Second Edition, Springer-Verlag, New York. [409] Karatzas, I. and Shreve, S.E. (1998). Methods of Mathematical Finance, Springer-Verlag, New York. 448 Bibliography [410] Keeler, E., Spence, M. and Zeckhauser, R.J. (1971). The optimal control of pollution. Journal of Economic Theory^ 4, 19-34. [411] Keller, H.B. (1968). Numerical Methods for Two-Point Boundary Value Problems, Blaisdell, Waltham, Mass. [412] Kemp, M.C. and Long, N.V. (1977). Optimal control problems with inte grands discontinuous with respect to time, Economic Record, 53, 405-420. [413] Kemp, M.C. and Long, N.V. (Eds.) (1980). Exhaustible Resources, Opti- mality, and Trade, North-Holland, Amsterdam, [414] Kendrick, D.A. (1981). Stochastic Control for Economic Models, McGraw-Hill, New York. [415] Khmelnitsky, E. and Kogan, K. (1994). Necessary optimality conditions for a generalized problem of production scheduling, Optimal Control Ap plications & Methods, 15, 215-222. [416] Khmelnitsky, E. and Kogan, K. (1996). Optimal policies for aggregate production and capacity planning under an arbitrary demand. Interna tional Journal of Production Research, 34, 7, 1929-1941. [417] Khmelnitsky, E., Kogan, K. and Maimon, O. (1995). A maximum prin ciple based combined method for scheduling in a flexible manufacturing system. Discrete Event Dynamic Systems, 5, 343-355. [418] Khmelnitsky, E., Kogan, K. and Maimon, O. (1997). Maximum principle- based methods for scheduling FMS with partially sequence dependent setups, International Journal of Production Research, 35, 10, 2701-2712. [419] Kilkki, P. and Vaisanen, U. (1969). Determination of optimal policy for forest stands by means of dynamic programming. Acta Forestalia Fen- nica, 102, 100-112. [420] Kirby, B.J. (Ed.) (1974). Optimal Control Theory and its Applications, Lecture Notes in Economics and Mathematical Systems, Part I k, H, Vols. 105 & 106, Springer-Verlag, Berlin. [421] Kirk, D.E. (1970). Optimal Control Theory: An Introduction, Prentice- Hall, Englewood ChfFs, NJ. [422] Klein, C.F. and Gruver, W.A. (1978). On the optimal control of a single- server queueing system: comment. Journal of Optimization Theory and Applications,2%, 457-462. [423] Kleindorfer, P.R. (1978). Stochastic control models in management sci ence: theory and computation, TIMS Studies in the Management Sci ences,9, 69-88. Bibliography 449 [424] Kleindorfer, P.R., Kriebel, C.H., Thompson, G.L. and Kleindorfer, G.B. (1975). Discrete optimal control of production plans, Management Sci ence, 22, 261-273. [425] Kleindorfer, P.R. and Lieber, Z. (1979). Algorithms and planning horizon results for production planning problems with separable costs. Operations Research, 27, 874-887. [426] Knobloch, H.W. (1981). Higher Order Necessary Conditions in Optimal Control Theory, Lecture Notes in Control and Information Sciences, Vol. 34, Springer-Verlag, Berlin. [427] Knowles, G. (1981). An Introduction to Applied Optimal Control, Aca demic Press, New York. [428] Kogan, K. and Khmelnitsky, E. (1996). An optimal control model for continuous time production and setup scheduling, International Journal of Production Research, 34, 3, 715-725. [429] Kogan, K., Khmelnitsky, E., Shtub, A. and Maimon, O. (1997). Optimal flow control of flexible manufacturing systems: setup localization by an iterative procedure. International Journal of Production Economics, 51, 37-46. [430] Kotowitz, Y. and Mathewson, F. (1979a). Informative advertising and welfare, American Economic Review, 69, 284-294. [431] Kotowitz, Y. and Mathewson, F. (1979b). Advertising, consumer infor mation, and product quality. Bell J. Economics, 10, 566-588. [432] Kreindler, E. (1982). Additional necessary conditions for optimal con trol with state-variable inequality constraints. Journal of Optimization Theory and Applications, 38, 241-250. [433] Krelle, W. (1984). Economic growth with exhaustible resources and en vironmental protection, Zeitschrift fiir ges. Staatswiss, 140, 399-429. [434] Krichagina, E., Lou, S., Sethi, S.P. and Taksar, M.I. (1993). Production control in a failure-prone manufacturing system: diffusion approximation and asymptotic optimality. Annals of Applied Probability, 3, 2, 421-453. [435] Krichagina, E., Lou, S., Sethi, S.P. and Taksar, M.I. (1995). Diffusion approximation for a controlled stochastic manufacturing system with av erage cost minimization. Mathematics of Operations Research, 20, 4, 895- 922. [436] Krouse, C.G. (1972). Optimal financing and capital structure programs for the firm. Journal of Finance, 27, 1057-1071. 450 Bibliography [437] Krouse, C.G. and Lee, W.Y. (1973). Optimal equity financing of the corporation, Journal of Financial and Quantitative Analysis, 8, 539-563. [438] Kugelmann, B. and Pesch, H.J. (1990). New general guidance method in constrained optimal control, part 1: numerical method. Journal of Optimization Theory and Applications, 67, 3, 421-435. [439] Kurawarwala, A.A. and Matsuo, H. (1996). Forecasting and inventory management of short life-cycle products. Operations Research, 44, 1, 131-150. [440] Kurcyusz, S. and Zowe, J. (1979). Regularity and stability for the math ematical programming problem in Banach spaces. Applied Mathematics & Optimization, 5, 49-62. [441] Kushner, H.J. (1971). Introduction to Stochastic Control, Holt, Rinehart & Winston, New York. [442] Kushner, H.J. (1977). Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Academic Press, New York. [443] Kydland, F.E. and Prescott, E.G. (1977). Rules rather than discretion: the inconsistency of optimal plans. Journal of Political Economy, 85, 473-493. [444] Lagunov, V.N. (1985). Introduction to Differential Games and Control Theory, Hildermann, Berlin. [445] Lansdowne, Z.F. (1970). The theory and applications of generalized lin ear control processes, technical report no. 10, Department of Operations Research, Stanford University. [446] Lasdon, L.S., Mitter, S.K., and Warren, A.D. (1967). The conjugate gradient method for optimal control problems, IEEE Transactions on Automatic Control, AC-12, 132-138. [447] Leban, R. and Lesourne, J. (1983). Adaptive strategies of the firm through a business cycle, J. of Economic Dynamics and Control, 5, 201- 234. [448] Lee, E.B. and Markus, L. (1968). Foundations of Optimal Control Theory, Wiley, New York. [449] Legey, L., Ripper, M. and Varaiya, P.P. (1973). Effects of congestion on the shape of the city. Journal of Economic Theory, 6,162-179. [450] Lehoczky, J.P., Sethi, S.P., Soner H.M. and Taksar, M.L (1991). An asymptotic analysis of hierarchical control of manufacturing systems un der uncertainty. Mathematics of Operations Research, 16, 3, 596-608. Bibliography 451 [451] Leitmann, G. (Ed.) (1962). Optimization Techniques with Applications to Aerospace Systems, Academic Press, New York. [452] Leitmann, G. (1966). An Introduction to Optimal Control, McGraw-Hill, New York. [453] Leitmann, G. (Ed.) (1967). Topics in Optimization, Academic Press, New York. [454] Leitmann, G. (1968). Sufficiency theorems for optimal control. Journal of Optimization Theory and Applications, 2, 285-292. [455] Leitmann, G. (1974). Cooperative and Non-Cooperative Many Players Differential Games, Springer-Verlag, Wien. [456] Leitmann, G. (Ed.) (1976). Multicriteria Decision Making and Differen tial Games, Plenum Press, New York. [457] Leitmann, G. (1981). The Calculus of Variations and Optimal control, Plenum Press, New York. [458] Leitmann, G. and Liu, P.-T. (1974). A differential game model of labor- management negotiation during a strike. Journal of Optimization Theory and Applications, 13, 427-444. [459] Leland, H.E. (1972). The dynamics of a revenue maximizing firm, Inter national Economic Review, 13, 376-385. [460] Lele, M.M., Jacobson, D.H. and McCabe, J.L. (1971). Qualitative ap plication of a result in control theory to problems of economic growth, International Economic Review, 12, 209-226. [461] Leonard, D. and Long, N.V.(1992). Optimal Control Theory and Static Optimization in Economics, Cambridge University Press, Cambridge, UK. [462] Lesourne, J. (1973). Croissance Optimale des Entreprises, Dunod, Paris. [463] Lesourne, J. and Leban, R. (1982). Control theory and the dynamics of the firm: a survey, OR-Spektrum, 4, 1-14. [464] Levine, J. and Thepot, J. (1982). Open loop and closed loop equilibria in a dynamic duopoly, in Optimal Control Theory and Economic Analysis, G. Feichtinger (Ed.), North-Holland, Amsterdam, 143-156. [465] Lewis, T.R. and Schmalensee, R. (1977). Non-convexity and optimal exhaustion of renewable resources. International Economic Review, 18, 535-552. 452 Bibliography [466] Lewis, T.R. and Schmalensee, R. (1979). Non-convexity and optimal har vesting strategies for renewable resources, Canadian J. of Economics, 12, 677-691. [467] Lewis, T.R. and Schmalensee, R. (1982). Optimal use of renewable re sources with nonconvexities in production, in Essays in the Economics of Renewable Resources, L.J. Mirman and P.F. Spulber (Eds.), North- Holland, Amsterdam, 95-111. [468] Li, G. and Rajagopalan, S. (1997). A learning curve model with knowl edge depreciation, European Journal of Operational Research, 105, 1, 143-154. [469] Li, G. and Rajagopalan, S. (1998). Process improvement, quality and learning effects. Management Science, 4A, 1517-1532. [470] Lieber Z. (1973). An extension of Modigliani and Hohn's planning horizon results. Management Science, 20, 319-330. [471] Lieber Z. and Barnea, A. (1977). Dynamic optimal pricing to deter entry under constrained supply. Operations Research, 25, 696-705. [472] Lignell, J. and Tuominen, M.P.T. (1983). An advertising control model of two state variables, European Journal of Operations Research, 24, 1, 77-84. [473] Lintner, J. (1963). The cost of capital and optimal financing of corporate growth. Journal of Finance, 23, 292-310. [474] Lions, J.L. (1971). Optimal Control of Systems Governed by Partial Dif ferential Equations, Springer-Verlag, New York. [475] Little, J.D.C. (1979). Aggregate advertising models: the state of the art, Operations Res., 27, 4, 629-667. [476] Little, J.D.C. (1986). Comment on "Advertising pulsing policies..." by V. Mahajan and E. Muller, Marketing Science, 5, 2, 107-108. [477] Liu, P.-T. (1980). Dynamic Optimization and Mathematical Economics, Plenum Press, New York. [478] Liu, P.-T. and Roxin, E.G. (Eds.) (1979). Differential Games and Control Theory III, Marcel Dekker, Inc., New York. [479] Liu, P.-T. and Sutinen, J.G. (Eds.) (1979). Control Theory in Mathemat ical Economics, Marcel Dekker, Inc., New York. [480] Long, N.V. and Vousden, N. (1977). Optimal control theorems, in Appli cations of Control Theory in Economic Analysis, J.D. Pitchford and S.J. Turnovsky (Eds.), North-Holland, Amsterdam, 11-34. Bibliography 453 [481] Loon, P.J.J.M., van (1983). A Dynamic Theory of the Firm: Production, Finance and Investment, Lecture Notes in Economics and Mathematical Systems, Vol. 218, Springer-Verlag, Berlin. [482] Lou, S., Sethi, S.P. and Zhang, Q. (1994). Optimal feedback produc tion planning in a stochastic two-machine fiowshop, European Journal of Operational Research, 73, 331-345. [483] Lucas, Jr., R.E. (1971). Optimal management of a research and develop ment project. Management Science, 17, 679-697. [484] Lucas, Jr., R.E. (1981). Optimal investment with rational expectations, in Rational Expectations and Economic Practice, R.E. Lucas, Jr. and T.J. Sargent (Eds.), G. Allen & Unwin, London, 55-66. [485] Luenberger, D.G. (1969). Optimization by Vector Space Methods, Wiley, New York. [486] Luenberger, D.G. (1972). Mathematical programming and control the ory: trends of interplay, in Perspectives on Optimization, A.M. Geoffrion (Ed.), Addison-Wesley, Reading, Mass. [487] Luenberger, D.G. (1973). Introduction to Linear and Nonlinear Program- ming, Addison-Wesley, Reading, Mass. [488] Luenberger, D.G. (1975). A nonlinear economic control problem with a linear feedback solution, IEEE Transactions on Automatic Control, AC- 20, 184-191. [489] Luenberger, D.G. (1979). Introduction to Dynamic Systems: Theory, Models, and Applications, Wiley, New York. [490] Luhmer, A., Steindl, A., Feichtinger, G., Hartl, R.F. and Sorger, G. (1988). ADPULS in continuous time, European Journal of Operational Research, 34, 2, 171-177. [491] Lundin, R.A. and Morton, T.E. (1975). Planning horizons for the dy namic lot size model: protective procedures and computational results, Operations Research, 23, 711-734. [492] Luptacik, M. (1982). Optimal price and advertising policy under atom istic competition, Journal of Economic Dynamics and Control, 4, 57-71. [493] Luptacik, M. and Schubert, U. (1982). Optimal investment policy in pro ductive capacity and pollution abatement processes in a growing econ omy, in Optimal Control Theory and Economic Analysis, G. Feichtinger (Ed.), North-Holland, Amsterdam, 231-243. 