Applied Functional Analysis Lecture Notes Spring, 2010

Dr. H. T. Banks Center for Research in Scientiﬁc Computation Department of Mathematics N. C. State University

April 12, 2010 8 Example 6: Cantilever Beam

We introduce another example which is ubiquitous in structural applications and is ideal for use in illustrating our weak formulations that follow.

8.1 The Beam Equation We consider the beam equation given by ∂2y ∂y ∂2 + + M = f(t, ) (1) ∂t2 ∂t ∂2

∂y where the term ∂t represents the external damping (the air or viscous damping), M is the internal moment, is the mass density, and f is the external force applied. For a development of this equation from basic principles, see [BSW, BT]. Before discussing the equation further, we discuss boundary and initial conditions. Boundary Conditions We consider a beam with one end ﬁxed and one end free. Fixed end: y(t, 0) = 0 zero displacement, ∂y ∂ (t, 0) = 0 zero slope. Free end: M(t, l) = 0 zero moment, ∂M ∂ (t, l) = 0 zero shear. ∂M We observe that the shear force is represented by S(t, ) = − ∂ (t, ). Initial Conditions We consider a beam with initial displacement and velocity given by Φ and Ψ, respectively.

y(0, ) = Φ() y˙(0, ) = Ψ()

We observe that we have an equation (1) given with two unknowns; however we couple these with a constitutive relationship given by

M = M(y).

Using Hooke’s law ( = E where is the stress and is the strain, E is the Young’s modulus), we obtain (using basic principles [BT]) for linear elastic systems the relationship ∂2y M = EI ∂2

1 where I represents the moment of inertia of the cross-sectional area. We remark that either E and/or I may be a function of , i.e., E = E() and/or I = I(). We remark that the above derivations use the assumptions of linear elasticity and small displacements. Without small displacements, M could be a nonlinear function of y. For further discussions, see [BSW, BT]. We also have internal damping involved for which we must account. We will assume Kelvin-Voigt damping so that M(t, ) is given by

∂2y ∂3y M(t, ) = EI + c I . (2) ∂2 D ∂2∂t

Combining equations (1) and (2), we have the beam equation given by

∂2y ∂y ∂2 ∂2y ∂3y + + EI + c I = f(t, ), ∂t2 ∂t ∂2 ∂2 D ∂2∂t with the modiﬁed boundary conditions

∂2y ∂3y EI ∂2 + cDI ∂2∂t ∣=l = 0,

h ∂ ∂2y ∂3y i EI 2 + cDI 2 = 0. ∂ ∂ ∂ ∂t =l

8.2 The Beam Equation in the form x˙ = Ax + F We wish to write the beam equation in the formx ˙ = Ax + F . Let x be represented by y(t, ⋅) x(t, ⋅) = y˙(t, ⋅)

2 and take as state space X = HL(0, l) × L2(0, l) where

2 2 ′ HL(0, l) = {' ∈ H (0, l)∣ '(0) = 0,' (0) = 0}.

2 Thus for elements in HL(0, l), the function and the derivative both vanish at the left boundary. We rewrite the equation using the abbreviation ∂ = ∂ ∂ ∂ , ⋅ = ∂t , and have 1 1 y¨ = −∂2(EI∂2y) − ∂2(c I∂2y˙) − y˙ + f. D

2 We rewrite this in ﬁrst order vector form as

d y 0 I y 0 = + 1 . dt y˙ −A −B y˙ f Let 0 I A = −A −B where we deﬁne A and B in the following way:

1 2 2 A = ∂ (EI∂ )

1 2 2 B = ∂ (cDI∂ ) + . Then we have the formx ˙ = Ax + F with

2 D(A) = {(', ) ∈ X∣ ∈ HL(0, l), A' + B ∈ L2(0, l), 2 2 2 2 [EI∂ ' + cDI∂ ]l = 0, [∂(EI∂ ' + cDI∂ )]l = 0}.

8.2.1 A is an Inﬁnitesimal Generator

We claim that A is an infinitesimal generator of a C0 semigroup in Xℰ where Xℰ is an associated energy space. Specifically, we define Xℰ as X with the energy product

R l 2 2 R l ⟨('1, 1), ('2, 2)⟩ℰ = 0 EI∂ '1∂ '2d + 0 1 2d

R l 2 2 = 0 EI∂ '1∂ '2d + ⟨ 1, 2⟩.

If 0 < 1 ≤ () ≤ 2 < ∞, then ⟨ 1, 2⟩ is equivalent to the norm in 2 L2(0, l) where the norm for H (0, l) is deﬁned by

2 2 ′ 2 ′′ 2 ∣'∣H2 = ∣'∣L2 + ∣' ∣L2 + ∣' ∣L2 . An equivalent norm is

2 ′ ′′ 2 ∣'∣∼ = ∣'(a)∣ + ∣' (b)∣ + ∣' ∣L2 ,

2 2 where a,b are any values in [0, l]. In particular, on HL, we can take ∣'∣H2 ≡ ′ ′′ 2 ∣'(0)∣ + ∣' (0)∣ + ∣' ∣L2 . This deﬁnes a Hilbert space which is topologically equivalent to X so that we thus have that if A generates a C0 semigroup in Xℰ , it is also a C0 semigroup in X.

3 8.2.2 Poincare (First) Inequality To argue that A is a generator, we need a standard and well-known [A] inequality given by Suppose there exists a set Ω which is a bounded subset of Rn. Then there l,2 l exists a c = c(Ω), such that for all ' ∈ W0 (Ω) = H0(Ω), one has Z 2 X s 2 ∣'∣Hl(Ω) ≤ c(Ω) ∣∂ '∣ ∣s∣=l Ω where s = (s1, . . . , sn).

8.2.3 Dissipativeness of A

2 2 We argue that A is dissipative in Xℰ . Recalling that M = EI∂ '+cDI∂ , we have Z l Z l 2 2 2 2 2 ⟨A(', ), (', )⟩ℰ = EI∂ ∂ ' + −∂ (EI∂ ' + cDI∂ ) − 0 0 Z l Z l Z l 2 2 2 2 2 2 = EI∂ ∂ ' − ∂ (EI∂ ' + cDI∂ ) − ) 0 0 0 Z l Z l 2 2 2 2 2 2 = EI∂ ∂ ' − (EI∂ ' + cDI∂ )∂ + 0 0 l l − ∂M ∣0 + M∂ ∣0 Z l Z l 2 2 2 = − cDI∣∂ ∣ − ∣ ∣ 0 0 2 ≤ −k∣ ∣H2(0,l) ≤ 0 for k =min {∣cDI∣, ∣ ∣}. Here we have integrated by parts twice and used the fact that (', ) ∈ D(A) so that (0) = ∂ (0) = 0 and M(t, l) = ∂M(t, l) = 0.

8.2.4 ℛ(I − A) = X for some To verify the range statement of Lumer Phillips, we need to show for some that (', ) − A(', ) = (ℎ, g) (3)

4 can be solved for (', ) for a given (ℎ, g) ∈ X. However, (3) reduces to 2 ﬁnding, for any (ℎ, g) ∈ X = HL(0, l) × L2(0, l), a solution (', ) in D(A) that satisﬁes − + ' = ℎ (4) A' + B + = g (5) 2 2 where ℎ ∈ HL and g ∈ L . We can rewrite (4) to obtain

= ' − ℎ (6)

By substituting (6) into (5), we can reduce the above question to one of solving 2' + A' + B' = g + ℎ + Bℎ (7) 2 for ', given any (ℎ, g) in HL(0, l)×L2(0, l). After solving for ', we can then obtain using (6). If we obtain (', ) ∈ D(A), then we have proven the range statement of Lumer-Phillips. To complete the argument we will use the concepts and results for sesquilinear forms to be discussed below.

5 9 Gelfand Triple and Sesquilinear Forms

An easy way to solve the above problem is to use the Lax-Milgram theorem; however, we ﬁrst introduce Gelfand triples. For relevant material, see also [Sh, T, W].

9.1 Concept of Gelfand Triple The usual notation for a Gelfand triple is “V,→ H,→ V ∗ with pivot space H”. This notation stands for V,H, complex Hilbert spaces, such that V ⊂ H and V is densely and continuously embedded in H. That is, V is a dense subset of H and ∣v∣H ≤ k∣v∣V for all v ∈ V and some constant k. Therefore, you can identify elements in V with elements in H with an injection operator, i, where i is continuous and i(V ) is a dense subset of H. We denote by V ∗ the conjugate dual of V . That is, V ∗ consists of all conjugate linear continuous functionals on V . (Note that we use V ′ to denote the algebraic dual of V and V ∗ to denote the topological dual. Frequently one encounters exactly the opposite notation in the literature.) For ℎ ∈ H, we deﬁne '(ℎ) ∈ V ∗ by

'(ℎ)(v) = ⟨ℎ, v⟩H for v ∈ V . We claim that ' : H → '(H) ⊂ V ∗ is continuous, linear, one-to-one and onto. Moreover, '(H) is dense in V ∗ in the V ∗ topology. By the Riesz theorem, every ℎ∗ ∈ H∗ can be represented by ∗ ˆ∗ ℎ (ℎ) = ⟨ℎ , ℎ⟩H ℎ ∈ H for some ℎˆ∗ ∈ H. Hence, it is readily argued that H∗ is isomorphic to H, H∗ ≃ H. That is, we may identify H∗ with H through '(H) = H˜ ≃ H∗ ∗ and write ℎ(v) = ⟨ℎ, v⟩H for ℎ ∈ H = H . This construction is commonly written as V,→ H =∼ H∗ ,→ V ∗ for the pivot space H.