454 Bibliography [494] Luus, R. (1993). Piecewise linear continuous optimal control by iterative dynamic programming, Industrial and Engineering Chemistry Research, 32, 859-865. [495] Macki, J. and Strauss, A. (1982). Introduction to Optimal Control Theory, Springer-Verlag, New York. [496] Magat, W.A., McCann, J.M. and Morey, R.C. (1986). When does lag structure really matter in optimizing advertising expenditures?, Man- agement Science, 32, 2, 182-193. [497] Magat, W.A., McCann, J.M. and Morey, R.C. (1988). Reply to when does lag structure really matter ... Indeed?, Management Science, 34, 7, 917-918. [498] Magill, M.J.P. (1970). On a General Economic Theory of Motion, Springer-Verlag, New York. [499] Mahajan, V. and Muller, E. (1979). Innovation diffusion and new product growth models in marketing. Journal Marketing, 43, 55-68. [500] Mahajan, V. and Muller, E. (1986). Advertising pulsing policies for gen erating awareness for new products. Marketing Science, 5, 2, 89-111. [501] Mahajan, V. and Peterson, R.A. (1978). Innovation diffusion in a dy namic potential adopter population. Management Science, 24, 1589- 1597. [502] Mahajan, V. and Peterson, R.A. (1985). Models for Innovation Diffusion, Sage, Beverly Hills. [503] Mahajan, V. and Wind, Y. (Eds.) (1986). Innovation Diffusion Models of New Product Acceptance, Ballinger, Cambridge, MA. [504] Maimon, O., Khmelnitsky, E. and Kogan, K. (1998). Optimal Flow Con trol in Manufacturing Systems, Kluwer Academic Publishers, Boston. [505] Majumdar, M. and Mitra, T. (1983). Dynamic optimization with a non- convex technology: the case of a linear objective function. Review of Economic Studies, 50, 143-151. [506] Malinowski, K. (1997). Sufficient optimality conditions for optimal con trol problems subject to state constraints, SIAM Journal on Control and Optimization, 35, 205-227. [507] Malliaris, A.G. and Brock, W.A. (1982). Stochastic Methods in Eco nomics and Finance, North-Holland, Elsevier Science Publishers, New York. Bibliography 455 [508] Mangasarian, O.L. (1966). Sufficient conditions for the optimal control of nonlinear systems, SI AM Journal on Control^ 4, 139-152. [509] Mangasarian, O.L. (1969). Nonlinear programming, McGraw-Hill, New York. [510] Mangasarian, O.L. and Fromovitz, S. (1967). A maximum principle in mathematical programming, in Mathematical Theory of Control, A.V. Balakrishman and L.W. Neustadt (Elds.), Academic Press, New York. [511] Manh-Hung, N. (1974). Essays on the optimal dynamic exploitation of natural resources and the social rate of discount, Ph.D. Dissertation, University of Toronto. [512] Martirena-Mantel, A.M. (1971). Optimal inventory and capital policy under certainty. Journal of Economic Theory, 3, 241-253. [513] Masse, P. (1962). Optimal Investment Decisions, Prentice- Hall, Engle- wood Cliffs, NJ. [514] Maurer, H. (1976). Numerical solution of singular control problems us ing multiple shooting techniques. Journal of Optimization Theory and Applications, 18, 235-257. [515] Maurer, H. (1977). On optimal control problems with bounded state variables and control appearing linearly, SIAM Journal on Control and Optimization, 15, 345-362. [516] Maurer, H. (1979). Differential stability in optimal control problems. Ap plied Mathematics & Optimization, 5, 283-295. [517] Maurer, H. (1981). First and second order sufficient optimality conditions in mathematical programming and optimal control. Math. Programming Stud., 14, 163-177. [518] Maurer, H. and Wiegand, M. (1992). Numerical solution of a drug dis placement problem with bounded state variables, Optimal Control Ap plications & Methods, 13, 43-55. [519] Maurer, H., Buskens, C. and Feichtinger, G. (1998). Solution techniques for periodic control problems: a case study in production planning. Op timal Control Applications & Methods, 19, 185-203. [520] Maurer, H. and Gillessen, W. (1975). Application of multiple shooting to the numerical solution of optimal control problems with bounded state variables. Computing, 15, 105-126. 456 Bibliography [521] Mayne, D.Q. and Polak, E. (1987). An exact penalty function algorithm for control problems with state and control constraints, IEEE Transac tions on Automatic Control, 32, 380-387. [522] Mclntyre, J., and Paiewonsky, B. (1967). On Optimal Control with Bounded State Variables, in Advances in Control Systems 5, C.T. Leon- des/indexLeondes, C.T. (Ed.), Academic Press, New York. [523] Mehra, R.K. (1975). An optimal control approach to national settlement system planning, RM-75-58, International Institute of Applied Systems Analysis, Laxenburg, Austria. [524] Mehra, R.K. and Davis, R.E. (1972). Generahzed gradient method for optimal control problems with inequality constraint and singular arcs, IEEE Transactions on Automatic Control, AC-17, 69-79. [525] Mehlmann, A. (1985). State transformations and the derivation of Nash closed-loop equilibria for non-zero-sum differential games. Applied Math ematical Modelling, 9, 353-357. [526] Mehlmann, A. (1988). Applied Differential Games, Plenum, New York. [527] Mehlmann, A. (1997). Wer Gewinnt Das Spiel? - Spieltheorie in Fabeln und Paradoxa, Vieweg, Braunschweig. [528] Mehrez, A. (1983). A note on the comparison of two different formulations of a risky R&D model, O. R. Letters, 2, 249-251. [529] Merton, R.C. (1969). Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Economics & Statistics, 51, 247-257. [530] Merton, R.C. (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3, 373-413. [531] Merton, R.C. (1973). An intertemporal capital asset pricing model, Econometrica, 5, 867-888. [532] Merton, R.C. (1982). On the microeconomic theory of investment under uncertainty, in Handbook of Mathematical Economics, K.J. Arrow and M.D. Intriligator (Eds.), Vol. II, North-Holland, Amsterdam, 601-669. [533] Mesak, H.I. (1985). On modeling advertising pulsing decisions. Decision Sciences, 16, 25-42. [534] Mesak, H.I. and Darrat, A.F. (1992). On comparing alternative advertis ing policies for pulsation. Decision Sciences, 23, 541-564. Bibliography 457 [535] Michel, P. (1981). Choice of projects and their starting dates: an exten sion of Pontryagin's maximum principle to a case which allows choice among different possible evolution equations, Journal of Economic Dy namics and Control, 3, 97-118. [536] Michel, P. (1982). On the transversality condition in infinite horizon op timal control problems, Econometrica, 50, 975-985. [537] Michel, P. (1985). Application of optimal control theory to disequilib rium analysis, in Optimal Control Theory and Economic Analysis 2, G. Feichtinger (Ed.), North-Holland, Amsterdam, 417-427. [538] Miele, A. (1962). Extremization of linear integral equations by Green's theorem, in Optimization Techniques, G. Leitmann (Ed.), Academic Press, New York. [539] Miller, M.H. and Modigliani, F. (1961). Dividend policy, growth, and valuation of shares, The Journal of Business, 34, 411-433. [540] Miller, R.E. (1979). Dynamic Optimization and Economic Applications, McGraw-Hill, New York. [541] Mirman, L.J. and Spulber, P.F. (Eds.) (1982). Essays in the Economics of Renewable Resources, North-Holland, Amsterdam. [542] Mirrlees, J. (1967). Optimum growth when technology is changing. Re view of Economic Studies, 34, 95-124. [543] Mirrlees, J. (1972. The optimum town, Swedish J. of Economics, 74, 114-135. [544] Modigliani, F. and Hohn, F. (1955). Production planning over time, Econometrica, 23, 46-66. [545] Moiseev, N.N. (1971). Numerical Methods in Optimal Control Theory, Nauka, Moscow. [546] Monahan, G.E. (1984). A pure birth model of optimal advertising with word-of-mouth, Marketing Sc, 3, 2, 169-178. [547] Mond, B. and Hanson, M. (1968). Duality for control problems, SI AM Journal on Control, 6, 114-120. [548] Morton, T.E., Mitchell, A. and Zemel, E. (1982). A discrete maximum principle approach to a general dynamic market response model, TIMS Studies in the Management Sciences, North-Holland, Amsterdam, 18, 117-140. 458 Bibliography [549] Morton, T.E. (1978). Universal planning horizons for generalized convex production scheduling, Operations Research, 26, 1046-1057. [550] Motta, M. and F. Rampazzo (1996). Dynamic programming for nonUnear systems driven by ordinary and impulsive controls, SIAM Journal on Control and Optimization, 34, 1, 199-225. [551] MuUer, E. (1983). Trial/awareness advertising decision: a control prob lem with phase diagrams with non-stationary boundaries, J. Economic Dynamics and Control, 6, 333-350. [552] Murata, Y. (1982). Optimal Control Methods for Linear Discrete-Time Economic Systems, Springer-Verlag, New York. [553] Murray, D.M. and Yakowitz, S.J. (1984). Differential dynamic program ming and Newton's methods for discrete optimal control problems, Jour nal of Optimization Theory and Applications, 43, 3, 395-414. [554] Muzicant, J. (1980). Systeme mit verteilten Parametern in der Biookonomie: Ein Maximumprinzip zur Kontrolle altersstrukturierter Modelle, Diss., Inst. f. Unternehmensforschung, Techn. University Wien. [555] Nahorski, Z., Ravn, H.F. and Vidal, R.V.V. (1984). The discrete-time maximum principle: a survey and some new results. International J. of Control, 40, 533-554. [556] Naik, RA., Mantrala, M.K. and Sawyer, A. (1998). Planning pulsing media schedules in the presence of dynamic advertising quality. Marketing Science, 17, 3, 214-235. [557] Naslund, B. (1966). Simultaneous determination of optimal repair policy and service life. The Swedish Journal of Economics, 68, 63-73. [558] Naslund, B. (1969). Optimal rotation and thinning. Forest Science, 15, 466-451. [559] Naslund, B. (1979). Consumer behavior and optimal advertising, J. of the Operational Research Society, 20, 237-243. [560] Neck, R. (1982). Dynamic systems with several decision-makers, in Oper ations Research in Progress, G. Feichtinger and P. Kail (Eds.), D. Reidel Publishing Company, Dordrecht, Holland, 261-284. [561] Neck, R. (1984). Stochastic control theory and operational research, Eu ropean Journal of Operational Research, 17, 283-301. [562] Nelson, R.T. (1960). Labor assignment as a dynamic control problem, Operations Research, 14, 369-376. Bibliography 459 [563] Nepomiastchy, P. (1970). Application of optimal control theory and penalty function techniques to solution of a particular scheduling prob lem, Zhurnal VychifliteVnoi Matemtiki i Matematichefkoi Fiziki (in Rus sian), 10, No. 4. [564] Nerlove, M. and Arrow, K.J. (1962). Optimal advertising policy under dynamic conditions, Economica, 29, 129-142. [565] Neustadt, L.W. (1976). Optimization: A Theory of Necessary Conditions, Princeton University Press, Princeton, NJ. [566] Nguyen, D. (1997). Marketing Decisions under Uncertainty, Kluwer Aca demic Publishers, Boston. [567] Norstrom, C.J. (1978). The continuous wheat trading model reconsid ered. An application of mathematical control theory with a state con straint, W.P. 58-77-78, GSIA, Carnegie Mellon University, Pittsburgh, Pa. [568] Novak, A. and Feichtinger, G. (1993). Optimal treatment of cancer dis eases, International Journal of Systems Science, 24, 1253-1263. [569] Oberle, H.J. (1979). Numerical computation of singular control problems with application to optimal heating and cooling by solar energy, AppL Math. & Optimiz., 5, 297-314. [570] Oberle, H.J. (1986). Numerical solution of minimax optimal control prob lems by multiple shooting technique. Journal of Optimization Theory and Applications, 50, 331-358. [571] Oberle, H.J. and Grimm, W. (1989). BNDSCO: A Program for the Nu merical Solution of Optimal Control Problems, Report 515, Institut for Flight Systems Dynamics, German Aerospace Research Establishment DLR, OberpfafFenhofen, Germany. [572] Oberle, H.J. and Sothmann, B.(1999). Numerical computation of optimal feed rates for a fed-batch fermentation model. Journal of Optimization Theory and Applications, 100, 1, 1-13. [573] Oguztoreh, M.N. and Stein, R.B. (1983). Optimal control or antagonistic muscles, Biol. Cybern., 48, 91-99. [574] Oksendal, B.K. (1998). Stochastic Differential Equations: An Introduc tion With Applications, Springer-Verlag, New York. [575] Olsder, G.J. (1976). Some thoughts about simple advertising models as dijBFerential games and the structure of coalitions, in Directions in Large- Scale Systems, Many-Person Optimization and Decentralized Control, Y.- C. Ho and S.K. Mitter (Eds.), Plenum Press, New York, 187-205. 