9.2 Duality Pairing With a Gelfand triple, one frequently utilizes the duality pairing denoted by ⟨⋅, ⋅⟩V ∗,V given by the extension by continuity of the H inner product from H × V to V ∗ × V . That is, for v∗ ∈ V ∗, ∗ ∗ v (v) = ⟨v , v⟩V ∗,V = lim⟨ℎn, v⟩H n

6 ∗ ∗ where ℎn ∈ H, ℎn → v in V . Note that we have ⟨ℎ, v⟩V ∗,V = ⟨ℎ, v⟩H if ℎ ∈ V ∗ also satisﬁes ℎ ∈ H.

9.3 Sesquilinear Forms

Deﬁnition 1 Let H1 and H2 be two complex Hilbert spaces, and let : H1 × H2 → C. Then we call a sesquilinear form if it satisﬁes 1. is linear/conjugate linear;

2. is continuous. More precisely, there exists > 0 such that

∣(x, y)∣ ≤ ∣x∣1∣y∣2

for x ∈ H1 and y ∈ H2.

9.4 Norm of a Sesquilinear Form The norm of a sesquilinear form is deﬁned by

∣(x, y)∣ ∣∣ = sup x,y∕=0 ∣x∣1∣y∣2 for x ∈ H1 and y ∈ H2.

9.5 Representation We consider two diﬀerent representations:

1. For y ∈ H2, x → (x, y) is continuous and linear on H1. Therefore, by the Riesz Theorem, there exists a unique z ∈ H1 such that

(x, y) = ⟨x, z⟩H1 .

Thus there exists a mapping B ∈ ℒ(H2,H1) deﬁned by By = z. There- fore,

(x, y) = ⟨x, z⟩H1 = ⟨x, By⟩H1 .

We ﬁnd that ∣∣ = ∣B∣ℒ(H2,H1). To see this we argue as follows.

7 For x in H1 and y in H2, we have

∣(x,y)∣ ∣⟨x,By⟩H1 ∣ ∣∣ = sup ∣x∣ ∣y∣ = sup ∣x∣ ∣y∣ x,y∕=0 1 2 1 2

∣x∣1∣By∣1 ∣By∣1 ≤ sup ∣x∣ ∣y∣ = sup ∣y∣ 1 2 y∕=0 2

= ∣B∣ℒ(H1,H2).

Therefore, we have ∣∣ ≤ ∣B∣ℒ(H2,H1). Thus we need only argue that

∣∣ ≥ ∣B∣ℒ(H2,H1). However, we have

∣(x,y)∣ ∣⟨x,By⟩H1 ∣ ∣∣ = sup ∣x∣ ∣y∣ = sup ∣x∣ ∣y∣ x,y∕=0 1 2 1 2

∣⟨By,By⟩H ∣ ∣By∣1 ≥ sup ∣By∣ ∣y∣ = sup ∣y∣ x=By 1 2 2

= ∣B∣ℒ(H2,H1).

Hence we have ∣∣ = ∣B∣ℒ(H2,H1).

2. For x ∈ H1, y → (x, y) is continuous and antilinear on H2, i.e., ∗ (x, ⋅) ∈ H2 . Therefore, by the Riesz Theorem, there exists a unique z ∈ H2 such that

(x, y) = ⟨z, y⟩H2 .

Thus there exists a mapping A ∈ ℒ(H1,H2) deﬁned by Ax = z. There- fore,

(x, y) = ⟨z, y⟩H2 = ⟨Ax, y⟩H2 , with ∣A∣ = ∣∣. Deﬁne ˜(y, x) = (x, y), and interchange the roles of H1 and H2 in the above arguments to argue ∣A∣ = ∣∣.

We wish use our Gelfand triple and sesquilinear form formulations to consider elliptic equations of the form Ax = f. However, ﬁrst, we begin by discussing operators of the form A : H1 → H2 such that A is bounded and continuous. This is very useful in integral equations. However, we will need to modify this formulation to account for the unbounded nature of PDE’s. We can think of our equation Ax = f in the form (x, y) = ⟨f, y⟩H for all y in H where is equivalent to A. In other words, ⟨Ax − f, y⟩H = 0 for all y in H. The useful results are given in the famous Lax-Milgram theorems, in bounded and unbounded formulations.

8 10 Lax-Milgram(bounded form)

Theorem 1 (Lax-Milgram Theorem (bounded form)) Suppose : H ×H → C is continuous and linear/conjugate linear, i.e., it is a sesquilinear form with ∣(x, y)∣ ≤ ∣x∣H ∣y∣H for all x and y in H, and is strictly positive, i.e.,

∣(x, x)∣ ≥ ∣x∣2 for all x in H where and are positive constants. Then there exists A : H → H,A ∈ ℒ(H,H) deﬁned by

(x, y) = ⟨Ax, y⟩ with ∣A∣ = ∣∣ ≤ for all x and y in H. Moreover,

(A−1x, y) = ⟨x, y⟩

−1 1 with ∣A ∣ ≤ for all x and y in H.

Proof of Theorem By the Riesz Theorem, we know there exists A ∈ ℒ(H) such that ∣A∣ = ∣∣ ≤ where (x, y) = ⟨Ax, y⟩H for all x and y in H. We claim that A is one-to-one. For this we need to show that Ax = 0 implies x = 0. Suppose Ax = 0. Then, by the deﬁnition of A and the assumptions given, we have

0 = ∣⟨Ax, x⟩∣ = ∣(x, x)∣ ≥ ∣x∣2.

Therefore, ∣x∣2 ≤ 0 implies ∣x∣ = 0, and hence x = 0. In other words, A is one-to-one. Since A is one-to-one, we know that A−1 exists on ℛ(A) ⊂ H. We next claim that A−1 is bounded on ℛ(A). By the proposition that was strictly positive and the deﬁnition of A, we have

∣x∣2 ≤ ∣(x, x)∣ = ∣⟨Ax, x⟩∣ ≤ ∣Ax∣∣x∣.

Therefore, ∣Ax∣ ≥ ∣x∣ for all x in H. In particular, for x = A−1y, we have

∣y∣ ≥ ∣A−1y∣.

9 Therefore, 1 ∣A−1y∣ ≤ ∣y∣ for all y in ℛ(A). In other words, A−1 is bounded. Finally all we have to show is that ℛ(A) = H. By assumption, we know that A is a continuous operator and we have argued that A−1 is bounded on ℛ(A). Therefore, ℛ(A) is closed. Now suppose ℛ(A) ∕= H. This implies there exists z ∕= 0 such that z ⊥ ℛ(A). In other words, ⟨Ax, z⟩ = 0 for all x in H. In particular, choosing x = z, we have 0 = ⟨Az, z⟩ = (z, z) ≥ ∣z∣2. This implies z = 0; therefore, we have a contradiction. So, ℛ(A) = H.

10.1 Discussion of Ax = f with A bounded The bounded form of Lax-Milgram is very useful in linear integral equations. The Fredholm integral equations with kernel k ∈ L2([a, b] × [a, b]) are given by Z b k(x, y)'(y)dy = f(x) x ∈ [a, b] (first kind) a Z b '(x) − k(x, y)'(y)dy = f(x) x ∈ [a, b] (second kind). a These can be written as operator equations in H = L2(a, b), A' = f ' − A' = f to be solved for ', given f, where A is a bounded linear operator (discussed below) in H. It also has applications in scattering theory (acoustic and electromagnetic radiation), far field radiation, and single and double layer potential theory. For relevant material, see also [K], [CK1], and [CK2]. As we have noted, the bounded form of Lax-Milgram is adequate if one wants to solve Ax = f in H where A is bounded. Consider k ∈ L2(Ω) with Ω = [0, 1] × [0, 1] with H = L2(0, 1). Then define A : H → H by Z 1 (A')(t) = k(t, )'()d. 0

10 We can see that this operator is bounded, i.e,

Z 1 Z 1 2 Z 1 k(t, )'()d dt < ∣'()∣2d. 0 0 0 Therefore, there is a sesquilinear form such that A corresponds as in the Lax-Milgram above with

R 1 R 1 (', ) = 0 0 k(t, )'()d (t)dt

= ⟨A', ⟩L2(0,1). We can show that all the requirements of Lax-Milgram (bounded form) are met with this operator A; therefore, Lax-Milgram (bounded form) is applicable.

10.2 Example - The steady state heat equation Let Ω = [0, 1] × [0, 1]. Then the heat equation is given by ∂u = ∇ ⋅ (D∇u) + f ∂t where ∇ = ∂ ˆi + ∂ ˆj with ( , ) ∈ Ω. We assume boundary conditions ∂1 ∂2 1 2 1 that require u ∈ H0 (Ω) Then the steady state equation is given by −∇ ⋅ (D∇u) = f (8) or ∂ ∂u ∂ ∂u (D ) + (D ) = f(1, 2). ∂1 ∂1 ∂2 ∂2 In the weak or variational form, we have (by integration by parts and use of the boundary conditions)

⟨D∇u, ∇'⟩L2 = ⟨f, '⟩L2 (9)

1 1 1 for a test function ' ∈ H0 (Ω). We deﬁne on H0 × H0 by ( , ') =

⟨D∇ , ∇'⟩L2 with ∣D∣ ≤ , i.e., D is bounded. Then is not continuous 1 on H = L2(Ω). On the other hand, for ', ∈ H0

∣( , ')∣ ≤ ∣∇ ∣L2 ∣∇'∣L2

≤ ∣ ∣ 1 ∣'∣ 1 H0 H0

11 1 implies is continuous on V = H0 (Ω). Moreover, we do have some type of positivity. If ∣D(1, 2)∣ ≥ , then

∣(', ')∣ = ∣⟨D∇', ∇'⟩∣

2 ≥ ∣∇'∣ ≥ ˜∣'∣ 1 . L2 H0 (Ω)

1 In other words, in the V = H0 (Ω) norm, we would have both continuity and strict positivity of the sesquilinear form. But we don’t have continuity and strict positivity in the H = L2(Ω) sense. Thus if we choose our H space to 1 be H (Ω), we would be guaranteed a unique solution of ⟨Au − f, '⟩ 1 = 0. 0 H0 −1 1 1 −1 That is, u = A f in the H0 sense (and of course u ∈ H0 since A is 1 1 bounded from H0 to H0 . This also requires the input function f to be 1 in H0 (Ω) which is a strong requirement for f. Therefore, it would not be 1 useful in some cases to choose our H space to be H0 (Ω) in the Lax-Milgram theorem. Instead, we need an extension of Lax-Milgram to treat the heat equation in a more reasonable manner.

12 11 Lax-Milgram (unbounded form)

Let V,→ H,→ V ∗ be a Gelfand triple.

Deﬁnition 2 A sesquilinear form is said to be V -continuous if

∣(', )∣ ≤ ∣'∣V ∣ ∣V for all ' and in V .

Consequences of V -continuity For a fixed ' in V , we consider the mapping → (', ) for a V -continuous . This mapping is continuous by the definition of continuity above, and its a conjugate linear mapping into C. Therefore, → (', ) is in V ∗. This implies there exists an operator A ∈ ℒ(V,V ∗) such that (', ) = ⟨A', ⟩V ∗,V . Conversely, if A ∈ ℒ(V,V ∗), we can define : V × V → C by (', ) = (A') so that is V -continuous and linear/conjugate linear. In other words, if we have a V -continuous sesquilinear form , there is a one-to-one correspondence between and A ∈ ℒ(V,V ∗). While the operator is not bounded in V , it will be useful in considering the equation Au = f in V ∗ will be useful. (Note the weak requirements on f in this case.)

Deﬁnition 3 A sesquilinear form is V -coercive if there exists a constant > 0 such that 2 ∣(', ')∣ ≥ ∣'∣V for ' ∈ V .

Theorem 2 (Lax-Milgram Theorem (unbounded form)) Let V,→ H,→ V ∗ be a Gelfand triple. Let : V × V → C be a V -continuous, V -coercive sesquilinear form. Then A : V → V ∗ given by

(', ) = ⟨A', ⟩V ∗,V is a linear (topological) isomorphism between V and V ∗. Moreover, A−1 is continuous from V ∗ to V with

−1 1 ∣A ∣ ∗ ≤ . ℒ(V ,V )

13 Proof of Theorem Let R : V ∗ → V be a Riesz isomorphism, so that for v∗ ∈ V ∗ we have

∗ ∗ ∗ v (v) = ⟨v , v⟩V ∗,V = ⟨Rv , v⟩V . (10)

Then R−1 : V → V ∗ is continuous. Let : V × V → C be a V -continuous, V -coercive sesquilinear form. Then → (', ) in V ∗ implies there exists a z in V such that (', ) = ⟨z, ⟩V . From the bounded version of Lax-Milgram, this implies there exists A ∈ ℒ(V ) such that (', ) = ⟨A', ⟩V (11) −1 for all ' and in V , with A one-to-one, A onto, ∣A∣ℒ(V ) ≤ and ∣A ∣ℒ(V ) ≤ 1 . In other words, A is an isomorphism V → V . We have that R−1 : V → V ∗ is also an isomorphism; therefore, R−1A : V → V ∗ is an isomorphism. The claim is that A = R−1A. By (10) we have for all ', ∈ V −1 ⟨R A', ⟩V ∗,V = ⟨A', ⟩V . However, by (11), this implies

−1 ⟨R A', ⟩V ∗,V = (', ).

Therefore, by deﬁnition of A : V → V ∗, A must be given by A = R−1A. So, we have ⟨A', ⟩V ∗,V = (', ) ≤ ∣'∣V ∣ ∣V (12) and 2 ⟨A', '⟩V ∗,V ≥ ∣'∣V . (13) −1 Recall that sup∣ ∣=1 ∣⟨A', ⟩V ∗,V ∣ = ∣A'∣V ∗ . Letting ' = A in (13) and combining this with (12) and (13), we have

∣'∣V ≤ ∣A'∣V ∗ ≤ ∣'∣V . (14)

Therefore, we have ∣A∣ℒ(V,V ∗) ≤ and −1 1 ∣A ∣ ∗ ≤ . ℒ(V ,V )

14 Implications of Lax-Milgram(unbounded form) For A as in the Lax-Milgram Theorem 2 and f ∈ V ∗, we consider Au = f in V ∗. Then Lax-Milgram implies there exists a unique solution u = A−1f in V that depends continuously on f. More precisely, −1 ∣u∣V = ∣A f∣V ≤ ∣f∣V ∗ . We revisit the steady-state heat equation (8) in the example above with D ∈ ∞ 1 L (Ω) and V = H0 (Ω). This, of course, allows discontinuous coeﬃcients. Then for any f in V ∗ = H−1(Ω) there exists a unique u ∈ V satisfying

∗ ∗ Au = f. That is, ⟨Au − f, '⟩V ,V = (u, ') − ⟨f, '⟩V ,V = ⟨D∇u, ∇'⟩L2 − ⟨f, '⟩V ∗,V = 0 for all ' ∈ V . Thus we say u ∈ V satisﬁes Au = f in the sense of V ∗. Hence (8) holds in the V ∗ sense which is precisely (9). This is also sometimes referred to as u satisfying (8) in the “sense of distributions.”

11.1 The concept of DA We assume that we have a Gelfand triple V,→ H,→ V ∗ and a continuous sesquilinear form : V × V → C. As usual, f( ) = ⟨f, ⟩H deﬁnes, for f ∈ H, an element f ∈ V ∗. Considering (A')( ) = (', ) for ' and in V , we deﬁne DA = {' ∈ V ∣A' ∈ H}. ∗ That is, DA is the set of ' ∈ V such that A' ∈ V has the representation (A')( ) = ⟨',˜ ⟩H , for in V and for some' ˜ in H. We denote this element' ˜ by −A' =' ˜, i.e., A is linear from DA ⊂ V into H and given by (', ) = ⟨−A', ⟩H for in V and ' in DA. We note that the above can be interpreted as: ' ∈ DA ⊂ V if and ∗ ∼ only if ' ∈ V and A' ∈ H = H so that (A')( ) = ⟨',˜ ⟩H , for all in V , and for some' ˜ ∈ H. Alternatively, we may write DA = {' ∈ V ∣∣(A')( )∣ = ∣(', )∣ ≤ k∣ ∣H , ∈ V } for some k. Moreover, we could ∗ write DA = {' ∈ V ∣ → (', ) is in H , i.e, continuous on H (continuous in the H sense) }. Note that we also have

(', ) = (A')( ) = ⟨A', ⟩V ∗,V for all ' and in V . If we restrict ' ∈ DA then this also equals ⟨−A', ⟩H . Theorem 3 If is a V -continuous V -coercive sesquilinear form on V , then DA is dense in V and, hence, dense in H.

15 Proof of Theorem Deﬁne ˜(', ) = ( , ') (called the “adjoint” sesquilinear form) If is V -coercive and V continuous, then ˜ is also V -coercive and V continuous. In other words, there exists an operator A˜ : V → V ∗ such that A˜ ∈ ℒ(V,V ∗) with

˜(', ) = ⟨A˜', ⟩V ∗,V = (A˜')( ) for all ' and in V . Hence, ℛ(A˜) = V ∗ by the Lax-Milgram theorem.

Next we need to show that DA is dense in V . It suﬃces to show that if ∗ f ∈ V and f(v) = 0 for all v ∈ DA, then f ≡ 0. ∗ ∗ ∗ Let f ∈ V such that f(v) = 0 for all v ∈ DA. Since f ∈ V and ℛ(A˜) = V , then there exists ' ∈ V such that f = A˜'. For v ∈ DA, we have

(−Av)(') = (Av)(') = (v, ')

= ˜(', v) = ⟨A˜', v⟩V ∗,V

= ⟨f, v⟩V ∗,V = f(v)

= 0.

Therefore, (Av)(') = 0 for all v ∈ DA. ∼ ∗ ∗ But, we know ℛ(A∣DA ) = H = H . Then, for every ℎ ∈ H , ℎ(') = 0 . However, as H∗ is dense in V ∗, then this implies ' = 0. Moreover, A˜' = f implies f = 0.