460 Bibliography [576] Oniki, H. (1973). comparative dynamics (sensitivity analysis) in optimal control theory, Journal of Economic Theory, 6, 265-283. [577] Oren, S.S. and Powell, S.G. (1985). Optimal supply of a depletable re source with a backstop technology. Operations Research, 33, 277-292. [578] Osayimwese, I. (1974). Rural-urban migration and control theory. Geo graphical Analysis, 4, 147-161. [579] Ozga, S. (1960). Imperfect markets through lack of knowledge. Quarterly Journal of Economics, 74, 29-52. [580] Palda, K.S. (1964). The Measurement of Cumulative Advertising Effects, Prentice-Hall, Englewood Cliffs, NJ. [581] Pantoja, J.F. and Mayne, D.Q. (1991). Sequential quadratic program ming algorithm for discrete optimal control problems with control in equality constraints. International Journal on Control, 53, 4, 823-836. [582] Parlar, M. (1983). Optimal forest fire control with limited reinforcements, Optimal Control Application & Methods, 4, 185-191. [583] Parlar, M. (1984). Optimal dynamic service rate control in time depen dent M/M/S/N queues, International Journal of Systems Science, 15, 107-118. [584] Parlar, M. (1986). A problem in jointly optimal production and advertis ing decisions, International Journal Systems Science, 17, 9, 1373-1380. [585] Parlar, M. and Vickson, R.G. (1980). An optimal control problem with piecewise quadratic cost functional containing a 'dead zone'. Optimal Control Application & Methods, 1, 361-372. [586] Parlar, M. and Vickson, R.G. (1982). Optimal forest fire control: an extension of Park's model. Forest Science, 28, 345-355. [587] Pauwels, W. (1977). Optimal dynamic advertising policies in the presence of continuously distributed time lags. Journal of Optimization Theory and Applications, 22, 1, 79-89. [588] Pekelman, D. (1974). Simultaneous price-production decision, Operations Research, 22, 788-794. [589] Pekelman, D. (1975). Production smoothing with fluctuating price. Man agement Science, 21, 576-590. [590] Pekelman D. (1979). On optimal utihzation of production processes. Op erations Research, 27, 260-278. Bibliography 461 [591] Pekelman D. and Rausser, G.C. (1978). Adaptive control: survey of methods and applications, in Applied Optimal Control, TIMS Studies in Management Sciences, A. Bensoussan et al. (Eds.), Vol. 9, North- Holland, Amsterdam, 89-120. [592] Pekelman, D. and Sethi, S.P. (1978). Advertising budgeting, wearout and copy replacement. Journal of the Operational Research Society, 29, 651-659. [593] Pepyne, D.L. and Cassandras, C.G. (1999). Performance optimization of a class of discrete event dynamic systems using calculus of variations techniques. Journal of Optimization Theory and Applications, 100, 3, 599-622. [594] Pesch, H.J. (1989). Real-time computation of feedback controls for con strained optimal control problems, part 1: neighboring extremals. Opti mal Control Applications & Methods, 10, 2, 129-145. [595] Pesch, H.J. (1989). Real-time computation of feedback controls for con strained optimal control problems, part 2: a correction method based on multiple shooting. Optimal Control Applications & Methods, 10, 2, 147-171. [596] Pesch, H.J. (1994). A practical guide to the solution of real-life optimal control problems. Control and Cybernetics, 23, 1/2, 7-60. [597] Pesch, H.J. and Bulirsch, R. (1994). The maximum principle. Bellman's equation, and Canatheodory's Work, Journal of Optimization Theory and Applications, 80, 1, 203-229. [598] Peterson, D.W. (1973). The economic significance of auxiliary functions in optimal control. International Economic Review, 14, 234-252. [599] Peterson, D.W. and Zalkin, J.H. (1978). A review of direct sufficient conditions in optimal control theory. International Journal Control, 28, 589-610. [600] Peterson, P.M. and Fisher, A.C. (1977). The exploitation of extractive resources: a survey. Economic Journal, 87, 681-721. [601] Petrov, lu.P. (1968). Variational Methods in Optimum Control Theory, Academic Press, New York. [602] Pierskalla, W.P. and Voelker, J.A. (1976). Survey of maintenance mod els: the control and surveillance of deteriorating systems. Naval Research Logistics Quarterly 23, 353-388. [603] Pindyck, R.S. (Ed.) (1978a). Advances in the Economics of Energy and Resources, II, J.A.I. Press, Greenwich, Conn. 462 Bibliography [604] Pindyck, R.S. (1978b). The optimal exploration and production of non renewable resources, Journal of Political Economy, 86, 841-862. [605] Pindyck, R.S. (1978c). Gains to producers from the cartelization of ex haustible resources. Rev. Economics & Statistics, 60, 238-251. [606] Pindyck, R.S. (1978b). The optimal exploration and production of non renewable resources. Journal of Political Economy, 86, 841-862. [607] Pindyck, R.S. (1982). Adjustment costs, uncertainty, and the behavior of the firm, American Economic Review, 72, 3, 415-427. [608] Pitchford, J.D. and Turnovsky, S.J. (Eds.) (1977). Applications of Control Theory to Economic Analysis, North-Holland, Amsterdam. [609] Pohjola, M. (1984). Threats and bargaining in capitalism: a differential game view. Journal of Economic Dynamics and Control, 8, 291-302. [610] Polak, E. (1971). Computational Methods in Optimization, Academic Press, New York. [611] Polak, E. (1973). A historical survey of computational methods in optimal control, SIAM Review, 15, 553-584. [612] Polak, E., Yang, T.H. and Mayne, D.Q. (1993). A method of centers based on barrier functions for solving optimal control problems with continuous state and control constraints, SIAM Journal on Control and Optimization, 31, 159-179. [613] Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V. and Mischenko, E.F. (1962). The mathematical theory of optimal processes, Wiley, New York. [614] Presman, E., Sethi, S.P. and Suo, W. (1997a). Optimal feedback produc tion planning in stochastic dynamic jobshops, in Mathematics of Stochas tic Manufacturing Systems, G. Yin and Q. Zhang (Eds.); Lectures in Ap plied Mathematics, Vol. 33, American Mathematical Society, Providence, RI, 235-252. [615] Presman, E., Sethi, S.P. and Suo, W. (1997b). Existence of optimal feed back production plans in stochastic iV-machine flowshops with limited buffers, Automatica, 33, 4, 1899-1903. [616] Presman, E., Sethi, S.P., Zhang, H. and Zhang, Q. (1998a). Analysis of average cost optimality for an unreliable two-machine flowshop. Proceed ings of the Fourth International Conference on Optimization Techniques and Applications, Curtin University of Technology, Perth, Australia, 94- 112. Bibliography 463 [617] Presman, E., Sethi, S.R, Zhang, H. and Zhang, Q. (1998b). Optimahty of zero-inventory poHcies for an unreliable manufacturing system pro ducing two part types, Dynamics of Continuous, Discrete and Impulsive Systems, 4, 4, 485-496. [618] Presman, E., Sethi, S.P. and Zhang, Q. (1995). Optimal feedback pro duction planning in a stochastic TV-machine flowshop, Automatica, 31, 9, 1325-1332. [619] Prskawetz, A., Feichtinger, G. and Luptacik, M. (1998). The accomplish ment of the Maastricht criteria with respect to initial debt. Journal of Economics, 68, 93-110. [620] Pytlak, R., and Vinter, R.B. (1993). PH2S0L Solver: An 0(N) Imple mentation of an Optimization Algorithm for a General Optimal Control Problem, Research Report C93-36, Centre for Process Systems Engineer ing, Imperial College, London, United Kingdom. [621] Pytlak, R., and Vinter, R.B. (1999). Feasible direction algorithm for opti mal control problems with state and control constraints: implementation, Journal of Optimization Theory and Applications, 101, 3, 623-649. [622] Raman, K. (1990). Stochastically optimal advertising policies under dy namic conditions: the ratio rule, Optimal Control Applications & Meth ods, 11, 283-288. [623] Raman, K. and Chatterjee, R. (1995). Optimal monopolist pricing under demand uncertainty in dynamic markets, Management Science, 41, 1, 144-162. [624] Ramsey, F.P. (1928). A mathematical theory of savings. Economic Jour nal, 38, 543-559. [625] Rao, R.C. (1970). Quantitative Theories in Advertising, Wiley, New York. [626] Rao, R.C. (1984). Advertising decisions in oligopoly: an industry equi librium analysis, Optimal Control Applications & Methods, 5, 4, 331-344. [627] Rao, R.C. (1985). A note on optimal and near optimal price and adver tising strategies. Management Science, 31, 3, 376-377. [628] Rao, R.C. (1986). Estimating continuous time advertising-sales models, Marketing Science, 5, 2, 125-142. [629] Rao, R.C. (1990). Impact of competition on strategic marketing decisions, in Interface of Marketing and Strategy, G. Day, B. Weitz and R. Wens ley (Eds.), JAI Press, Greenwich, CT. 464 Bibliography [630] Rapoport, A. (1980). Mathematische Methoden in den Sozialwis- senschaften^ Physica-Verlag, Wiirzburg. [631] Rapp, B. (1974). Models for Optimal Investment and Maintenance Deci sions, Almqvist & Wiksell, Stockholm; Wiley, New York. [632] Rausser, G.C. and Hochman, E. (1979). Dynamic Agricultural systems: Economic Prediction and Control, North-Holland, New York. [633] Raviv, A. (1979). The design of an optimal insurance policy. The Amer ican Economic Review, 69, 84-96. [634] Ray, A. and Blaquiere, A. (1981). Sufficient conditions for optimality of threat strategies in a differential game. Journal of Optimization Theory and Applications, 33, 99-109, [635] Reinganum, J.F. (1982). A dynamic game of R & D: patent protection and competitive behavior, Econometrica, 50, 671-688. [636] Rempala, R. (1980). On the multicommodity Arrow-Karlin inventory model, in Proceedings First International Symposium on Inventories, Hungarian Academy of Science, Budapest. [637] Rempala, R. (1982). On the multicommodity Arrow-Karlin inventory model, part H: horizon and horizontal solution, in Proceedings Second International Symposium on Inventories, Hungarian Academy of Science, Budapest. [638] Rempala, R. (1986). Horizon for the dynamic family of wheat trading problems. Proceedings Fourth International Symposium on Inventories, Budapest. [639] Rempala, R. and Sethi, S.R (1988). Forecast horizons in single product inventory models, in Optimal Control Theory and Economic Analysis 3, G. Feichtinger (Ed.), North-Holland, Amsterdam, 225-233. [640] Rempala, R. and Sethi, S.R (1992). Decision and forecast horizons for one-dimensional optimal control problems: existence results and appli cations, Optimal Control Applications & Methods, 13, 179-192. [641] Richard, S.F. (1979). A generalized capital asset pricing model, TIMS Studies in the Management Sciences, 11, 215-232. [642] Ringbeck, J. (1985). Mixed quality and advertising strategies under asymmetric information, in Optimal Control Theory and Economic Anal ysis, G. Feichtinger (Ed.), Vol. 2, North-Holland, Amsterdam, 197-214. Bibliography 465 [643] Rishel, R.W. (1965). An extended Pontryagin principle for control sys tems whose control laws contain measures, Journal of Soc. Industrial & AppL Math. Control, 3, 191-205. [644] Rishel, R.W. (1985). A partially observed advertising model, in Optimal Control Theory and Economic Analysis, G. Feichtinger (Ed.), 2, 253-262. [645] Roberts, S.M. and Shipman, J.S. (1972). Two-Point Boundary Value Problems: Shooting Methods, Elsevier, New York. [646] Robinson, B. and Lakhani, C. (1975). Dynamic price models for new- product planning. Management Science, 21, 10, 1113-1122. [647] Robson, A.J. (1981). Sufficiency of the Pontryagin conditions for optimal control when the time horizon is free. Journal of Economic Theory, 24, 438-445. [648] Robson, A.J. (1985). Optimal control of systems governed by partial dif ferential equations: economic applications, in Optimal Control Theory and Economic Analysis 2, G. Feichtinger (Ed.), North-Holland, Amster dam, 105-118. [649] Rockafeller, R.T. (1970). Convex Analysis, Princeton University Press, Princeton, NJ. [650] Rockafeller, R.T. (1972). State constraints in convex control problems of Bolza, SIAM Journal on Control, 19, 691-715. [651] Russak, B. (1976). Relations among the multipliers for problems with bounded state constraints, SIAM Journal on Control and Optimization, 14, 1151-1155. [652] Russell, D.L. (1965). Penalty functions and bounded phase coordinates, SIAM Journal on Control, 2, 409-422. [653] Sage, A.P. (1968). Optimum Systems Control, Prentice-Hall, Englewood Cliffs, NJ. [654] Salukvadze, M.E. (1979). Vector-Valued Optimization Problems in Con trol Theory, Academic Press, New York. [655] Samuelson, P.A. (1972). The general saddle point property of optimal- control motions. Journal of Economic Theory, 5, 102-120. [656] Sarma, V.V.S. and Alam, M. (1975). Optimal maintenance policies for machines subject to deterioration and intermittent breakdowns, IEEE Transactions on Systems, Man, and Cybernetics, SMC-5, 396-398. 466 Bibliography [657] Samaratunga, C, Sethi, S.P. and Zhou, X. (1997). Computational eval uation of hierarchical production policies for stochastic manufacturing systems. Operations Research, 45, 2, 258-274. [658] Sasieni, M. (1989). Optimal advertising strategies. Marketing Science, 8, 358-370. [659] Schaefer, M.B. (1957). Some considerations of population dynamics and economics in relation to the management of marine fisheries. Journal of Fisheries Research Board of Canada, 14, 669-681. [660] Scalzo, R.C. (1974). N person linear quadratic differential games with constraints, SIAM Journal on Control, 12, 419-425. [661] Schijndel, G.-J.C.Th.,van (1986). Dynamic shareholder behaviour under personal taxation: a note, in Operations Research Proceedings 1985, L. Streitferdt et al. (Eds.), Springer-Verlag, Berhn, 488-495. [662] Schilling, K. (1985). On optimization principles in plant ecology, in Dy- namics of Macrosystems, Lecture Notes in Economics and Mathematical Systems, J.-P. Aubin et al. (Eds.), Vol. 257, Springer-Verlag, Berlin, 63-71. [663] Seidman, T.L, Sethi, S.P. and Derzko, N.A. (1987). Dynamics and opti mization of a sales advertising model with population migration. Journal of Optimization Theory and Applications, 52, 3, 443-462. [664] Seierstad, A. (1982). Differentiability properties of the optimal value function in control theory. Journal of Economic Dynamics and Control, 4, 303-310. [665] Seierstad, A. (1985). Existence of an optimal control with sparse jumps in the state variable. Journal of Optimization Theory and Applications, 45, 265-293. [666] Seierstad, A. and Sydsseter, K. (1977). Sufficient conditions in optimal control theory. International Economic Review, 18, 367-391. [667] Seierstad, A. and Sydsaeter, K. (1983). Sufficient conditions applied to an optimal control problem of resource management. Journal of Economic Theory, 31, 375-382. [668] Seierstad, A. and Sydsaeter, K. (1987). Optimal Control Theory with Eco nomic Applications, North-Holland, Amsterdam. [669] Selten, R. (1975). Reexamination of the perfectness concept for equilib rium points in extensive games. International Journal of Game Theory, 4, 25-55. Bibliography 467 [670] Sethi, S.R (1973a). Optimal control of the Vidale-Wolfe advertising model, Operations Research, 21, 998-1013. [671] Sethi, S.P. (1973b). Simultaneous optimization of preventive maintenance and replacement policy for machines: a modern control theory approach, AIIE Transactions, 5, 2, 156-163. [672] Sethi, S.P. (1973c). An application of optimal control theory in forest management, Journal of Management Science and Applied Cybernetics (New Delhi), 2, 9-16. [673] Sethi, S.P. (1973d). A note on modeling simple dynamic cash balance problems. Journal of Financial and Quantitative Analysis, 8, 685-687. [674] Sethi, S.P. (1974a). Sufficient conditions for the optimal control of a class of systems with continuous lags. Journal of Optimization Theory and Applications, 13, 542-552. [675] Sethi, S.P. (1974b). Some explanatory remarks on the optimal control of the Vidale-Wolfe advertising model. Operations Research, 22, 1119-1120. [676] Sethi, S.P. (1974c). Quantitative guidelines for communicable disease control program: a complete synthesis. Biometrics, 30, 681-691. [677] Sethi, S.P. (1975). Optimal control of a logarithmic advertising model, Operations Research Quarterly, 26, 317-319. [678] Sethi, S.P. (1977a). Dynamic optimal control models in advertising: a survey, SIAM Review, 19, 4, 685-725. [679] Sethi, S.P. (1977b). Nearest feasible paths in optimal control problems: theory, examples, and counterexamples. Journal of Optimization Theory and Applications, 23, 4, 563-579. [680] Sethi, S.P. (1977c). Optimal advertising for the Nerlove-Arrow model under a budget constraint. Operational Research Quarterly, 28, 3, 683- 693. [681] Sethi, S.P. (1977d). A linear bang-bang model of firm behavior and water quality, IEEE Transactions on Automatic Control, AC-22, 706-714. [682] Sethi, S.P. (1978a). A survey of management science applications of the deterministic maximum principle, TIMS Studies in the Management Sci ences, 9, 33-67. [683] Sethi, S.P. (1978b). Optimal equity financing model of Krouse and Lee: corrections and extensions. Journal of Financial and Quantitative Anal ysis, 13, 3, 487-505. 468 Bibliography [684] Sethi, S.P. (1978c). A note on modeling simple dynamic cash balance problem: errata, Journal of Financial and Quantitative Analysis, 13, 585-586. [685] Sethi, S.P. (1978d). Optimal quarantine programs for controlling an epi demic spread, Journal of Operational Research Society, 29, 3, 265-268. [686] Sethi, S.P. (1979a). Optimal depletion of exhaustible resources. Applied Mathematical Modelling, 3, 367-378. [687] Sethi, S.P. (1979b). Optimal pilfering policy for dynamic continuous thieves, Management Science, 25, 6, 535-542. [688] Sethi, S.P. (1979c). Optimal advertising policy with the contagion model, Journal of Optimization Theory and Applications, 29, 4, 615-627. [689] Sethi, S.P. (1979d). A note on the Nerlove-Arrow model under uncer tainty, Operations Research, 27, 4, 839-842; Erratum, 28, 4, July-August 1980, 1026-1027. [690] Sethi, S.P. (1983a). Optimal long-run equilibrium advertising level for the Blattberg-Jeuland model. Management Science, 29, 12, 1436-1443. [691] Sethi, S.P. (1983b). Deterministic and stochastic optimization of a dy namic advertising model. Optimal Control Application and Methods, 4, 2, 179-184. [692] Sethi, S.P. (1983c). Applications of optimal control to management sci ence problems, Proceedings of the 1983 World Conference on Systems, Caracas, Venezuela. [693] Sethi, S.P. (1984). Application of the maximum principle to production and inventory problems, Proceedings Third International Symposium On Inventories, Budapest, 753-756. [694] Sethi, S.P. (1990). Decision and forecast horizons in dynamic optimiza tion, Systems and Control Encyclopedia Supplementary, M.G. Singh (Ed.), Pergamon Press, Oxford, U.K., Vol. 1, 192-198. [695] Sethi, S.P. (1996). When does the share price equal the present value of future dividends? - a modified dividend approach. Economic Theory, 8, 307-319. [696] Sethi, S.P. (1997a). Optimal Consumption and Investment with Bankruptcy, Kluwer Academic Publishers, Boston. [697] Sethi, S.P. (1997b). Some insights into near-optimal plans for stochastic manufacturing systems, Mathematics of Stochastic Manufacturing Sys tems, G. Yin and Q. Zhang (Eds.); Lectures in Applied Mathematics, Vol. 33, American Mathematical Society, Providence, RI, 287-315. Bibliography 469 [698] Sethi, S.P. (1998). Optimal consumption - investment decisions allow ing for bankruptcy: a survey, in Worldwide Asset and Liability Model ing, W.T. Ziemba and J.M. Mulvey (Eds.), Cambridge University Press, Cambridge, U.K., 387-426 [699] Sethi, S.P. and Chand, S. (1979). Planning horizon procedures in machine replacement models. Management Science, 25, 140-151; Erratum, 26, 3, 1980, 342. [700] Sethi, S.P. and Chand, S. (1981). Multiple finite production rate dynamic lot size inventory models. Operations Research, 29, 5, 931-944. [701] Sethi, S.P., Derzko, N.A. and Lehoczky, J.P. (1982). Mathematical anal ysis of the Miller-Modigliani theory, O.R. Letters, 1, 148-152. [702] Sethi, S.P. and Lee, S.C. (1981). Optimal advertising for the Nerlove- Arrow Model under a replenishable budget. Optimal Control Applications & Methods, 2, 2, 165-173. [703] Sethi, S.P. and Lehoczky, J.P. (1981). A comparison of Ito and Stratonovich formulations of problems in finance. Journal of Economic Dynamics and Control, 3, 343-356. [704] Sethi, S.P. and McGuire, T.W. (1977). Optimal skill mix: an application of the maximum principle for systems with retarded controls. Journal of Optimization Theory and Applications, 23, 245-275. [705] Sethi, S.P. and Morton, T.E. (1972). A mixed optimization technique for the generalized machine replacement problem. Naval Research Logistics QuaHerly, 19, 471-481. [706] Sethi, S.P. and Sorger, G. (1989). Concepts of forecast horizons in stochastic dynamic games. Proceedings of the 28th Conference on De cision and Control, Tampa, Florida, 195-197. [707] Sethi, S.P. and Sorger, G. (1990). An exercise in modeling of consump tion, import, and export of an exhaustible resource. Optimal Control Applications & Methods, 11, 191-196. [708] Sethi, S.P. and Sorger, G. (1991). A theory of rolling horizon decision making. Annals of O.R., 29, 387-416. [709] Sethi, S.P. and Staats, P.W. (1978). Optimal control of some simple deterministic epidemic models. Journal of Operational Research Society, 29, 2, 129-136. [710] Sethi, S.P., Suo, W., Taksar, M.L and Yan, H. (1998). Optimal produc tion planning in a multiproduct stochastic manufacturing systems with long-run average cost. Discrete Event Dynamic Systems, 8, 1, 37-54. 470 Bibliography [711] Sethi, S.R, Suo, W., Taksar, M.I. and Zhang, Q. (1997). Optimal produc tion planning in a stochastic manufacturing system with long-run average cost, Journal of Optimization Theory and Applications, 92, 1, 161-188. [712] Sethi, S.P. and Taksar, M.I. (1988). Deterministic and stochastic control problems with identical optimal cost functions, Analysis and Optimiza tion of Systems, A. Bensoussan and J.L. Lions (Eds.), Lecture Notes in Control and Information Sciences, Springer-Verlag, New York, 641-645. [713] Sethi, S.P. and Taksar, M.I. (1990). Deterministic equivalent for a continuous-time linear-convex stochastic control problem. Journal of Op timization Theory and Applications, 64, 1, 169-181. [714] Sethi, S.P. and Thompson, G.L. (1970). Applications of mathematical control theory to finance. Journal of Financial and Quantitative Analysis, 5, 4 fe 5, 381-394. [715] Sethi, S.P. and Thompson, G.L. (1977). Christmas toy manufacturer's problem: an application of the stochastic maximum principle, Opsearch, 14, 161-173. [716] Sethi, S.P. and Thompson, G.L. (1981). Simple models in stochastic production planning, in Applied Stochastic Control in Econometrics and Management Science, A. Bensoussan, P.R. Kleindorfer and C.S. Tapiero (Eds), North-Holland, Amsterdam, 295-304. [717] Sethi, S.P. and Thompson, G.L. (1981). A tutorial on optimal control theory. Information Systems and Operational Research, 19, 4, 279-291. [718] Sethi, S.P. and Thompson, G.L. (1982). Planning and forecast horizons in a simple wheat trading model. Operations Research in Progress, G. Fe- ichtinger and P. Kail (Eds.), D. Reidel Publishing Company, Dordrecht, Holland, 203-214. [719] Sethi, S.P., Thompson, G.L. and Udayabhanu, V. (1985). Profit max imization models for exponential decay processes, European Journal of Operational Research, 22, 101-115. [720] Sethi, S.P. and Zhang, H. (1999a). Average-cost optimal policies for an unreliable flexible multiproduct machine, I.J.F.M.S., 11, 147-157. [721] Sethi, S.P. and Zhang, H. (1999b). Hierarchical production controls for a stochastic manufacturing system with long-run average cost: asymptotic optimality, in Stochastic Analysis, Control, Optimization and Applica tions: A Volume in Honor of W.H. Fleming, W.M. McEneaney, G. Yin and Q. Zhang (Eds.), Birkhauser, Boston, 621-637. Bibliography 471 [722] Sethi, S.R, Zhang, H. and Zhang, Q. (1997). Hierarchical production control in a stochastic manufacturing system with long-run average cost, Journal of Mathematical Analysis and Applications, 214, 151-172. [723] Sethi, S.R, Zhang, H. and Zhang, Q. (1998). Minimum average-cost production planning in stochastic manufacturing systems. Mathematical Models and Methods in Applied Sciences, 8, 7, 1251-1276. [724] Sethi, S.R and Zhang, Q. (1994a). Hierarchical Decision Making in Stochastic Manufacturing Systems, in series Systems and Control: Foun dations and Applications, Birkhauser, Boston. [725] Sethi, S.P. and Zhang, Q. (1994b). Hierarchical controls in stochastic manufacturing systems, SIAG/CST Newsletter, 2, 1, 1-5. [726] Sethi, S.R and Zhang, Q. (1994c). Asymptotic optimal controls in stochastic manufacturing systems with machine failures dependent on production rates, Stochastics and Stochastics Reports, 48, 97-121. [727] Sethi, S.P. and Zhang, Q. (1995a). Hierarchical production and setup scheduling in stochastic manufacturing systems, IEEE Transactions on Automatic Control, 40, 5, 924-930. [728] Sethi, S.P. and Zhang, Q. (1995b). Multilevel hierarchical decision mak ing in stochastic marketing - production systems, SIAM Journal on Con trol and Optimization, 33, 2, 538-553. [729] Sethi, S.P. and Zhang, Q. (1998a). Asymptotic optimality of hierarchical controls in stochastic manufacturing systems: a review, Operations Re search: Methods, Models, and Applications, J.E. Aronson and S. Zionts (Eds.), Quoram Books, Westport, CT, 267-294. [730] Sethi, S.P. and Zhang, Q. (1998b). Optimal control of dynamic systems by decomposition and aggregation, Journal of Optimization Theory and Applications, 99, 1, 1-22. [731] Sethi, S.R, Zhang, Q. and Zhou, X. (1997). Hierarchical production con trols in a stochastic two-machine flowshop with a finite internal buffer, IEEE Transactions on Robotics and Automation, 13, 1, 1-13. [732] Sethi, S.P. and Zhou, X. (1996a). Optimal feedback controls in determin istic dynamic two-machine flowshops, O. R. Letters, 19, 5, 225-35. [733] Sethi, S.R and Zhou, X. (1996b). Asymptotic optimal feedback con trols in stochastic dynamic two-machine flowshops, in Recent Advances in Control and Optimization of Manufacturing Systems, Lecture Notes in Control and Information Sciences, Vol. 214, G. Yin and Q. Zhang (Eds.), Springer-Verlag, New York, 147-180. 