Thus we ﬁnd that any V -continuous V -coercive sesquilinear form on V gives rise to a densely deﬁned operator A on D(A) = DA with

(', ) = ⟨A', ⟩V ∗,V ', ∈ V = ⟨−A', ⟩H ' ∈ DA, ∈ V.

11.2 V-elliptic Deﬁnition 4 A sesquilinear form on V is V -elliptic if there exists a constant > 0 such that

2 Re (', ') ≥ ∣'∣V ' ∈ V.

16 Discussion of V -elliptic We note that is V -elliptic implies is V -coercive. Of course, if we are working in real spaces, Re = and V -elliptic is equivalent to V -coercive. We also remark that the terminology among various authors is not standard. Some authors (e.g., Wloka in [W]) use our deﬁnition for V -coercive as the deﬁnition of V -elliptic and then use

2 2 ∣(', ') + k∣'∣H ∣ ≥ ∣'∣V ' ∈ V, > 0 as the deﬁnition of V -coercive. Some of the terminology and usage of sesquilinear forms derives directly from that for PDE’s of the form ∂y X ∂ ∂y X ∂y = (a ) + b , t > 0, ∈ G ⊂ Rn. ∂t ∂ ij ∂ j ∂ i,j i j j j

Deﬁnition 5 In classical PDE’s, an “operator” {aij} is said to be strongly elliptic on G if there exists > 0 such that for ∈ G

X X 2 Re aij()qiq¯j ≥ ∣qi∣ i,j i for all q ∈ Cn.

An associated sesquilinear form on V = H1(G) can be deﬁned by ⎡ ⎤ Z ¯ X ∂' ∂ X ∂' ¯ (', ) = ⎣ aij() + bk ⎦ d ∂i ∂j ∂k G i,j k

Discussion of Strongly elliptic

It is a standard result that {aij} strongly elliptic implies there exists > 0 such that for suﬃciently large

2 2 Re (', ') + ∣'∣H ≥ ∣'∣V ,' ∈ V.

Hence if {aij} is strongly elliptic, then for some 0 suﬃciently large,

˜(', ) = (', ) + 0⟨', ⟩H is V -elliptic and hence V -coercive. A most useful result is that continuous V -elliptic forms give rise to operators that are inﬁnitesimal generators of C0 (actually analytic) semigroups.

17 12 Some Useful Theorems

Deﬁnition 6 A semigroup T (t) on H is called an analytic semigroup if t → T (t)' is analytic for each ' in H.

Theorem 4 Let V , H be complex Hilbert spaces with V,→ H,→ V ∗ and suppose that : V × V → C is continuous and V -elliptic; i.e.

∣(', )∣ ≤ ∣'∣V ∣ ∣V ', ∈ V, and 2 Re (', ') ≥ ∣'∣V > 0,' ∈ V. Deﬁne A : D(A) ⊂ V → H by

D(A) = {' ∈ V ∣ there exists K' > 0 such that (', ) ≤ K'∣ ∣H , ∈ V } and (', ) = ⟨−A', ⟩H ,' ∈ D(A), ∈ V. Then D(A) is dense in H and A is the inﬁnitesimal generator of a contraction semigroup in H that actually is an analytic semigroup. The proof of this theorem will come later.

Theorem 5 Suppose all the assumptions of Theorem 4 hold except that the V -ellipticity condition for is replaced by

2 2 Re (', ') + 0∣'∣H ≥ ∣'∣V for some 0, > 0,' ∈ V . Then defining A as in Theorem 4, we have that A is densely defined and is the infinitesimal generator of an analytic semigroup in H.

12.1 Return to Example 1 The system is given by

∂y ∂ ∂y = D() ∂t ∂ ∂

∂y y(t, 0) = 0, ∂ (t, l) = 0

y(0, ) = Φ().

18 We choose the state space X = L2(0, l) as before. To obtain the weak variational form and the space V , we work backwards by multiplying the equation by a ”test” function ' and integrating.

R l R l ′ ′ 0 y'˙ = 0(Dy ) '

R l ′ ′ ′ l = 0 −Dy ' + Dy '∣0 Therefore we have

′ ′ ′ l ⟨y˙(t),'⟩ + ⟨Dy (t),' ⟩ − Dy (t)'∣0 = 0 (15) where ⟨⋅, ⋅⟩ is the usual inner product in L2. However, (15) is equivalent to

⟨y˙(t),'⟩ + ⟨Dy′(t),'′⟩ = 0

1 1 ′ if ' ∈ HL(0, l) = {' ∈ H (0, l)∣'(0) = 0} and Dy (t, l) = 0. 1 Deﬁning V = HL(0, l) and on V × V by

(', ) = ⟨D'′, ′⟩, we may write the equation in weak form as: ﬁnd y(t) ∈ V satisfying

⟨y˙(t),'⟩ + (y(t),') = 0 for all ' ∈ V . This equation is equivalent to the original system whenever y(t) ∈ V T H2(0, l) by using the reverse of the above arguments. Next we consider the ﬂux boundary condition of the original problem. Suppose y(t) is a weak solution, i.e.,

⟨y˙(t),'⟩ + (y(t),') = 0 ∀ ' ∈ V

y(0) = Φ() and y in H2(0, l). Then

Z l ⟨y˙(t),'⟩ + Dy′'′ = 0. 0 Integrating by parts, we ﬁnd that the above equation is equivalent to

Z l (y ˙ − (Dy′)′)' + Dy′(t, l)'(l) = 0 (16) 0

19 1 1 1 for all ' ∈ HL. However, H0 ⊂ HL; therefore, Z l (y ˙ − (Dy′)′)' = 0 (17) 0

1 1 ′ ′ for all ' ∈ H0 . Since H0 is dense in L2(0, l), this impliesy ˙ − (Dy ) = 1 0. However, if we choose ' ∈ HL such that '(l) ∕= 0, then (16) implies Dy′(t, l) = 0, i.e., the flux boundary condition is satisfied. If we define the V -inner product as

Z l ′ ′ ⟨', ⟩V = ' , 0

∗ and set H = X = L2(0, l), then we readily see V,→ H,→ V . Note that 1 1 the V norm is equivalent to the usual H norm on HL(0, l). Furthermore, we have ∣(', )∣ = ∣⟨D'′, ′⟩∣

′ ′ ≤ ∣D∣∞∣' ∣L2 ∣ ∣L2

= ∣D∣∞∣'∣V ∣ ∣V . Also, ′ ′ ′ 2 2 Re (', ') = Re ⟨D' ,' ⟩ ≥ ∣' ∣L2 = ∣'∣V so that is bounded and V -elliptic. We can deﬁne A : V → V ∗ by

′ ′ ⟨A', ⟩V ∗,V = (', ) = ⟨D' , ⟩.

Note that A' ∈ H,→ V ∗ if and only if ⟨D'′, ′⟩ = ⟨w, ⟩ for all ∈ V for some w ∈ H. However, integrating by parts we have

R l ′ ′ R l ′ ′ ′ l 0 D' = − 0(D' ) + D' ∣0

= ⟨−(D'′)′, ⟩ + D(l)'′(l) (l)

= ⟨−(D'′)′, ⟩

′ ′ ′ if ' (l) = 0 and (D' ) ∈ L2(0, l). Thus we may deﬁne

A' = (D'′)′

20 on 1 ′ ′ ′ D(A) = {' ∈ HL(0, l)∣(D' ) ∈ L2(0, l),' (l) = 0} and obtain A' = −A' ∈ H exactly whenever ' ∈ D(A). The above results hence guarantee that A generates a C0-semigroup (actually an analytic semigroup) T (t) on H = X = L2(0, l).

12.2 Return to Example 2 We consider again the transport equation given by

∂y ∂ ∂ ∂y ∂t + ∂ (y) = ∂ (D ∂ ) − y y(t, 0) = 0 ∂y (D ∂ − y)∣=l = 0 y(0, ) = Φ().

We can rewrite the transport equation by

′ ′ yt = (Dy − y) − y.

Multiplying by a test function and integrating from 0 to l, we have

R l ′ ′ ⟨yt,'⟩ = 0 ((Dy − y) ' − y') d

′ ′ ′ l = −⟨Dy − y, ' ⟩ + (Dy − y)'∣0 − ⟨y, '⟩.

1 If we choose H = X = L2(0, l) and V = HL(0, l) as in Example 1 above, with the same V inner product, we have

′ ′ ⟨yt,'⟩ = −⟨Dy − y, ' ⟩ − ⟨y, '⟩.

As before, we have V,→ H,→ V ∗. Then we can deﬁne the sesquilinear form : V × V → C by

(', ) = ⟨D'′ − ', ′⟩ + ⟨', ⟩.

Therefore, we have the equation

⟨y,˙ '⟩ + (y, ') = 0.

Briefly, we discuss the various possibilities for boundary conditions and the effects on the choice of V . If we had a no flux boundary condition at 1 = 0, we would choose V = HR(0, l). On the other hand, if we had essential

21 boundary conditions at both boundaries, i.e., y = 0 at = 0 and = l, we 1 would need to choose V = H0 (0, l). A third possibility is if we had the no ﬂux boundary conditions at both boundaries, = 0 and = l. In that case, as both boundary conditions were natural, we would choose V = H1(0, l). The V -continuity of is established by arguing

′ ′ ′ ∣(', )∣ ≤ ∣D∣∞∣' ∣H ∣ ∣H + ∣∣∞∣'∣H ∣ ∣H + ∣∣∞∣'∣H ∣ ∣H

2 ≤ ∣D∣∞∣'∣V ∣ ∣V + ∣∣∞k∣'∣V ∣ ∣V + ∣∣∞k ∣'∣V ∣ ∣V

2 = (∣D∣∞ + k∣∣∞ + k ∣∣∞)∣'∣V ∣ ∣V . As is V -continuous, we have

(', ) = ⟨A', ⟩V ∗,V ' ∈ V = ⟨−A', ⟩H ' ∈ D(A), where D(A) is deﬁned by

D(A) = {' ∈ H2(0, l)∣'(0) = 0, (D'′ − ') ∈ H1(0, l), (D'′ − ')(l) = 0}.