472 Bibliography [734] Shapiro, A. (1997). On uniqueness of Lagrange multipliers in optimiza tion problems subject to cone constraints, SI AM Journal on Optimiza tion, 7, 2, 508-518. [735] Shapiro, C. (1982). Consumer information, product quality, and seller reputation. Bell Journal of Economics, 13, 20-25. [736] Shell, K. (Ed.) (1967). Essays on the Theory of Optimal Economic Growth, The MIT Press, Cambridge, MA. [737] Shell, K. (1969). Applications of Pontryagin's maximum principle to eco nomics, in Mathematical Systems Theory and Economics, H.W. Kuhn and G.P. Szego (Eds.), Springer-Verlag, Berlin. [738] Siebert, H. (1985). Economics of the Resource-Exporting Country: In tertemporal Theory of Supply and Trade, JAI Press, Greenwich, Conn. [739] Silva, G.N. and Vinter, R.B. (1997). Necessary conditions for optimal impulsive control problems, SI AM Journal on Control and Optimization, 35, 6, 1829-1846. [740] Simaan, M. and Cruz, J.B., Jr. (1975). Formulation of Richardson's model of arms race from a differential game viewpoint. Review of Eco nomic Studies, 42, 67-77. [741] Simon, H.A. (1982). ADPULS: an advertising model with wearout and pulsation. Journal of Marketing Research, 19, 352-363. [742] Simon, H.A. (1956). Dynamic programming under uncertainty with a quadratic criterion function, Econometrica, 24, 74-81. [743] Singh, M.G. (1980). Dynamical Hierarchical Control, North-Holland, Amsterdam. [744] Singh, M.G., Tith, A. and Mahnowski, K. (1985). Decentralized control design: an overview, Large Scale Systems, 9, 215-230. [745] Skiba, A.K. (1978). Optimal growth with a convex-concave production function, Econometrica, 46, 527-539. [746] Smith, V.L. (1972). Dynamics of waste accumulation: disposal versus recycling. Quarterly Journal of Economics, 86, 600-616. [747] Snower, D.J. (1982). Macroeconomic policy and the optimal destruction of vampires. Journal of Political Economy, 90, 647-655. [748] Solow, R.M. (1974). Intergenerational equity and exhaustible resources, Review of Economic Studies, 41, 29-45. Bibliography 473 [749] Solow, R.M. and Wan, F.Y. (1976). Extraction costs in the theory of exhaustible resources, Bell Journal of Economics, 7, 359-370. [750] Sorenson, H. (1966). Kalman filtering, in Advance in Control Systems, C.T. Leondes (Ed.), Vol. 3, Academic Press, New York, 219-292. [751] Sorger, G. (1989). Competitive dynamic advertising: a modification of the case game. Journal of Economic Dynamics and Control, 13, 55-80. [752] Southwick, L. and Zionts, S. (1974). An optimal-control-theory approach to the education-investment decision. Operations Research, 22, 1156- 1174. [753] Spence, M. (1981). The learning curve and competition. Bell Journal of Economics, 12, 49-70. [754] Spiegel, M.R. (1971). Schaum's Outline of Theory and Problems of Cal culus of Finite Differences and Difference Equations, McGraw-Hill, New York. [755] Spremann, K. (1985). The signaling of quality by reputation, in Optimal Control Theory and Econ. Analysis, G. Feichtinger (Ed.), Vol. 2, North- Holland, Amsterdam, 235-252. [756] Sprzeuzkouski, A.Y. (1967). A problem in optimal stock management, Journal of Optimization Theory and Applications, 1, 232-241. [757] Srinivasan, V. (1976). Decomposition of a multi-period media scheduling model in terms of single period equivalents. Management Science, 23, 349-360. [758] Stalford, H. and Leitmann, G. (1973). Sufficiency conditions for Nash equilibria in N-person differential games, in Topics in Differential Games, A. Blaquiere (Ed.), North-Holland, Amsterdam, 345-376. [759] Starr, A.W. and Ho, Y.-C. (1969). Nonzero-sum differential games. Jour nal of Optimization Theory and Applications, 3, 184-206. [760] Steindl, A., Feichtinger, G., Hartl, R.F. and Sorger, G. (1986). On the optimality of cyclical employment policies: a numerical investigation, Journal of Economic Dynamics and Control, 10, 457-466. [761] Stepan, A. (1977). An application of the discrete maximum principle to designing ultradeep drills - the casing optimization. International Journal of Production Research, 15, 315-327. [762] Stern, L.E. (1984). Criteria of optimality in the infinite-time optimal control problem. Journal of Optimization Theory and Applications, 44, 497-508. 474 Bibliography [763] Stiglitz, J.E. and Dasgupta, P. (1982). Market structure and the resource depletion: a contribution to the theory of intertemporal monopoHstic competition, Journal of Economic Theory^ 28, 128-164. [764] Stoer, J. and Bulirsch, R. (1978). Einfuhrung in die Numerische Mathe- matik 11^ Heidelberger Taschenbiicher, 114, Springer-Verlag, Berlin. [765] Stoppler, S. (1975). Dynamische Produktionstheorie, Westdeutscher- Verlag, Opladen, West Germany. [766] Stoppler, S. (1985). Der Einflufider Lagerkosten auf die Produktionsan- passung bei zyklischem Absatz - Eine kontroUtheoretische Analyse, OR- Spektrum, 7, 129-142. [767] Swan, G.W. (1984). Applications of Optimal Control Theory in Biomedicine, M. Dekker, New York. [768] Sydsaeter, K. (1978). Optimal control theory and economics: some critical remarks on the literature, Scand. Journal of Economics, 80, 113-117. [769] Sydsaeter, K. (1981). Topics in Mathematical Analysis for Economists, Academic Press, London. [770] Sweeney, D.J., Abad, P.L. and Dornoff, R.J. (1974). Finding an optimal dynamic advertising policy. International Journal of Systems Science, 5, 10, 987-994. [771] Takayama, A. (1974). Mathematical Economics, The Dryden Press, Hins dale. [772] Tan, K.C. and Bennett, R.J. (1984). Optimal Control of Spatial Systems, George Allen Sz Unwin, London. [773] Tapiero, C.S. (1971). Optimal simultaneous replacement and main tenance of a machine with process discontinuities. Revue francaise d'informatique et recherche operationnelle, 2, 79-86. [774] Tapiero, C.S. (1973). Optimal maintenance and replacement of a se quence of machines and technical obsolescence, Opsearch, 19, 1-13. [775] Tapiero, C.S. (1977). Managerial Planning: An Optimum and Stochastic Control Approach, Gordon Breach, New York. [776] Tapiero, C.S. (1978). Optimum advertising and goodwill under uncer tainty, Operations Research, 26, 3, 450-463. [777] Tapiero, C.S. (1981). Optimum product quality and advertising. Infor mation Systems and Operational Research, 19, 4, 311-318. Bibliography 475 [778] Tapiero, C.S. (1982a). Optimum control of a stochastic model of advertis ing, in Optimal Control Theory and Econ. Analysis^ G. Feichtinger (Ed.), North-Holland, Amsterdam, 287-300. [779] Tapiero, C.S. (1982b). A stochastic model of consumer behavior and optimal advertising, Management Science, 28, 9, 1054-1064. [780] Tapiero, C.S. (1983). Stochastic diffusion models with advertising and word-of-mouth effects, European Journal of Operational Research, 12, 4, 348-356. [781] Tapiero, C.S. (1988). Applied Stochastic Models and Control in Manage ment, North-Holland, Amsterdam. [782] Tapiero, C.S., Eliashberg, J. and Wind, Y. (1987). Risk behaviour and optimum advertising with a stochastic dynamic sales response. Optimal Control Applications & Methods, 8, 3, 299-304. [783] Tapiero, C.S. and Farley, J.U. (1975). Optimal control of sales force effort in time. Management Science, 21, 9, 976-985. [784] Tapiero, C.S. and Farley, J.U. (1981). Using an uncertainty model to assess sales response to advertising. Decision Science, 12, 441-455. [785] Tapiero, C.S. and Soliman, M.A. (1972). Multi- commodities transporta tion schedules over time. Networks, 2, 311-327. [786] Tapiero, C.S. and Venezia, I. (1979). A mean variance approach to the optimal machine maintenance and replacement problem. Journal of Op- erational Research Society, 30, 457-466. [787] Tapiero, C.S. and Zuckermann, D. (1983). Optimal investment policy of an insurance firm. Insurance: Mathematics & Economics, 2, 103-112. [788] Taylor, J.G., (1972). Comments on a multiplier condition for problems with state variable inequality constraints, IEEE Transactions on Auto matic Control, AC-12, 743-744. [789] Taylor, J.G., (1974). Lanchester-type models of warfare and optimal con trol, Naval Research Logistics Quarterly, 21, 79-106. [790] Teng, J.-T. and Thompson, G.L. (1983). Oligopoly models for optimal advertising when production costs obey a learning curve. Management Science, 29, 9, 1087-1101. [791] Teng, J.-T. and Thompson, G.L. (1985). Optimal strategies for general price-advertising models, in Optimal Control Theory and Economic Anal ysis 2, G. Feichtinger (Ed.), North-Holland, Amsterdam, 183-195. 476 Bibliography [792] Teng, J.-T., Thompson, G.L. and Sethi, S.R (1984). Strong decision and forecast horizons in a convex production planning problem, Optimal Control Applications & Methods^ 5, 4, 319-330. [793] Teo, K.L., Goh, C.J. and Wong, K.H. (1991). A Unified Computational Approach to Optimal Control Problems^ Longman Scientific & Technical, E^ssex, England. [794] Teo, K.L. and Moore, E.J. (1977). Necessary conditions for optimality for control problems with time delays appearing in both state and control variables. Journal of Optimization Theory and Applications, 23, 413-427. [795] Terborgh, G. (1949). Dynamic Equipment Policy, McGraw-Hill, New York. [796] Thepot, J. (1983). Marketing and investment policies of duopolists in a growing industry, Journal of Economic Dynamics and Control, 5, 387- 404. [797] The Review of Economic Studies (1974). Symposium on the Economics of Exhaustible Resources, P. Dasgupta and G.M. Heal (Eds.), 41. [798] Thompson, G.L. (1968). Optimal maintenance policy and sale date of a machine. Management Science, 14, 543-550. [799] Thompson, G.L. (1981). An optimal control model of advertising pul sation and wearout, in ORSA/TIMS Special Interest Conf on Market Measurement and Analysis, J.W. Keon (Ed.), The Institute for Manage ment Sciences, Providence, RI, 34-43. [800] Thompson, G.L. (1982a). Continuous expanding and contracting economies, in Games, Economic Dynamics, Time Series Analysis, M. Deistler, E. Furst and G. Schwodiauer (Eds.), Physica-Verlag, Berhn, West Germany, 145-153. [801] Thompson, G.L. (1982b). Many country continuous expanding open economies with multiple goods, in Optimal Control Theory and Economic Analysis, G. Feichtinger (Ed.), North-Holland, New York, 157-168. [802] Thompson, G.L. and Sethi, S.P. (1980). Turnpike horizons for production planning. Management Science, 26, 229-241. [803] Thompson, G.L., Sethi, S.P. and Teng, J.-T. (1984). Strong planning and forecast horizons for a model with simultaneous price and production decisions, European Journal of Operational Research, 16, 378-388. [804] Thompson, G.L. and Teng, J.-T. (1984). Optimal pricing and advertising policies for new product oligopoly models. Marketing Science, 3, 2, 148- 168. Bibliography 477 [805] Tintner, G. (1937). Monopoly over time, Econometrica, 5, 160-170. [806] Tintner, G. and Sengupta, J.K. (1972). Stochastic Economics: Stochastic Processes, Control, and Programming, Academic Press, New York. [807] Tolwinski, B. (1982). A concept of cooperative equilibrium for dynamic games, Automatica, 18, 431-447. [808] Tousssaint, S. (1985). The transversality condition at infinity applied to a problem of optimal resource depletion, in Optimal Control Theory and Economic Analysis 2, G. Feichtinger (Ed.), North-Holland, Amsterdam, 429-440. [809] Tracz, G. S. (1968). A selected bibliography on the application of optimal control theory to economic and business systems, management science and operations research, Operations Research, 16, 174-186. [810] Tragler, G., Caulkins, J.R and Feichtinger, G. (2000). The impact of enforcement and treatment on illicit drug consumption. Operations Re search, forthcoming. [811] Treadway, A.B. (1970). Adjustment costs and variable inputs in the the ory of the competitive firm. Journal of Economic Theory, 2, 329-347. [812] Troch, I. (Ed.) (1978). Simulation of Control Systems with Special Em phasis on Modelling and Redundancy, Proceedings of the IMACS Sym posium, North-Holland, Amsterdam. [813] Tsurumi, H. and Tsurumi, Y. (1971). Simultaneous determination of market share and advertising expenditure under dynamic conditions: the case of a firm within the Japanese pharmaceutical industry. The Eco nomic Studies Quarterly, 22, 1-23. [814] Tu, P.N.V. (1969). Optimal educational investment program in an eco nomic planning model, Canadian Journal of Economics, 2, 52-64. [815] Tu, P.N.V. (1984). Introductory Optimization Dynamics, Springer- Verlag, Berlin. [816] Turner, R.E. and Neuman, C.P. (1976). Dynamic advertising strategy: a managerial approach. Journal of Business Administration, 7, 1-21. [817] Turnovsky, S.J. (1981). The optimal intertemporal choice of inflation and unemployment. Journal of Economic Dynamics and Control, 3, 357-384. [818] Tzafestas, S.G. (1982). Optimal and modal control of production- inventory systems, in Optimization and Control of Dynamic Operational Research Models, S.G. Tzafestas (Ed.), North-Holland, Amsterdam, 1-71. 478 Bibliography [819] Uhler, R.S. (1979). The rate of petroleum exploration and extraction, in Advances in the Economics of Energy and Resources^ R.S,. Pindyck (Ed.), Vol. 2, JAI Press, Greenwich, CT, 93-118. [820] Valentine, F.A. (1937). The problem of Lagrange with differential in equalities as added side conditions, in Contributions to the Theory of the Calculus of Variations, 1933-1937, University Chicago Press, Chicago, IL. [821] Van Hilten, O., Kort, P.M. and Van Loon, P.J.J.M. (1993). Dynamic Policies of the Firm: An Optimal Control Approach, Springer-Verlag, New York. [822] Vanthienen, L. (1975). Simultaneous price-reduction decision making with production adjustment costs, in the Series Proceedings XX Inter national Meeting of TIMS held in Tel Aviv, Israel on June 24-29, 1973, Jerusalem Academic Press, Jerusalem, Vol. 1, 249-254. [823] Varaiya, P.P. (1970). N-person nonzero-sum differential games with linear dynamics, SIAM Journal on Control, 8, 441-449. [824] Verheyen, P. A. (1985). A dynamic theory of the firm and the reaction on governmental policy, in Optimal Control Theory and Economic Analysis, G. Feichtinger (Ed.), Vol. 2, North-Holland, Amsterdam, 313-329. [825] Vickson, R.G. (1981). Schedule control for randomly drifting production sequences. Information Systems and Operational Research, 19, 330-346. [826] Vickson, R.G. (1982). Optimal control production sequences: a continu ous parameter analysis. Operations Research, 30, 659-679. [827] Vickson, R.G. (1985). Optimal conversion to a new production technol ogy under learning, lEE Transactions, 17, 175-181. [828] Vickson, R.G. (1986/7). A single product cycling problem under Brown- ian motion demand, erscheint in Management Science. [829] Vidale, M.L. and Wolfe, H.B. (1957). An operations research study of sales response to advertising. Operations Research, 5, 3, 370-381. [830] Villas-Boas, J.M. (1993). Predicting advertising policies in an oligopoly: a model and empirical test. Marketing Science, 12, 88-102. [831] Vinokurov, V.R. (1969). Optimal control of processes described by inte gral equations I, SIAM Journal on Control and Optimization, 7, 324-336. [832] Vinter, R.B. (1983). New global optimality conditions in optimal control theory, SIAM Journal on Control and Optimization, 21, 235-245. Bibliography 479 [833] Vousden, N. (1974). International trade and exhaustible resources: a theoretical model, International Economic Review, 15, 149-167. [834] Wagner, H.M. and Whitin, T.M. (1958). Dynamic version of the economic lot size model. Management Science, 5, 89-96. [835] Wang, P.K.C. (1964). Control of distributed parameter systems, in Ad vances in Control Systems 1, C.T. Leondes (Ed.), Academic Press, New York, 75-170. [836] Warga, J. (1972). Optimal Control of Differential and Functional Equa tions, Academic Press, New York. [837] Warschat, J. (1985a). Optimal control of a production-inventory system with state constraints and a quadratic cost criterion, RAIRO-OR, 19, 275-292. [838] Warschat, J. (1985b). Optimal production planning for cascaded pro duction inventory systems, in Toward the Factory of the Future, H.J. BulUnger and H.J. Warnecke (Eds.), Springer-Verlag, Berlin, 669-674. [839] Warschat, J. and Wunderlich, H.J. (1984). Time-optimal control poli cies for cascaded production-inventory systems with control and state constraints. International Journal of Systems Science, 15, 513-524. [840] Weinstein, M.C. and Zeckhauser, R.J. (1975). The optimum consumption of depletable natural resources. The Quarterly Journal of Economics, 89, 371-392. [841] Weizsacker, C.C. von (1980). Barriers to Entry: A Theoretical Treat ment, Lecture Notes in Economics and Mathematical Systems, Vol. 185, Springer-Verlag, Berlin. [842] Welam, U.P. (1982). Optimal and near optimal price and advertising strategies for finite and infinite horizons. Management Science, 28, 11, 1313-1327. [843] Westphal, L.C. (1974). Toward the synthesis of solutions of dynamic games, in the Series, Advances in Control Systems, C.T. Leondes (Ed.), Academic Press, New York, Vol. 11, 389-489. [844] Whittle, P. (1982/3). Optimization over Time: Dynamic Programming and Stochastic Control, Vol. I &: II, Wiley, Chichester. [845] Wickwire, K. (1977). Mathematical models for the control of pests and infectious diseases: a survey. Theoretical Population Biology, 11, 182-238. [846] Wiener, N. (1949). The Interpolation and Smoothing of Stationary Time Series, MIT Press, Cambridge, Mass. 480 Bibliography [847] Wirl, F. (1984). Sensitivity analysis of OPEC pricing polices, OPEC Rev., 8, 321-331. [848] Wirl, F. (1985). Stable and volatile prices: and explanation by dynamic demand, in Optimal Control Theory and Economic Analysis 2, G. Fe- ichtinger (Ed.), North-Holland, Amsterdam, 263-277. [849] Wonham, W.M. (1970). Random differential equations in control theory, in Probabilistic Methods in Applied Mathematics, Vol. 2, A.T. Bharucha- Reid (Ed.), Academic Press, New York. [850] Wright, C. (1974). Some political aspects of pollution control. Journal of Environmental Economics and Management, 1, 173-186. [851] Wright, S.J. (1993). Interior-point methods for optimal control of discrete-time systems. Journal of Optimization Theory and Applications, 77, 1, 161-187. [852] Yang, J., Yan, H. and Sethi, S.P. (1999). Optimal production planning in pull flow lines with multiple products, European Journal of Operational Research, 119, 3, 26-48. [853] Yin, G. and Zhang, Q. (1997). Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, Springer-Verlag. New York. [854] Young, L.C. (1969). Calculus of Variations and Optimal Control Theory, W.B. Saunders Co., Philadelphia, Pa. [855] Zeidan, V. (1984a). Extended Jacobi sufficiency criterion for optimal con trol, SIAM Journal on Control and Optimization, 22, 294-301. [856] Zeidan, V. (1984b). A modified Hamilton-Jacobi approach in the gener alized problem of Bolza, Appl Math. & Optimiz., 11, 97-109. [857] Zhou, X. and Sethi, S.P. (1994). A sufficient condition for near optimal stochastic controls and its application to manufacturing systems. Applied Mathematics & Optimization, 29, 67-92. [858] Ziemba, W.T. and Vickson, R.G. (Eds.) (1975). Stochastic Optimization Models in Finance, Academic Press, New York. [859] Ziolko, M. and Kozlowski, J. (1983). Evolution of body size: an optimal control model. Math. Biosci., 64, 127-143. [860] Zimin, I.N. and Ivanilov, Y.P. (1971). Solution of network planning prob lems by reducing them to optimal control problems, Zhurnal Vychifli- teVnoi Matematiki i Matematichefkoi Fiziki (in Russian), 11, 3. Bibliography 481 [861] Zoltners, A.A. (Ed.) (1982). Marketing Planning Models, TIMS Studies in the Management Sciences, Vol. 18, North-Holland, Amsterdam. Index Abad, P.L., 417, 474 Axsater, S., 418 adjoint equation, 31, 32, 36, 230 adjoint function, 322 Baar, T., 425 adjoint variables, 10, 32 backlogging of demand, 5, 348 adjoint vector, 30, 33 Bagchi, A., 419 admissible control, 24 Balachandran, B., 445 advertising model, 5, 6 Balakrishman, A.V., 455 affine function, 19 bang function, 16 Agnew, C.E., 417 bang-bang, 42, 78, 84, 85, 87, Alam, M., 249, 265, 417, 465 122, 132, 135, 190, 194, AUen, K.R., 286, 417 232, 244, 247, 257, 258, Amit, R., 154, 279, 418 313, 322, 328, 329, 333, Amoroso-Robinson relation, 187 334, 408 Anderson, R.M., 418 bankruptcy, 358 anti-difference operator, 376 Bamea, A., 452 Aoki, M., 345, 418 Basar, T., 308, 419, 420, 430, applications to biomedicine, 295 446 applications to finance, 119 Bass, F.M., 419 applications to marketing, 185 Bayes theorem, 342 Arnold, L., 344, 347, 348, 418 Bean, J.C, 182, 419 Aronson, J.E., 418, 471 begiiming game, 321 Arora, S.R., 243, 418 Bell, D.J., 42, 407, 419 Arrow, K.J., 10, 46, 58, 81, 82, Belhnan, R.E., 9, 27, 419 186, 188, 289, 290, 360, Bennett, R.J., 474 418, 456, 459 Bensoussan, A., 10, 58, 88, 154, Arthur, W.B., 304, 418 173, 182, 191, 315, 322, Arugaslan, O., viii 324, 360, 419, 420, 444, Arutyunov, A.V., 103, 418 461, 470 Aseev, S.M., 103, 418 bequest function, 7, 355 Aubin, J.-R, 418, 466 Berkovitz, L.D., 10, 27, 308, 420 autocorrelation function, 343 Bernoulli, Jacob, 8, 9, 384 autonomous, 52, 81 Bernoulli, Johann, 8, 9, 384 484 Index Bertsekas, D.P., 236, 346, 420, Bulirsch, R., 9, 108, 422, 461, 430 474 Bes, C, 173, 182, 420 Bullinger, H.J., 479 Beyer, D., viii, 421 Bultez, A.V., 315, 419, 422 Bharucha-Reid, A.T., 480 Burdet, C.A., 236, 239, 422 Bhaskaran, S., 182, 421 Burmeister, E., 290, 291, 422 bionomic eqmlibrium, 269, 314 Buskens, C., 455 Black, F., 355, 421 Butkowskiy, A.G., 317, 422 Blaquiere, A., 325, 326, 331, 421, Bylka, S., 182, 254, 423 464, 473 BUss point, 306 Gaines, P., 423 BUss, G.A., 9 calculus of variations, 8, 379 Boiteux problem, 241 Canon, M.D., 235, 423 Boiteux, M., 241, 421 canonical adjoints, 33 Boltyanskii, V.G., 9, 27, 421, canonical system of equations, 462 33 Bolza, 9 capital accumulation model, 290 Bolza form, 25, 26, 229, 235, 239 Caratheodory, C., 9 Bookbinder, J.H., vi, 10, 304, Carlson, D.A., 10, 82, 423-425 421 Carraro, C, 315, 425 boundary conditions, 61, 350 Carrillo, J., 154, 425 boundary interval, 105 Case, J.H., 308, 325, 425 Bourguignon, F., 421 Cass, D., 425 Brachistochrone problem, 8, 9, Cassandras, C.G., 236, 461 384 cattle ranching problem, 318 Breakwell, J.V., 190, 421 Caulkins, J.P., 477 Brekke, K.A., 360, 422 CeUina, A., 418 Brito, D.L., 304, 422 certainty equivalence, 403, 404 Brock, W.A., 360, 454 Cesari, L., 26, 425 Brockhoff, K., 420 chain of forests model, 276-278 broken extremal, 387 chain of machines, 254 Brotherton, T., 423 Chand, S., 182, 254, 425, 469 Brown, R.G., 164, 422 Chandra, T., 445 Brownian Motion, 356 Chang, S., 360, 418 Bryant, G.F., 27, 422 Charnes, A., 304, 425 Bryson, Jr., A.E., 33, 87, 105, Chatterjee, R., 463 108, 341, 401, 403, 404, Chen, S.F., 425 406, 422 Cheng, F., viii Buchanan, L.F., 345, 422 Chiarella, C, 426 Bucy, R., 345, 447 Chichilinsky, G., 426 Index 485 Chikan, A., 440 Cottle, R.W., 430 Chintagunta, P.K., 315, 426 Cowling, K., 444 Chow, G.C., 345, 426 critical points, 218 Clark, C.W., 10, 241, 267, 268, Crouhy, M., 173, 182, 419 273, 286, 312, 314, 426 Cruz, J.B., Jr., 472 Clarke, F.H., 27, 106, 167, 427 CuUum, CD., 235, 423 Clemhout, S., 427 current-value adjoint variables, closed-loop control, 346 70 closed-loop Nash solution, 311 current-value formulation, 58, Coddington, E.A., 193, 427 65, 239 Cohen, K.J., 36, 187, 427 current-value functions, 67 common-property fishery re current-value Hamiltonian, 66 sources, 312 current-value Lagrange multipli comparison lemma, 198 ers, 67 complimentary slackness condi current-value Lagrangian, 66 tions, 60 current-value maximum princi computational methods, vii, ple, 68, 111 108, 236 cycloid, 9 concave function, 18, 64 Cyert, R.M., 36, 187, 427 Connors, M.M., 10, 427 Conrad, K., 427 D'Autimie, A., 428 Constantinides, G.M., 427 Dantzig, G.B., 418, 427 constraint of rth order, 105 Darrat, A.F., 456 constraint qualification, 60, 105, Darrough, M.N., 427 226 Dasgupta, P., 279, 428, 474, 476 constraints, 24 Davis, B.E., 119, 129, 152, 360, consumption model, 7, 8 428 consumption-investment prob Davis, M.H.A., 346, 428 lem, 355 Davis, R.E., 456 contact time, 105 Dawid, H., 428 continuous wheat trading model, Day, G., 463 164 DDT, 299, 300, 303 control of pest infestations, 295 Deal, K.R., 311, 315, 337, 428 control trajectory, 2, 24 decision horizon, 173, 175, 177, control variable, 2, 24 179, 180 convex combination, 17 Deger, S., 428 convex function, 18, 64 Deissenberg, C, 428, 429 convex huU, 17 Deistler, M., 476 convex set, 17 derivative operator, 363 CorelDRAW, vii derived Hamiltonian, 45 486 Index Derzko, N.A., 45, 130, 279, 315, dynamic efficiency condition, 317, 360, 420, 429, 466, 291 469 dynamic programming, 27, 345, DeSarbo, W., 304, 445 393 Dhrymes, P.J., 213, 429 difference equation, 229, 341 economic applications, 289, 360 diflference operator, 375 economic interpretation, 34, 69, differential games, 308 138, 291, 322 differentiation with scalars, 12 educational policy, 21 differentiation with vectors, 12, eigenvalues, 370, 371 13 eigenvectors, 370, 371 diffusion process, 344 El-Hodiri, M., 431 Dirac delta function, 323, 344 EUashberg, J., 304, 431, 445, 475 direct adjoining method, 100 Elton, E., 119, 431 direct contribution, 35 Elzinga, D.J., 119, 152, 428 discount factor, 6 ending correction, 158 discount rate, 6 ending game, 321 entry time, 105 discrete maximum principle, environmental management, 315 217, 228, 229 EOQ, 153 discrete-time optimal control epidemic control, 295 problem, 228, 229 equilibrimn relation, 36 distributed parameter maximimi Erickson, G.M., 431 principle, 317 Erickson, L.E., 10, 154, 315, 443 distributed parameter systems, Euler, 8, 384 315 Euler equation, 380, 382, 387, Dixit, A.K., 429 388 Dobell, A.R., 290, 291, 422 EXCEL, vii, 48-50 Dockner, E.J., 10, 289, 304, 308, exhaustible resource model, 279 315, 429, 430, 432, 446 exit time, 105 Dohrmann, C.R., 236, 430 Dolan, R.J., 430, 445 factorial power, 375 Dorfman, R., 430 Fan, L.T., 154, 304, 431, 443 Dornoff, R.J., 474 Farley, J.U., 475 Drews, W., 430 Feichtinger, G., vi, viii, 10, 33, Dreyfus, S.E., 420 58, 69, 81, 99, 106, 108, dual variables, 36 113, 154, 167, 185, 194, Dubovitskii, A.Y., 430 211, 289, 304, 322, 427- Dunn, J.C, 236, 430 429, 431-434, 438-440, Durrett, R., 347, 431 442, 446, 451, 453, 455, Index 487 457^59, 463-465, 470, Furst, E., 476 473, 475^78, 480 Feinberg, F.M., 434 Gaandolfo, G., 436 Fel'dbaum, A.A., 31, 393, 398, Gaimon, C, 154, 249, 304, 322, 434 331, 425, 435, 436 Ferreira, M.M.A., 103, 434 Galileo, 384 Ferreyra, G., 434 Gamkrelidze, R.V., 9, 436, 462 Filar, J., 315, 425 Gandolfo, G., 432 Filipiak, J., 434 Gaskins, Jr., D.W., 436 filtering, 339, 341 Gaugusch, J., 437 finite diflference equations, 375 Gaussian, 341-343 first-order pure state con Geismar, N., viii straints, 106, 108 Gelfand, I.M., 379, 380, 382, Fischer, T., 434 387, 389, 437 Fisher, A.C., 461 general discrete maximum prin fishery management, 315 ciple, 234 fishery model, 268, 312 general inequality constraints, fishing mortality function, 314 97 fixed-end-point problem, 70 generalized bang-bang, 87, 132 Fleming, W.H., 346, 360, 434 generalized derivative, 344 Fletcher, R., 434 generalized Legendre-Clebsch Fomin, S.V., 379, 380, 382, 387, condition, 152, 407, 408 389, 437 forecast horizons, 173 Geoffrion, A.M., 453 forest fertilization model, 287 Gerchak, Y., 437 forest thinning model, 273, 276 Gfrerer, H., 437 forgetting coefficient, 5 Gibson, J.E., 408, 445 Forster, B.A., 435 Gihman, LI., 344, 353, 437 Fourgeaud, C, 435 Gillessen, W., 455 Francis, P.J., 295, 435 Girsanov, I.V., 437 Frankena, J.F., 435 Glad, S.T., 437 free-end-point problem, 70 goal level, 319 Friedman, A., 308, 435 GOAL SEEK, 48, 49 Fromovitz, S., 228, 455 Goh, B.