Note that V carries the essential boundary conditions, while the natural boundary conditions are found in D(A). To show that is V -coercive, if we assume D ≥ c1 > 0 and ⟨', '⟩ ≥ 2 −∣∣∞∣'∣H , then we have

2 2 ∣∣∞ 2 2 2 Re (', ') ≥ c1∣'∣V − 4 ∣'∣H − ∣'∣V − ∣∣∞∣'∣H

2 2 ∣∣∞ 2 = (c1 − )∣'∣V − ( 4 + ∣∣∞)∣'∣H .

c1 Hence, setting = 2 , we have c Re (', ') ≥ 1 ∣'∣2 − ∣'∣2 2 V 0 H

2 for = ∣∣∞ + ∣∣ . Thus we see that ˜ given by 0 2c1 ∞

˜(', ) = (', ) + 0⟨', ⟩ = ⟨−A', ⟩ + 0⟨', ⟩ = ⟨−(A − 0)', ⟩ is V -elliptic (indeed it is V coercive). We thus ﬁnd that A − 0, and hence A, is the generator of an analytic semigroup in H = X = L2(0, l).

22 12.3 Return to Example 6 We return to the beam equation. Recall the system is given by

2 ytt + yt + ∂ M = f 0 < < l, with ∂y y(t, 0) = 0 = (t, 0) ∂ M(t, l) = 0 = ∂M(t, l), 2 2 where M(t, ) = EI∂ y + cDI∂ yt. We choose as our basic space H =

L2(0, l) with the weighted inner product ⟨', ⟩H = ⟨', ⟩L2(0,l). Then the weak form becomes EI c I 1 ⟨y + y ,'⟩ + ⟨ ∂2y, ∂2'⟩ + ⟨ D ∂2y , ∂2'⟩ = ⟨ f, '⟩ tt t H H t H H

2 for all ' ∈ V = HL(0, l). We choose the weighted inner product for V given R l ′′ ′′ by ⟨', ⟩V = 0 EI' . We deﬁne the sesquilinear forms 1 and 2 on V × V → C by

Z l EI ′′ ′′ ′′ ′′ 1(', ) = ⟨ ' , ⟩H = EI' 0 I (', ) = ⟨c '′′, ′′⟩ + ⟨ ', ⟩ . 2 D H H The weak form of the equation is then f ⟨y ,'⟩ + (y, ') + (y ,') = ⟨ ,'⟩ tt H 1 2 t H for ' ∈ V . To write this in ﬁrst order vector form, we use the state space ∗ XE = ℋ = V × H with the space V = V × V , noting that V,→ H,→ V and V ,→ ℋ ,→ V∗ form Gelfand triples, where V∗ = V × V ∗.

Homework Exercises ∙ Ex. 9 : Explain why we have V∗ = V ×V ∗ in the Gelfand triple instead of V∗ = V ∗ × V ∗.

23 We deﬁne the sesquilinear form : V × V → C by (for = (', ), = (g, ℎ) in V)

( , ) = ((', ), (g, ℎ)) = −⟨ , g⟩V + 1(', ℎ) + 2( , ℎ).

Using the state variable w(t) = (y(t, ⋅), yt(t, ⋅)) in XE = ℋ, we can rewrite the equation as

⟨w˙ (t), ⟩ℋ + (w(t), ) = ⟨F (t), ⟩ℋ

1 for ∈ V, where F (t) = (0, f(t)). We readily argue that is bounded (continuous) and V-elliptic (actually, − ∣ ⋅ ∣2 is V-elliptic). Consider ﬁrst the boundedness argument: 0 XE

∣( , )∣ = ∣((', ), (g, ℎ))∣ = ∣ − ⟨ , g⟩V + 1(', ℎ) + 2( , ℎ)∣

≤ ∣ ∣V ∣g∣V + 1∣'∣V ∣ℎ∣V + 2∣ ∣V ∣ℎ∣V ≤ ∣ ∣V ∣∣V + 1∣ ∣V ∣∣V + 2∣ ∣V ∣∣V

= (1 + 1 + 2)∣ ∣V ∣∣V for , ∈ V. The arguments for V-ellipticity are also simple: for = (', ) ∈ V we ﬁnd

Re ( , ) = Re {−⟨ , '⟩V + 1(', ) + 2( , )}

= Re {−⟨', ⟩V + ⟨', ⟩V + 2( , )}

= Re 2( , )

2 ≥ 2∣ ∣V

2 2 2 = 2(∣'∣V + ∣ ∣V ) − 2∣'∣V

2 2 2 2 ≥ 2(∣'∣V + ∣ ∣V ) − 2(∣'∣V + ∣ ∣H )

2 2 = 2∣ ∣V − 2∣ ∣ℋ.

We thus ﬁnd that ( , ) = ⟨A˜ , ⟩V,V∗ gives rise to the inﬁnitesimal generator A of a C0 (indeed, analytic) semigroup on XE = ℋ. It is readily argued that ( , ) = ⟨−A , ⟩ℋ for ∈ D(A) = { = (', ) ∈ ℋ∣ ∈ V =

24 2 ′′ ′′ ′′ ′′ ′ HL(0, l),A1' + A2 ∈ H, (EI' + cDI )(l) = 0, (EI' + cDI ) (l) = 0} where 0 I A = −A1 −A2

2 EI 2 2 cDI 2 with A1' = ∂ ( ∂ ') and A2' = ∂ ( ∂ ').

Homework Exercises ∙ Ex. 10 : Some books deﬁne D(A) by

˜ 4 2 4 2 D(A) = (H (0, l) ∩ HL(0, l)) × (H (0, l) ∩ HL(0, l))

plus boundary conditions. We know A∣D(A) is an inﬁnitesimal generator of a C0 semigroup which, in turn, implies A∣D(A) is a closed operator. You can show A∣D˜(A) is not closed. Therefore, we claim that D(A) ∕= D˜(A) . Is this true? Look at both the damped and undamped cases.

25 13 Summary of Results on Analytic Semigroup Generation by Sesquilinear Forms

We summarize results available for the special cases of analytic semigroupo generation. For further details the reader can consult [BI, T]. Let V and H be complex Hilbert spaces with the Gelfand triple V,→ ∗ H,→ V . Let ⟨⋅, ⋅⟩V ∗,V be the duality product, and : V × V → C be a sesquilinear form such that is

1. V continuous, i.e., ∣(', )∣ ≤ ∣'∣V ∣ ∣V .

2 2. V -elliptic, i.e., Re (', ') ≥ ∣'∣V . (We can, if necessary, replace this 2 2 by a shift: Re (', ') + 0∣'∣H ≥ ∣'∣V .)

As before, let Aˆ ∈ ℒ(V,V ∗) (note that this is −A in our old notation) and A : DA ⊂ H → H be deﬁned such that

(', ) = ⟨−A',ˆ ⟩V ∗,V for all ', ∈ V = ⟨−A', ⟩H ' ∈ DA, ∈ V.

Then we have ℛ(Aˆ) = V ∗, ℛ(A) = H, and 0 ∈ (Aˆ). We can also note

ˆ 2 Re (', ') = Re ⟨−A', '⟩V ∗,V ≥ ∣'∣V .

ˆ 2 for all ' ∈ V . In other words, Re⟨A', '⟩ ≤ −∣'∣V ≤ 0. Similarly, for ' ∈ DA, Re ⟨A', '⟩H ≤ 0 which implies A is dissipative. By Lumer Phillips, because A is dissipative and ℛ(A) = H, A is the infinitesimal generator of a C0 semigroup of contractions S(t) on H. We recall the definition of dissipativeness in a Banach space X. An operator B ∈ D ⊂ X → X is dissipative if for each x ∈ D(B) there exists ∗ ∗ ∗ x ∈ F (x) ⊂ X such that Re⟨x , Bx⟩X∗,X ≤ 0 where F (x) is the duality set. Let’s apply this definition to X = V ∗, which is a reflexive Banach space in its own right, with the operator B = Aˆ, Aˆ : V ⊂ V ∗ → V ∗. We have Aˆ being dissipative in the Banach space V ∗ means for x ∈ V there ∗ ∗ ∗∗ ∗ exists x ∈ F (X) ⊂ X = V = V such that Re⟨x , Axˆ ⟩V,V ∗ ≤ 0 or ∗ ∗ ∗ Re⟨Ax,ˆ x ⟩V ∗,V ≤ 0. However, we have this holding for every x ∈ V ⊂ V . (In particular, we can find such a x∗ in the duality set.) Therefore, Aˆ : V = D(Aˆ) ⊂ V ∗ → V ∗ is dissipative. Using Lumer Phillips again we have Aˆ is an ∗ infinitesimal generator of a C0 semigroup of contractions Sˆ(t) on V where Sˆ(t)∣H = S(t).