-S., 108, 267, 437 Fruchter, G.E., 315, 435 Goh, C.J., 476 fuU-rank condition, 20, 60, 105, Goldberg, S., 375, 437 106 Golden Path, 82 Fuller, D., 280, 435 Golden Rule, 82, 93 fundamental lemma, 382 Goldstine, H.H., 437 Funke, U.H., 435 goodwill, 5, 186 488 Index goodwill elasticity of demand, 304, 317, 322, 428, 432, 188 433, 438-442, 453, 473 Gopalsamy, K., 437 Harvey, A.C., 442 Gordon's formula, 145 Haunschmied, J., 304, 433 Gordon, H.S., 268, 269, 437 Haurie, A., 10, 82, 173, 308, 315, Gordon, M.J., 145, 437 317, 322, 420, 424, 425, Gould, J.P., 191, 210, 214, 437 438, 442 Green's theorem, 185, 196, 198, Haussmann, U.G., 442 205, 211, 214, 270, 296, Heal, G.M., 279, 428, 443, 476 297, 305, 314 Heaps, T., 241, 443 Grimm, W., 438, 459 Heckman, J., 443 Gross, M., 438 Heineke, J.M., 427 Gruber, M., 119, 431 Hestenes, M.R., 10, 58, 62, 443 Gruver, W.A., 448 higher-order constraints, 105 HJB equation, 30, 31, 354 Hadley, G., 10, 58, 289, 438 HMMS model, 153 Hahn, M., 438 Ho, Y.-C., 33, 87, 105, 308, 311, Halkin, H., 27, 235, 438 341, 401, 403, 404, 406, Hamalainen, R.P., 315, 438 422, 443, 459, 473 Hamilton, 9 Hochman, E., 360, 464 Hamilton-Jacobi equation, 349 Hoffmann, K.H., 443 Hamilton-Jacobi-BeUman equa Hohn, F., 174, 457 tion, 27, 30, 345, 349 Holly, S., 443 Hamiltonian, 30, 35, 60, 230, Holt, C.C., 153, 443 291, 397 Holtzman, J.M., 235, 443 Hamiltonian maximizing condi homogeneous equation, 363 tion, 30, 34, 397 homogeneous equations of order Han, M., 438 n, 365 Hanson, M., 457 homogeneous equations of order Hanssens, D.M., 438 one, 364 Harris, F.W., 153, 438 homogeneous equations of order Harris, H., 303, 439 two, 364 Harrison, J.M., 357, 439 homogeneous function of degree Hartberger, R.J., 27, 396, 430, one, 19 439 homogeneous partial differential Hartl, R.F., viii, 10, 26, 27, 33, equations, 374 58, 62, 69, 81, 89, 99, Horsky, D., 443 100, 105, 106, 108, 113, Hotelling, H., 279, 443 115, 154, 167, 174, 185, Hung, N.M., 442 194, 211, 239, 243, 303, Hurst, Jr., E.G., 10, 58, 154, 419 Index 489 Hwang, C.L., 154, 443 Jazwinski, A.H., 444 Hyun, J.S., 438 Jedidi, K., 304, 445 Jennings, L.S., 445 Ijiri, Y., 153, 165, 444 Jeuland, A.P., 430, 445 Ilan, Y., 154, 418 Jiang, J., 445 imp, 17 Johnson, CD., 408, 445 impulse control, 16, 125, 202, Jones, P., 445 204, 322, 324, 325 J0rgensen, S., viii, 10, 154, 267, impulse control model, 88 289, 304, 308, 315, 429, impulse Hamiltonian, 326 430, 433, 440, 445, 446 impulse maximum principle, Joseph, P.D., 341, 404, 446 326, 332 jirnip conditions, 103, 108 impulse stochastic control, 360 jump Markov processes, 360 imputed value, 219 junction times, 105 indirect adjoining method, 98, 100, 104,111 Kaitala, V.T., 315, 433, 438, 446 indirect contribution, 35 Kalaba, R.E., 419 infinite horizon, 6, 80 Kalish, S., 304, 435, 446, 447 inhomogeneous partial differen KaU, P., 439, 458, 470 tial equation, 374 Kahnan filter, 339, 340, 342 instantaneous profit rate, 25 Kahnan, R.E., 341, 345, 447 interior interval, 105 Kaknan-Bucy filter, 339, 345 Intriligator, M.D., 290, 444, 456 Kamien, M.I., 10, 248, 253, 289, inventory control problem, 117 303, 447 loffe, A.D., 444 Kamien-Schwartz model, 249 Isaacs, R., 9, 133, 308, 444 Kaplan, W., 447 isoperimetric profit constraint, Karatzas, I., 340, 344, 347, 359, 214 360, 447 Ito stochastic differential equa Karreman, H.F., 421 tion, 344, 345 Keeler, E., 448 Ivanilov, Y.P., 304, 480 Keller, H.B., 299, 448 Kemp, M.C., 10, 58, 289, 426, Jabrane, A., 424 438, 448 Jacobi, 9 Kendrick, D.A., 448 Jacobson, D.H., 42, 108, 407, Keon, J.W., 476 419, 444, 451 Khmelnitsky, E., 10, 448, 449, Jacquemin, A.P., 188, 191, 444 454 Jagpal, S., 444 Kilkki, P., 273, 274, 448 Jain, D., 315, 426 Kirakossian, G.T., 304, 440 Jamshidi, M., 444 Kirby, B.J., 448 490 Index Kirk, D.E., 31, 448 L'Hospital, 384 Klein, C.F., 448 Lagrange, 8 Kleindorfer, G.B., 234, 449 Lagrange form, 25 Kleindorfer, P.R., 174, 182, 234, Lagrange multipliers, 57, 218 448, 449, 470 Lagrangian, 57, 60 Kleinschmidt, P., 440 Lagtmov, V.N., 450 Knobloch, H.W., 449 Lakhani, C, 465 Knowles, G., 449 Lansdowne, Z.F., 206, 450 Kogan, K., 10, 448, 449, 454 Lasdon, L.S., 450 Kopel, M., 428 Leban, R., 10, 289, 450, 451 Kort, P.M., 10, 154, 267, 289, Leclair, S.R., 445 303, 304, 433, 440, 441, Lee, E.B., 129, 450 446, 478 Lee, S.C, 213, 469 Kortanek, K., 304, 425 Lee, W.Y., 450 Kotowitz, Y., 449 left and right limits, 15 Kozlowski, J., 480 Legendre, 8 Krabs, W., 443 Legendre's conditions, 388 Kraft, D., 108, 422 Legey, L., 304, 450 Krarup, J., 430 Lehoczky, J.P., 130, 359, 447, Krauth, J., 304, 441 450, 469 Kreindler, E., 449 Leibniz, 8 KreUe, W., 420, 449 Leitmann, G., 27, 267, 308, 379, Krichagina, E., 449 425, 437, 438, 442, 451, Kriebel, C.H., 234, 449 457, 473 Kriendler, E., 108 Leizarowitz, A., 424, 425 Krouse, CO., 129, 449, 450 Leland, H.E., 451 Krutilla, J.V., 447 Lele, M.M., 108, 444, 451 Kugelmann, B., 450 Lele, P.T., 243, 418 Kuhn, H.W., 472 Lenclud, B., 435 Kuhn-Tucker conditions, 218, Leonard, D., 10, 289, 451 220, 228 Leondes, C.T., 422, 473, 479 Kuhn-Tucker constraint qualifi Lesourne, J., 10, 289, 360, 419, cation, 226, 227 420, 450, 451 Kumar, S., viii Lev, B., 436 Kurawarwala, A.A., 450 Levine, J., 451 Kurcyusz, S., 103, 450 Levinson, N.L., 193, 427 Kurz, M., 10, 46, 58, 81, 82, 188, Lewis, T.R., 451, 452 289, 290, 418 Li, G., 154, 452 Kushner, H.J., 450 Lieber, Z., 174, 182, 438, 449, Kydland, F.E., 450 452 Index 491 LigneU, J., 452 maintenance and replacement LiUen, G.L., 304, 446 model, 241, 242, 248, line integral, 197 331 linear differential equations, 363 Majumdar, M., 454 linear independence, 20 Malinowski, K., 115, 454, 472 linear Mayer form, 25, 26, 239 Malliaris, A.G., 360, 454 linear programming, 87, 132 Mangasarian, O.L., 46, 64, 226- linear-quadratic case, 85 228, 455 linear-quadratic problems, 401 Manh-Hung, N., 279, 455 linearly independent, 20 Mantrala, M.K., 345, 458 Lintner, J., 452 MAPI (Machinery and Applied Lions, J.L., 88, 317, 324, 360, Products Institute), 241 420, 444, 452, 470 MAPLE, 150 Little, J.D.C., 452 marginal cost, 36, 187 little-o notation, 15 marginal cost equals marginal Liu, P.-T., 418, 421, 428, 451, revenue, 36 452 marginal return, 398 Loewen, P.D., 106, 427 marginal revenue, 36 logarithmic Brownian Motion, Markus, L., 450 356 martingale problems, 360 Long, N.V., 10, 289, 308, 315, Martirena-Mantel, A.M., 455 426, 430, 448, 451, 452 Marzano, P., 432 long-rim stationary equilibrium, Masse, P., 241, 455 82 Mate, K., 443 Loon, P.J.J.M., van, 453 Mathematica, 150 Lou, S., 449, 453 mathematical requirements, 1 Lucas, Jr., R.E., 304, 453 Mathewson, P., 449 Luenberger, D.G., 228, 453 matrix Riccati equation, 345, Luhmer, A., 433, 453 403 limiped parameter systems, 315 Matsuo, H., 450 Lundin, R.A., 174, 182, 453 Maurer, H., viii, 108, 115, 118, Luptacik, M., 303, 441, 453, 463 455 Luus, R., 454 maximized Hamiltonian, 114 Lynn, J.W., 249, 417 maximum, 388 maximum likelihood estimate, Macki, J., 454 342 Magat, W.A., 454 maximum principle, 23, 33, 34, Magill, M.J.P., 454 57, 58, 67, 104,113, 217, Mahajan, V., 430, 447, 452, 454 393 Maimon, O., 10, 448, 449, 454 May, R.M., 418 492 Index Mayer form, 25, 397 model type (b), 85-87 Mayne, D.Q., 27, 108, 236, 422, model type (c), 85 456, 460, 462 model type (d), 85 McCabe, J.L., 451 model type (e), 85, 86 McCann, J.M., 454 model type (f), 85, 86, 251 McEneaney, W.M., 470 model types, 83, 85 McGuire, T.W., 304, 469 modeling "tricks", 86 Mclntyre, J., 108, 456 Modigliani, F., 130, 153, 174, McNicoU, G., 304, 418 443, 457 McShane, E.J., 10 Moiseev, N.N., 457 measurement noise, 340 Monahan, G.E., 457 Meech, J.A., 445 Mond, B., 457 Mehlmann, A., 304, 305, 308, Moore, E.J., 476 429, 433, 441, 456 Morey, R.C., 454 Mehra, R.K., 304, 456 Morton, T.E., vi, 174, 182, 254, Mehrez, A., 456 259, 264, 453, 457, 458, Merton, R.C., 355, 456 469 Mesak, H.I., 456 Motta, M., 324, 458 Michel, P., 428, 435, 457 Muller, E., 430, 452, 454, 458 middle game, 321 Mulvey, J.M., 469 Miele, A., 196, 457 Murata, Y., 458 MiUer, M.H., 130, 457 Murray, D.M., 236, 458 MiUer, R.E., 457 Muth, J.F., 153, 443 Milyutin, A.A., 430 Muzicant, J., 322, 458 minimax solution, 308, 309 minimum fuel problem, 238 Naert, P.A., 315, 419, 422 Mirman, L.J., 452, 457 Nahorski, Z., 458 Mirrlees, J., 457 Naik, P.A., 345, 458 miscellaneous applications, 303 Nash solutions, 308 Mischenko, E.F., 9, 462 Naslund, B., 10, 58, 154, 241, MitcheU, A., 457 278, 287, 419, 458 Mitra, T., 454 natural resources, 267, 360 Mitter, S.K., 459 necessary condition, 31, 33, 228 mixed constraints, 59, 104, 106 Neck, R., 458 mixed inequality constraints, 3, needle-shaped variation, 394, 57, 58 395 mixed optimization technique, neighborhood, 15 258 Nelson, R.T., 304, 458 model type (a), 84, 85, 87, 243, Nepomiastchy, P., 304, 459 244 Nerlove, M., 186, 188, 459 Index 493 Nerlove-Love advertising model, optimal long-run stationary 186 equilibriimi, 82, 189 Neuman, C.P., 186, 477 optimal path, 25 Neustadt, L.W., 10, 455, 459 optimal thinning, 274 Newton, 8, 384 optimal trajectory, 25 Nguyen, D., 459 order of the constraint, 105 Nissen, G., 322, 420 Oren, S.S., 460 nonlinear programming, 217, Osayimwese, I., 460 218, 227 overdraft, 124 nonzero-simi games, 310 Ozga model, 214 norm, 15 Ozga, S., 214, 460 Norstrom, C.J., vi, 125, 153, 170, 459 Paiewonsky, B., 108, 456 Norton, F.E., 345, 422 Palda, K.S., 186, 460 notation, 10 Pantoja, J.F., 236, 460 Novak, A., 304, 433, 434, 441, parametric linear programming, 459 87 Oakland, W.H., 304, 422 Parlar, M., 437, 460 Oberle, H.J., 438, 459 Parrish, B., 417 objective function, 2, 25, 398 Parsons, L.J., 438 Oettli, W., 422 partial differential equations, Oguztoreli, M.N., 459 372 oil driller's problem, 324 partial fractions, 350 0ksendal, B.K., 344, 360, 422, particular integral, 367 459 particular solutions, 366, 367 Okuguchi, K., 426 path of least time, 8 Olsder, G.J., 419, 459 Pauwels, W., 460 one-sector model, 291 Pekelman, D., 154, 174, 182, one-sided constraints, 71 241, 259, 460, 461 Oniki, H., 460 Pepyne, D.L., 236, 461 open access fishery, 269 Pesch, H.J., 9, 450, 461 open-loop Nash solution, 310 pessimal solution, 146 optimal consumption of an ini Peterson, D.W., 461 tial investment, 89 Peterson, F.M., 461 optimal control problem, 25 Peterson, R.A., 454 optimal control theory, 1, 379 Petrov, lu.P., 461 optimal economic growth mod phase diagram, 192, 293, 301 els, 289, 293 Phelps, E.S., 437 optimal financing model, 129 Pierskalla, W.P., 241, 461 494 Index Pindyck, R.S., 279, 360, 429, rank of a matrix, 20 461, 462, 478 Rao, R.C., 315, 463 Pitchford, J.D., 435, 452, 462 Rapoport, A., 464 Pliska, S.R., 357, 439 Rapp, B., 241, 464 Pohjola, M., 462 Rausser, G.C., 360, 461, 464 Polak, E., 108, 235, 423, 456, 462 Raviv, A., 304, 464 pollution control model, 299, 302 Ravn, H.F., 458 Pontryagin, L.S., 9, 10, 23, 27, Ray, A., 464 76, 106, 393, 462 reachable set, 3, 59 Powell, S.G., 460 Reeves, CM., 434 predator-prey relationships, 267 regional allocation of invest Prescott, E.G., 450 ment, 52 Presman, E., 462, 463 Reinganum, J.F., 464 price elasticity of demand, 187 Rempala, R., 174, 464 price shield, 174, 175, 177 Richard, S.F., 427, 464 principle of optimality, 27, 346, Ringbeck, J., 304, 464 358 Ripper, M., 304, 450 product rule for differentiation, Rishel, R.W., 346, 434, 465 14 Roberts, S.M., 33, 465 production fimction, 292 Robinett, R.D., 236, 430 production planning model, 153, Robinson, B., 465 339 Robson, A.J., 322, 465 production smoothing, 154 Rockafeller, R.T., 465 production-inventory model, 4, Roxin, E.G., 421, 423, 452 153, 154, 234 Russak, B., 465 Proth, J.-M., 173, 182, 419, 425 Russell, D.L., 427, 465 Prskawetz, A., 463 Riistem, B., 443 pure constraints, 116 Ruusunen, J., 315, 438 pure state variable inequality Ryu, Y., viii constraints, 3, 97, 98, 104 saddle point, 19, 309 Pytlak, R., 108, 463 Sage, A.P., 317, 465 Salukvadze, M.E., 465 quasiconcave fimction, 64 salvage value, 3, 25, 87 quasiconvex function, 64 Samaratunga, G., 466 Samuelson, P.A., 465 Rajagopalan, S., 154, 452 Sargent, T.J., 453 Raman, K., 360, 463 Sarma, V.V.S., 249, 265, 417, Rampazzo, F., 324, 458 465 Ramsey, P.P., 69, 289, 306, 463 Sasieni, M., 466 Index 495 sat fiinction, 16 476, 480 Sawyer, A., 345, 458 Sethi-Morton model, 254, 264, Scalzo, R.C., 466 331 Schaefer, M.B., 268, 466 shadow price, 10, 35, 219 Schijndel, G.-J.C.Th.,van, 304, Shapiro, A., 472 466 Shapiro, C, 103, 472 SchiUing, K., 466 Shell, K., 289, 425, 472 Schmalensee, R., 451, 452 Shipman, J.S., 33, 465 Scholes, M., 355, 421 shooting method, 182 Schubert, U., 303, 453 short-selling, 124 Schultz, R. L., 438 Shreve, S.E., 340, 344, 346, 347, Schwartz, N.L., 10, 248, 253, 359, 360, 420, 447 289, 303, 447 Shtub, A., 449 Schwodiauer, G., 476 Siebert, H., 472 second-order differential equa Silva, G.N., 324, 472 tions, 359 Simaan, M., 472 second-order variations, 388, 405 Simon, H.A., 153, 404, 443, 472 Segers, R., 430 Simon, L.S., 443 Seidman, T.I., 315, 466 simple cash balance problem, Seierstad, A., 10, 27, 45, 58, 62, 120 64, 81, 99, 113, 289, 466 simplest variational problem, Selten, R., 308, 466 379 Sen, S.K., 428, 447 Singh, M.G., 468, 472 Sengupta, J.K., 477 singular arcs, 407 separation principle, 403, 404 singular control, 42, 132, 140, Sethi, S.P., 10, 26, 27, 45, 58, 297, 407 62, 89, 99,100,105, 106, Skiba, A.K., 472 108, 113, 115, 119, 129, Skorohod, A.V., 344, 353, 437 130, 150, 153, 154, 173, Smith, B.L.R., 444 174, 182, 185, 190, 194- Smith, M., 445 196, 202, 206, 213-215, Smith, R.L., 182, 419 236, 239, 241, 247, 254, Smith, V.K., 447 259, 264, 270, 271, 276, Smith, V.L., 472 278, 279, 295, 298, 303, Snower, D.J., 472 304, 311, 315, 317, 322, sole owner fishery resource 337, 340, 347, 351, 352, model, 268 354, 357-360, 420-423, Soliman, M.A., 304, 475 425, 427-429, 433, 441, Solow, R.M., 279, 472, 473 442, 445, 447, 449, 450, Soner, H.M., 360, 434, 450 453, 461-464, 466-471, Sorenson, H., 341, 473 496 Index Sorger, G., viii, 10, 182, 254, stochastic advertising problem, 289, 308, 315, 423, 425, 352 430, 433, 434, 453, 469, stochastic calculus, 347 473 stochastic differential equations, Sothmann, B., 459 339, 344 Southwick, L., 303, 473 stochastic manufacturing prob special topics, 401 lems, 360 Spence, M., 299, 448, 473 stochastic optimal control, 339, Speyer, J.L., 108, 444 345, 346 Spiegel, M.R., 375, 473 stochastic production planning Spremarm, K., 473 model, 347 Sprzeuzkouski, A.Y., 154, 473 stockout cost, 5 Spulber, P.F., 452, 457 Stoer, J., 422, 474 Srinivasan, V., 186, 473 stopping time, 357 Sriskandarajah, C, viii Stoppler, S., 10, 154, 429, 474 Staats, P.W., 295, 469 Strauss, A., 454 Stalford, H., 473 Streitferdt, L., 466 strengthened Jacobi condition, standard adjoint variables, 65 389 standard Hamiltonian, 65 strengthened Legendre condi standard Lagrangian, 65 tion, 389 standard multipliers, 65 strengthened Legendre-Clebsch Starr, A.W., 308, 311, 473 condition, 406 starting correction, 158 strictly concave function, 18, 64 state equation, 24 strictly convex function, 64 state trajectory, 2, 24 strong forecast horizon, 174, state variable, 2, 24 177, 179 static efficiency condition, 291 strong maximum, 389, 390 stationarity assumption, 81 sufficiency conditions, 44, 46, 64, Stein, R.B., 459 113, 228 Steinberg, R., 431 sufficiency transversality condi Steindl, A., 453, 473 tion, 159 Steiner, P.O., 430 summary of transversality con Stepan, A., 473 ditions, 75 Stern, L.E., 473 Suo, W., viii, 462, 469, 470 Stiglitz, J.E., 474 surveys of appfications, 10 Stirling numbers of the first Sutinen, J.G., 428, 452 kind, 378 Swan, G.W., 295, 474 StirUng numbers of the second Sweeney, D.J., 417, 474 kind, 376 switching curves, 78 Index 497 switching point, 138 Tintner, G., 477 switching time, 80 TitU, A., 472 Sydsaeter, K., 10, 27, 45, 58, 62, Tolwinski, B., 308, 442, 477 64, 81, 99,113, 289, 466, total contribution, 35 474 Tou, J.T., 341, 404, 446 synthesis of optimal controls, 76, Toussaint, S., 477 133 TPBVP, 33, 48, 50, 182, 291 system, 2 Tracz, G.S., 477 system noise, 340 Tragler, G., 477 Szego, G.P., 472 transition matrix, 396 transversality condition, 32, 62, Takayama, A., 36, 289, 431, 474 67, 69, 75, 81 Taksar, M.I., 449, 450, 469, 470 Treadway, A.B., 303, 477 Tan, K.C., 474 Troch, I., 477 Tapiero, C.S., 10, 254, 264, 304, Tsurumi, H., 214, 477 322, 360, 420, 470, 474, Tsurumi, Y., 214, 477 475 Tu, P.N.V., 10, 303, 477 Taylor, J.G., 108, 304, 475 Tuominen, M.P.T., 452 Teichroew, D., 10, 427 Turner, R.E., 185, 477 Teng, J.-T., 182, 475, 476 Turnovsky, S.J., 435, 452, 462, Teo, K.L., 108, 445, 476 477 Terborgh, G., 241, 476 turnpike, 82, 158, 189, 207 terminal conditions, 32, 69 two person zero-sum games, 308 terminal inequality constraints, two-point bovmdary value prob 59 lem, 33, 48, 372 terminal time, 4, 25, 62 two-reservoir system, 116 Thepot, J., 304, 451, 476 Tzafestas, S.G., 322, 432, 477 Thisse, J., 444 Thompson's maintenance Udayabhanu, V., 470 model, 331 Uhler, R.S., 478 Thompson, G.L., 45, 119, 153, utility of consiunption, 7, 289, 165, 174, 182, 234, 241, 290, 355 249, 259, 264, 304, 311, 315, 317, 322, 331, 337, Vaisanen, U., 273, 274, 448 347, 351, 418, 428, 429, Valentine, F.A., 10, 478 436, 444, 449, 470, 475, value function, 27, 346 476 Van Hilten, O., 10, 289, 478 Tihomirov, V.M., 444 Van Loon, P.J.J.M., 10, 289, 478 time-optimal control problem, Vanthienen, L., 174, 478 76 Varaiya, P.P., 304, 308, 450, 478 498 Index variational equations, 395 Weinstein, M.C., 279, 479 Veinott, A.F., 418 Weitz, B., 463 Venezia, I., 475 Weizsacker, C.C. von, 479 Verheyen, P.A., 304, 478 Welam, U.P., 479 Verma, B., 445 WeU, K.H., 438 Vickson, R.G., 26, 27, 58, 62, 99, Wensley, R., 463 100, 105, 106, 108, 113, Westphal, L.C., 479 115, 154, 280, 420, 435, wheat trading model with no 442, 460, 478, 480 short-selling, 170 Vidal, R.V.V., 458 Whitin, T.M., 153, 254, 479 Vidale, M.L., 186, 194, 195, 478 Whittle, P., 479 Vidale-Wolfe advertising model, Wickwire, K., 10, 295, 479 194, 353 Wiegand, M., 118, 455 Vilcassim, N.J., 315, 426 Wiener process, 344, 346, 356 Villas-Boas, J.M., 478 Wiener, N., 479 Vincent, T.L., 267, 437 Wind, Y., 430, 447, 454, 475 Vinokurov, V.R., 478 Wirl, F., 434, 480 Vinter, R.B., 103, 108, 324, 434, Wolfe, H.B., 186, 194, 195, 478 463, 472, 478 Wong, K.H., 108, 476 Voelker, J.A., 241, 461 Wonham, W.M., 480 Vousden, N., 303, 452, 479 Wright, C, 303, 480 Wright, S.J., 236, 480 Wagner, H.M., 153, 254, 479 Wunderlich, H.J., 479 Wagner-Whitin framework, 258 Wagner-Whitin solution, 262 Yakowitz, S.J., 236, 458 Wan, F.Y., 473 Yan, H., viii, 469, 480 Wan, Jr., H.Y., 427 Yang, J., 480 Wang, C.-S., 304, 431 Yang, T.H., 108, 462 Wang, P.K.C., 479 Yeh, D., viii warehousing constraint, 175 Yin, G., 360, 462, 468, 470, 471, Warga, J., 479 480 Warnecke, H.J., 479 Young, L.C., 379, 399, 480 Warschat, J., 479 weak forecast horizon, 174, 175 Zaccour, G., 154, 446 weak maximum, 389 Zalkin, J.H., 461 Weierstrass, 9 Zarrop, M.B., 443 Weierstrass necessary condition, Zeckhauser, R.J., 279, 299, 448, 389, 391 479 Weierstrass-Erdmann corner Zeidan, V., 480 conditions, 388 Zemel, E., 457 Index 499 Zhang, H., viii, 462, 463, 470, 471 Zhang, Q., viii, 340, 360, 453, 462, 463, 468, 470, 471, 480 Zhou, X., 466, 471, 480 Ziemba, W.T., 469, 480 Zimin, I.N., 304, 480 Ziolko, M., 480 Zionts, S., 303, 471, 473 Zoltners, A.A., 445, 481 Zowe, J., 103, 450 Zuckermann, D., 475 List of Figures 1.1 The Brachistochrone Problem 9 1.2 A Concave Function 18 1.3 An Illustration of a Saddle Point 19 2.1 An Optimal Path in the State-Time Space 28 2.2 Optimal State and Adjoint Trajectories for Example 2.1 . 38 2.3 Optimal State and Adjoint Trajectories for Example 2.2 . 39 2.4 Optimal Trajectories for Examples 2.3 and 2.4 41 2.5 Optimal Control for Example 2.5 44 2.6 Solution of TPBVP by EXCEL 50 2.7 Water Reservoir of Example 2.12 53 3.1 State and Adjoint Trajectories in Example 3.3 73 3.2 Minimum Time Optimal Response for Problem (3.63) . . 79 4.1 State and Adjoint Trajectories in Example 4.1 102 4.2 Infeasible State Space and Optimal State Trajectory for Example 4.3 109 4.3 Adjoint Trajectory for Example 4.3 Ill 4.4 Two-Reservoir System of Exercise 4.6 116 5.1 Optimal Policy Shown in (Ai, A2) Space 123 5.2 Optimal Policy Shown in (^, A2/A1) Space 124 5.3 Adjoint Variables and Lagrange Multipliers for Example 5.1 128 5.4 Case A: g 6.1 Optimal Production and Inventory Levels 161 6.2 Solution of Example 6.1 with IQ = 10 163 6.3 Solution of Example 6.1 with IQ = 50 164 6.4 Solution of Example 6.1 with Io=^30 165 6.5 The Price Trajectory (6.49) 168 6.6 Adjoint Variable, Optimal Policy and Inventory in the Wheat Trading Model 169 6.7 Adjoint Trajectory and Optimal Policy for the Wheat Trading Model 173 6.8 Decision Horizon and Optimal Policy for the Wheat Trading Model 175 6.9 Optimal Policy and Horizons for the Wheat Trading Model with Warehouse Constraint 177 6.10 Optimal Policy and Horizons for Example 6.3 179 6.11 Optimal Policy and Horizons for Example 6.4 180 6.12 The Flow Chart for Exercise 6.9 183 7.1 Optimal Policies in the Nerlove-Arrow Model 189 7.2 A Case of a Time-Dependent Turnpike and the Nature of Optimal Control 190 7.3 Phase Diagram of System (7.18) for Problem (7.13) .... 192 7.4 Feasible Arcs in (^, a;)-Space 197 7.5 Optimal Trajectory for Case 1: XQ < x^ and x^ > XT - - • 200 7.6 Optimal Trajectory for Case 2: XQ < x^ and x^ < XT - - - 200 7.7 Optimal Trajectory for Case 3: XQ > x^ and x^ > XT - - - 201 7.8 Optimal Trajectory for Case 4: XQ > x^ and x^ < XT • • - 201 7.9 Optimal Trajectory (Solid Lines) 202 7.10 Optimal Trajectory When T Is Small in Case 1: XQ < x^ 8indxT 8.1 Shortest Distance from a Point to a Semi-Circle 224 8.2 Graph of Example 8.5 224 8.3 Kuhn-Tucker Constraint Qualification 226 8.4 Discrete-Time Conventions 229 8.5 Sketch of x^ and A^ 233 List of Figures 503 9.1 Optimal Maintenance and Machine Resale Value 247 9.2 Sat Function Optimal Control 249 10.1 Optimal Policy for the Sole Owner Fishery Model 271 10.2 Singular Usable Timber Volume x{t) 275 10.3 Optimal Policy for the Forest Thinning Model when XQ < x{to) 276 10.4 Optimal Policy for the Chain of Forests Model when T>i 277 10.5 Optimal Policy for the Chain of Forests Model when T f 285 10.9 Optimal Price Trajectory for T < f 285 11.1 Phase Diagram for the Optimal Growth Model 293 11.2 Optimal Trajectory when XT > x^ 298 11.3 Optimal Trajectory when XT 12.1 Region D with Boundaries Fi and r2 316 12.2 A Partition of Region D 320 12.3 Solution of Equation 12.52 323 12.4 Boundary of No-Drilling Region 328 12.5 Drilling Time 329 12.6 Value of-Q + A(t+)[l-x(t)] 330 12.7 Optimal Maintenance Policy 333 12.8 Replacement Time ti and Maintenance Policy 335 12.9 The Case ^1-^2 336 13.1 Autocorrelation Function for a Scalar Process 343 13.2 A Sample Path of Xt with Xo = a;o > 0 and 5 > 0 . . . . 352 B.l Examples of Admissible Functions for the Problem .... 380 B.2 Variation about the Solution Function 381 B.3 A Broken Extremal with Corner at r 387 504 List of Figures C.l Needle-Shaped Variation 394 C.2 Trajectories x*{t) and x{t) in a One-Dimensional Case. . . 394 List of Tables 1.1 The Product ion-Inventory Model of Example 1.1 4 1.2 The Advertising Model of Example 1.2 6 1.3 The Consumption Model of Example 1.3 8 3.1 Summary of the Transversality Conditions 75 3.2 State Trajectories and Switching Curves 78 3.3 Objective, State, and Adjoint Equations for Various Model Types 85 5.1 Characterization of Optimal Controls 135 A.l Examples of Homogeneous Equations of Order Two . . . 364 A.2 General Solution Forms for Second-Order Linear Homogeneous Equations, Constant Coefficients 365 A.3 Examples of Homogeneous E]quations of Order n 366 A.4 General Solution Forms for Multiple Roots of Auxiliary Equation 366 A.5 Particular Solution Forms for Various Forcing Functions . 367 A.6 Particular Integrals in Specific Examples 368 A.7 General Solution Forms for Some Homogeneous Partial Differential Equations 374