26 ˆ Recall DA = {x ∈ V ∣Axˆ ∈ H}. We define DÂ = {x ∈ V ∣Axˆ ∈ V } ˆ ˆ and define the operator Aˆ = A∣ ˆ in V . We have ℛ(Aˆ) = V ; therefore DÂ ˆ the range statement needed for Lumer Phillips holds for Aˆ. However, for ˆ ˆ Aˆ to be dissipative in V , we must have for each x ∈ DÂ ⊂ V there exists ∗ ∗ ∗ ˆ x ∈ F (x) ⊂ V such that Re⟨x , Axˆ ⟩V ∗,V ≤ 0. We do not directly have ˆ that Aˆ is dissipative in V . To pursue this, we need to consider the Tanabe estimates.

13.1 Tanabe Estimates (on “Regular Dissipative Operators”) Suppose a sesquilinear form (witℎassociatedoperatorAˆ) is V continuous −1 and V -elliptic. Then for Re ≥ 0 and ∕= 0,R(Aˆ) = (I − Aˆ) ∈ ℒ(V ∗,V ), and

ˆ 1 ∗ ˆ 1. ∣R(A)'∣V ≤ ∣'∣V ∗ for ' ∈ V . (In other words, ∣R(A)∣ℒ(V ∗,V ) ≤ 1 .)

ˆ M0 2. ∣R(A)'∣H ≤ ∣∣ ∣'∣H for ' ∈ H where M0 = 1 + . (In other words, ˆ M0 ∣R(A)∣ℒ(H) ≤ ∣∣ .)

ˆ M0 ∗ ˆ 3. ∣R(A)'∣V ∗ ≤ ∣∣ ∣'∣V ∗ for ' ∈ V . (In other words, ∣R(A)∣ℒ(V ∗) ≤ M0 ∣∣ .)

ˆ M0 ˆ M0 4. ∣R(A)'∣V ≤ ∣∣ ∣'∣V for ' ∈ V . (In other words, ∣R(A)∣ℒ(V ) ≤ ∣∣ .) We give the arguments for 4. since it is a very strong and useful estimate. Deﬁne the dual or adjoint operator in the usual manner: deﬁne ∗ ∗ ∗ Aˆ ∈ ℒ(V,V ) by (', ) = ⟨', −Aˆ ⟩V,V ∗ for ', ∈ V . Then

∗(', ) = (', ) and ∗ ∗ (', ) = ⟨', −Aˆ ⟩V,V ∗ implies ˆ∗ ˆ∗ ⟨', −A ⟩V,V ∗ = ⟨', −A ⟩V,V ∗ .

27 Therefore, Aˆ∗ also satisﬁes the estimates 1.-3. above because ∗ is V continuous and V -elliptic. Applying 3 to Aˆ∗ gives us for Re ≥ 0, ∕= 0, ', ∈ V

∗ ∣⟨R(Aˆ)', ⟩V,V ∗ ∣ = ∣⟨', R(Aˆ ) ⟩V,V ∗ ∣

∗ ≤ ∣'∣V ∣R(Aˆ ) ∣V ∗

M0 ≤ ∣'∣V ∣∣ ∣ ∣V ∗ .

ˆ M0 Therefore, ∣R(A)'∣V ≤ ∣∣ ∣'∣V .

We can show that the C0 semigroups from above are actually analytic. The theorem below gives a useful condition for analyticity.

Theorem 6 Let T (t) be a C0 semigroup on a Hilbert space X with inﬁnites- imal generator A, with 0 ∈ (A). Then a semigroup is analytic on X if there exists a constant c such that c ∣R (A)∣ ≤ +i ℒ(X) ∣∣ for > 0, ∕= 0 where = + i. See [P, Theorem II.5.2(b)].

ˆ c c c From the Tanabe estimates, we have ∣R(A)∣ℒ(X) ≤ = √ ≤ ∣∣ 2+ 2 ∣∣ for X chosen as V,H, or V ∗. Therefore, our estimates suﬃce to provide analyticity of the associated semigroups. Thus we have the following theorem.

Theorem 7 Let V,→ H,→ V ∗ be a Gelfand triple. Assume the sesquilin- ˆ ear form is V continuous and V -elliptic. Let A,ˆ A, and Aˆ be deﬁned as above. Then

∙ Aˆ is an inﬁnitesimal generator of an analytic semigroup Sˆ(t) of contractions on V ∗.

∙ A is an inﬁnitesimal generator of an analytic semigroup S(t) of contractions on H. ˆ ˆ ∙ Aˆ is an inﬁnitesimal generator of an analytic semigroup Sˆ(t) of contractions on V .

28 We also have

∙ domV ∗ (Aˆ) = V .

∙ domH (Aˆ) = DA = {x ∈ V ∣Axˆ ∈ H}. ˆ ∙ domV (Aˆ) = DÂ = {x ∈ V ∣Axˆ ∈ V }. This is usually stated as A or Aˆ generate an analytic semigroup of contractions on V ,H, V ∗. Remark: The results of this section will be of fundamental importance in subsequent discussion on feedback control problems for infinite dimensional systems such as parabolic and strongly damped hyperbolic partial differential equations as well as functional differential equations. These problems can be conveniently and profitably formulated in an abstract setting with the systems defined in terms of Gelfand triples and where the associated algebraic Riccati equations for the feedback gains are formulated in an appropriately defined space V . In later sections we will discuss some of the results as developed in [BKcontrol, BI, 32] and summarized partially in [BSW].

13.2 Inﬁnitesimal Generators in a General Banach Space

Recall that if A is an infinitesmal generator of a C0 semigroup T (t) in a ∗ Hilbert space X, then S(t) = T (t) is a C0 semigroup in X with infinitesimal generator A∗. Thus, if A is an infinitesimal generator, D(A) is dense in X. Similarly, if A∗ is an infinitesimal generator, D(A∗) is also dense in X. We can generalize this result in a general Banach space.

Theorem 8 If X is a reflexive Banach space and A is an infinitesimal ∗ generator of a C0 semigroup T (t) in X, then A is an infinitesimal generator ∗ ∗ ∗ of a C0 semigroup S(t) in X and S(t) = T (t) = (T (t)) . In other words, (eA∗t on X∗)∗ = eAt on X.

∗ Corollary 1 If Aˆ is an infinitesimal generator of a C0 semigroup on V , then Aˆ∗ is an infinitesimal generator on V ∗∗ = V for any reflexive Banach space V .

29 In the formulations of the previous sections, we know Aˆ ∈ ℒ(V,V ∗) and ∗ ∗∗ ∗ ∗ Aˆ ∈ ℒ(V ,V ) = ℒ(V,V ) are infinitesimal generators of C0 semigroups ∗ ∗ Aˆ∗t of contractions on V . In other words, Sˆ (t) = e is a C0 semigroup of contractions on V ∗. Applying the previous corollary, we have (Sˆ∗(t) on V ∗)∗ = Sˆ(t) on V . However, Sˆ∗(t) ∈ ℒ(V ∗,V ∗) implies (Sˆ∗(t))∗ ∈ ˆ ℒ(V ∗∗,V ∗∗) = ℒ(V,V ). Since V is a reflexive Hilbert space, Sˆ(t) defined ˆ previously is exactly Sˆ(t)∣V (here Aˆ = A∣ ˆ is the infinitesimal generator of DÂ ˆ Sˆ(t) in V .

30 14 General Second Order Systems

14.1 Introduction to Second Order Systems The ideas in Example 6 can be used to treat general second order systems. Consider the general abstract second order system

y¨(t) + A2y˙(t) + A1y(t) = f(t) or, in variational form

⟨y¨(t),'⟩V ∗,V + 1(y(t),') + 2(y ˙(t),') = ⟨f(t),'⟩V ∗,V (18) where H is a complex Hilbert space. As usual, we assume that 1 and 2 are sesquilinear forms on V where V,→ H,→ V ∗ is a Gelfand triple. We also assume that 1 is continuous, V -elliptic and symmetric (1(', ) = 1( , ')). We assume that 2 is continuous and satisﬁes a weakened ellipticity condition which we formally call H-semiellipticity.

Deﬁnition 7 A sesquilinear form on V is H-semielliptic if there is a constant b ≥ 0 such that

2 Re (', ') ≥ b∣'∣H for all v ∈ V. Note that b = 0 is allowed in this deﬁnition.

∗ Since 1 and 2 are continuous, we have that there exists Ai ∈ ℒ(V,V ), i = 1, 2, such that

i(', ) = ⟨Ai', ⟩V ∗,V for all ', ∈ V, i = 1, 2.

Following the ideas of Example 6, we define spaces V = V × V and ℋ = V × H and rewrite our second order system as a first order vector system. Defining, for = (', ), = (g, ℎ) ∈ V, the sesquilinear form

( , ) = ((', ), (g, ℎ)) = −⟨ , g⟩V + 1(', ℎ) + 2( , ℎ), we can write our system for x(t) = (y(t), y˙(t)) as

⟨x˙(t), ⟩ℋ + (x(t), ) = ⟨F (t), ⟩ℋ ∈ V where F (t) = (0, f(t)). This is formally equivalent to the system

x˙(t) = Ax(t) + F (t)

31 where A is given by

D(A) = {x = (', ) ∈ ℋ∣ ∈ V and A1' + A2 ∈ H} (19) and 0 I A = . (20) −A1 A2

We ﬁrst note that is V continuous. To see this, we observe that 1 and 2 being V continuous implies

1(', ℎ) ≤ 1∣'∣V ∣ℎ∣V and 2(', ℎ) ≤ 2∣ ∣V ∣ℎ∣V . 2 2 2 2 2 2 We also have ∣ ∣V = ∣'∣V + ∣ ∣V and ∣∣V = ∣g∣V + ∣ℎ∣V . Putting all of this together, we have

∣((', ), (g, ℎ))∣ ≤ ∣ ∣V ∣g∣V + 1∣'∣V ∣ℎ∣V + 2∣ ∣V ∣ℎ∣V ≤ ∣ ∣V ∣∣V + 1∣ ∣V ∣∣V + 2∣ ∣V ∣∣V = (1 + 1 + 2)∣ ∣V ∣∣V .

This indeed implies that is V continuous. As is V continuous, A is the negative of the restriction to D(A) of the ∗ operator A˜ ∈ ℒ(V, V ) deﬁned by ( , ) = ⟨A˜ , ⟩V∗,V so that ( , ) = ⟨−A , ⟩ℋ for ∈ D(A), ∈ V.

14.2 Results for 2 V -elliptic

If both 1 and 2 are V -elliptic and 1 is the same as the V inner product, then we have exactly the case of Kelvin-Voigt damping in Example 6. We proved with these assumptions, is V-elliptic. ( Actually, we proved (⋅, ⋅)+ 0⟨⋅, ⋅⟩ℋ is V-elliptic.) Therefore, we have A is the inﬁnitesimal generator of an analytic semigroup (not of contractions since 0 > 0) on ℋ. Even if the V inner product and 1 are not the same, this result is true. 2 Since 1 is continuous, we have ∣1(', ')∣ ≤ 1∣'∣V while 1 is symmetric (i.e., 1(', ) = 1( , ')) implies Re 1(', ') = 1(', '). Thus, 1 being 2 V -elliptic is equivalent to 1 is V -coercive: 1(', ') ≥ ∣'∣V . Hence, 1 and the inner product are equivalent. We may thus deﬁne V1 as the space V with 1 as inner product, obtaining a space that is setwise equal and topologically equivalent to V . In the space ℋ1 = V1 ×H the operator A is now associated (1) with the V1 = V1 × V1-elliptic form ( , ) = ⟨−A , ⟩ℋ1 that (as we

32 argued in Example 6) satisﬁes the conditions of our theorem. Hence, A generates an analytic semigroup in ℋ1 and hence an analytic semigroup in the equivalent space ℋ. Thus we have

∗ Theorem 9 Let V,→ H,→ V and suppose that 1 and 2 of (18) are V continuous and V -elliptic sesquilinear forms on V and that 1 is symmetric. Then the operator A deﬁned in (19) and (20) is the inﬁnitesimal generator of an analytic semigroup in ℋ = V × H.

14.3 Results for 2 H-semielliptic

If 2 is not V -elliptic, then we will not, in general, obtain an analytic solution semigroup for our system. We will obtain a C0 semigroup, but must work a little more to obtain such. So assume that 2 is only H-semielliptic. Then we have A deﬁned in (19) and (20) is dissipative in ℋ1 since

Re ⟨Ax, x⟩ℋ1 = Re {1( , ') − 1(', ) − 2( , )}

= Re {1(', ) − 1(', ) − 2( , )}

= −Re 2( , )

2 ≤ −b∣ ∣H ≤ 0. To argue that A is a generator, we use the Lumer Phillips theorem; thus we need to argue that for some > 0, the range of I − A is ℋ1. Thus, given = (g, ℎ) ∈ ℋ1, we wish to solve ( − A) = for = (', ) ∈ D(A). So we consider the equation

( − A)(', ) = (g, ℎ) for (g, ℎ) ∈ V1 × H.

This is equivalent to

' − = g (21) + A1' + A2 = ℎ. If we formally solve the ﬁrst equation for = ' − g and substitute this into the second equation, we obtain

2 ' − g + A1' + A2(' − g) = ℎ or 2 ' + A1' + A2' = ℎ + g + A2g. (22)

33 This equation must be solved for ' ∈ V1 ( and then deﬁned by = '−g will also be in V1). These formal calculations suggest that we deﬁne for > 0 the associated sesquilinear form on V × V → C

2 (', ) = ⟨', ⟩H + 1(', ) + 2(', ).

Since 1 is V -elliptic and 2 is H-semielliptic we have

2 2 Re (', ') = ∣'∣H + Re 1(', ') + Re 2(', ')

2 2 2 2 ≥ ∣'∣H + c1∣'∣V + b∣'∣H

2 2 = ( + b)∣'∣H + c1∣'∣V

2 > c1∣'∣V ˜ for = ( + b) > 0. Hence is V -elliptic and (22) is solvable for ' ∈ V by Lax-Milgram. It follows that (21) is solvable for (', ) ∈ D(A), i.e., ℛ( − A) = ℋ1. Thus we have that A generates a contraction semigroup in ℋ1 and a C0 semigroup in ℋ.

∗ Theorem 10 Let V,→ H,→ V and suppose that 1 and 2 of (18) satisfy: 1 is V -elliptic, V continuous and symmetric, 2 is V continuous and H- semielliptic. Then A deﬁned by (19) and (20) generates a C0 semigroup in ℋ.

14.4 Stronger Assumptions for 2 If we strengthen the assumption on the damping form, we can obtain a stronger result.

Theorem 11 Suppose 1 is V -elliptic, V continuous, and symmetric and 2 is H-elliptic, V continuous, and symmetric. Then A is the inﬁnitesimal generator of a C0 semigroup T (t) in ℋ = V ×H that is exponentially stable, −!t i.e., ∣T (t) ∣ℋ ≤ Me ∣ ∣ℋ for some ! > 0.

To motivate the arguments used to establish this result, we consider for ! > 0 the change of dependent variable y(t) = e−!tr(t) in the equation

y¨ + A2y˙(t) + A1y(t) = 0. (23)

34 Upon substitution, we obtain

r¨(t) + Aˆ2r˙(t) + Aˆ1r(t) = 0 (24) where 2 Aˆ1 = A1 − !A2 + ! I

Aˆ2 = A2 − 2!I. This suggests that we deﬁne the sesquilinear forms

2 ˆ1(', ) = 1(', ) − !2(', ) + ! ⟨', ⟩H

ˆ2(', ) = 2(', ) − 2!⟨', ⟩H so that ˆi(', ) = ⟨Aˆi', ⟩V ∗,V , i = 1, 2, and the transformed variational form of (23) is

⟨r¨(t),'⟩V ∗,V + ˆ1(r(t),') + ˆ2(r ˙(t),') = 0 for ' ∈ V . We observe that ˆ1, ˆ2 are continuous and ˆ1 is symmetric since both 1 and 2 are. Since 2 is symmetric (hence 2(', ') is real) and continuous 2 with 2(', ') ≤ k2∣'∣V , we have for ' ∈ V

Re ˆ1(', ') = ˆ1(', ') 2 2 = 1(', ') − !2(', ') + ! ∣'∣H 2 2 2 2 ≥ c1∣'∣V − ! 2∣'∣V + ! ∣'∣H 2 ≥ (c1 − ! 2)∣'∣V . Hence ˆ is V -elliptic if ! > 0 is chosen so that ! < c1 . 1 2 Moreover, we ﬁnd that ˆ2 is H-semielliptic if ! is chosen properly since

2 2 Re ˆ2(', ') = Re 2(', ') − 2!∣'∣H ≥ (b − 2!)∣'∣H .

b Therefore, ˆ2 is H-semielliptic if ! < 2 . Thus, if we choose ! > 0 as ! = 1 min { b , c1 }, we ﬁnd that ˆ and ˆ 2 2 2 1 2 satisfy the assumptions of Theorem 10. By the arguments preceding that theorem, we see that 0 I Aˆ = −Aˆ1 −Aˆ2

(see (19) and (20)) generates a contraction semigroup Tˆ(t) on ℋˆ1 = Vˆ1 × H where Vˆ1 is V taken with ˆ1 as inner product (Vˆ1 is equivalent to V ).

35 Now let T (t) be the C0-semigroup generated by A (see (19), (20) and y(t) r(t) Theorem 10). If x(t) = and w(t) = are solutions of y˙(t) r˙(t) y0 (23) and (24) respectively, we have x(t) = T (t)x0 where x0 = and w0 −wt wt wt w(t) = Tˆ(t)w0. Since y(t) = e r(t) andy ˙(t) = −we r(t) + e r˙(t), we see that x(t) = ewtΓw(t) where

1 0 Γ = −w 1 and w = Γ−1x . It follows since ∣Tˆ(t)∣ ≤ 1 that 0 0 ℋˆ1

∣T (t)x ∣ ≤ e−wt∣ΓTˆ(t)Γ−1x ∣ 0 ℋˆ1 0 ℋˆ1

≤ Me−wt∣x ∣ . 0 ℋˆ1

Since ℋˆ1 and ℋ = V × H are norm equivalent, we thus ﬁnd that the semigroup T (t) is exponentially stable in ℋ.

36 15 Abstract Cauchy Problem

It is of great practical as well as theoretical interest to know when, and in what sense, solutions of the abstract equations

x˙(t) = Ax(t) + f(t) (25) x(0) = x0 exist. Moreover, representations of such solutions in terms of a variation of parameters formula and the semigroup generated by A will play a fundamental role in control and estimation formulations. We begin by summarizing results available in the standard literature on linear semigroups and abstract Cauchy problems. Consider the abstract Cauchy problem (ACP) given by (25) where A is the inﬁnitesimal generator of a C0-semigroup T (t) in a Hilbert space H. We deﬁne a mild solution xm of (25) as functions given by

Z t xm(t) = T (t)x0 + T (t − s)f(s)ds (26) 0 whenever this entity is well defined (i.e., f is sufficiently smooth). We say that x : [0,T ] → H is a strong solution of (ACP) if x ∈ C([0,T ],H) T C1((0,T ],H), x(t) ∈ D(A) for t ∈ (0,T ], and x satisfies (25) on [0,T ]. We have the following series of results from the literature.

Theorem 12 If f ∈ L1((0,T ),H) and x0 ∈ H, there is at most one strong solution of (25). If a strong solution exists, it is given by (26).

1 Theorem 13 If x0 ∈ D(A) and f ∈ C ([0,T ],H), then xm given by (26) provides the unique strong solution of (25).

Theorem 14 If x0 ∈ D(A), f ∈ C([0,T ],H), f(t) ∈ D(A) for each t ∈ [0,T ] and Af ∈ C([0,T ],H), then (26) provides the unique strong solution of (25).

Theorem 15 Suppose A is the inﬁnitesimal generator of an analytic semigroup T (t) on H. Then if x0 ∈ H and f is Ho¨lder continuous (i.e. ∣f(t) − f(s)∣ ≤ k∣t−s∣ for some ≤ 1), then xm of (26) provides the unique strong solution of (25).

37 Unfortunately, all of these powerful results are too restrictive for use in many applications, including control theory, where typically f(t) = Bu(t) is not continuous, let alone Hölder continuous or C1. For this reason, a weaker formulation is more appropriate. For this, we follow the presentations of Lions, Wolka, and Tanabe which are developed in the context of sesquilinear forms and Gelfand triples, V,→ H,→ V ∗, where V,H,V ∗ are Hilbert spaces. We define the solution space W(0,T ) by dg W(0,T ) = {g ∈ L ((0,T ),V ): ∈ L ((0,T ),V ∗)} 2 dt 2 with scalar product Z T Z T dg dℎ ⟨g, ℎ⟩W = ⟨g(t), ℎ(t)⟩V dt + ⟨ (t), (t)⟩V ∗ dt. 0 0 dt dt Then it can be shown that W(0,T ) is a Hilbert space which embeds continuously into C([0,T ],H). Assume : V × V → C satisfies for ', ∈ V

2 2 Re (', ') ≥ c1∣'∣V − 0∣'∣H c1 ≥ 0, 0 real, for all ' ∈ V,

∣(', )∣ ≤ ∣'∣V ∣ ∣V for all ', ∈ V. Then, as already discussed, we have A ∈ ℒ(V,V ∗) such that (', ) = ⟨A', ⟩V ∗,V = ⟨−A', ⟩H where A is the densely deﬁned restriction of −A to the set DA = {' ∈ V ∣A' ∈ H}. We have moreover, that A is the inﬁnitesimal generator of an analytic semigroup T (t) on H. In fact, from Theorem 8 we have that −A is the generator of an analytic semigroup T (t) in V,H and V ∗ and T (t) agrees with T (t) on V and H. We may consider solutions of (25) in the sense of V ∗, i.e., in the sense

⟨x˙(t), ⟩ ∗ + (x(t), ) = ⟨f(t), ⟩ ∗ for ∈ V, V ,V V ,V (27) x(0) = x0. By a strong solution of (25) in the V ∗ sense (also quite frequently called a weak or variational or distributional solution), we shall mean a function ∗ x ∈ L2((0,T ),V ) such thatx ˙ ∈ L2((0,T ),V ) and (27) (or equivalently x˙(t) + Ax(t) = f(t)) holds almost everywhere on (0,T ). Similarily, mild ∗ solutions xm ∈ V are given by the analogue of (26) Z t xm(t) = T (t)x0 + T (t − s)f(s)ds. (28) 0 We then have the fundamental existence and uniqueness theorem.

38 ∗ Theorem 16 Suppose x0 ∈ H and f ∈ L2((0,T ),V ). Then (27) has a unique strong solution in the V ∗ or variational sense and this is given by the mild solution (28).

Proof ∞ Let {i}1 ⊂ V be a linearly independent total subset (i.e., a basis) of V . k P We deﬁne the “Galerkin” approximations by xk(t) = wi(t)'i where the i=1 coeﬃcients {wi} are chosen so that

⟨x˙ k(t),'j⟩H + (xk(t),'j) = ⟨f(t),'j⟩V ∗,V (29) for j = 1, . . . , k, satisfying the intial condition

xk(0) = xk0 where k X xk0 = wi0 'i → x0 i=1 in H as k → ∞. Equivalently, (29) can be written as

k k X X w˙ i(t)⟨'i,'j⟩ + wi(t)('i,'j) = Fj(t) i=1 i=1 where Fj(t) = ⟨f(t),'j⟩V ∗,V for j = 1, . . . , k. Therefore, w1, . . . , wk are unique solutions to a vector ordinary diﬀerential equation system. Multiplying (29) by wj and summing over j = 1, . . . , k, we obtain

⟨x˙ k(t), xk(t)⟩H + (xk(t), xk(t)) = ⟨f(t), xk(t)⟩V ∗,V

with xk(0) = xk0 → x0 in H. Therefore

1 d 2 ∣x (t)∣ + (x (t), x (t)) = ⟨f(t), x (t)⟩ ∗ . (30) 2 dt k H k k k V ,V Integrating (30), we obtain

Z t Z t 1 2 1 2 ∣xk(t)∣H − ∣xk(0)∣H + (xk(s), xk(s))ds = ⟨f(s), xk(s)⟩V ∗,V ds. 2 2 0 0 Using the fact that is V -elliptic, we have

39 Z t Z t 1 2 2 1 2 ∣xk(t)∣H + c1 ∣xk(s)∣V ds ≤ ∣xk(0)∣H + ∣⟨f(s), xk(s)⟩V ∗,V ∣ds 2 0 2 0 Z t 1 2 1 2 2 ≤ ∣xk(0)∣H + ( ∣f(s)∣V ∗ + ∣xk(s)∣V )ds. 2 0 4 Therefore,

Z t Z t 1 2 2 1 2 1 2 ∣xk(t)∣H + (c1 − ) ∣xk(s)∣V ds ≤ ∣xk(0)∣H + ∣f(s)∣V ∗ ds (31) 2 0 2 0 4 or Z t 1 2 2 1 2 1 2 ∣x (t)∣ + (c − ) ∣x (s)∣ ds ≤ ∣x (0)∣ + ∣f∣ ∗ . k H 1 k V k H L2((0,t),V ) 2 0 2 4

This implies we have {xk} bounded in C((0,T ),H) and in L2((0,T ),V ).

Since L2((0,T ),V ) is a Hilbert space, we can choose {xkn ∣xkn ⇀ x˜ ∈ L2((0,T ),V )} to be a convergent subsequence of xk. Without loss of gener- ality, we reindex and denote xkn by xk. Then the limitx ˜ is our candidate for a solution where xk ⇀ x˜ in L2((0,T ),V ). 1 Let (t) ∈ C (0,T ) with (T ) = 0 and (0) = 0 and deﬁne Ψj(t, ⋅) by Ψj(t, ⋅) = (t)'j. Multiplying (29) by (t) and integrating, we have

Z T (⟨x˙ k(t),'j⟩H (t) + (xk(t),'j) (t) − ⟨f(t),'j⟩V ∗,V (t)) dt = 0. (32) 0 Integrating by parts, we ﬁnd that (32) becomes

Z T Z T Z T − ⟨xk(t),'j⟩ ˙(t)dt + (xk(t),'j) (t) − ⟨f(t),'j⟩V ∗,V (t)dt = 0. 0 0 0 We can now let k → ∞ and pass the limit through term by term to obtain Z T Z T Z T −⟨x˜(t),'j⟩ ˙(t)dt + (˜x(t),'j) (t)dt − ⟨f(t),'j⟩V ∗,V (t)dt = 0, 0 0 0 (33) holding for all 'j ∈ V . Recall that {'j} is a total subset of V and observe that the s such as above are dense in L2(0,T ). Thus we have (33) holding for all Ψ = ' in L2((0,T ),V ). We can rewrite (˜x(t),') (t) as Ax˜(t)Ψ R T and 0 ⟨f(t),'⟩V ∗,V (t)dt as f(Ψ).

40 Therefore, (33) becomes d ⟨ x,˜ Ψ⟩ ∗ + (Ax˜ − f)Ψ = 0 dt V ,V dx˜ ∗ where Ψ ∈ L2((0,T ),V ), orx ˜ satisﬁes dt + Ax˜ − f = 0 in the L2((0,T ),V ) sense. However, we have the following needed theorem (details can be found in [E]).

Theorem 17 Let X be a reflexive Banach space. Then ∗ ∼ ∗ Lp((0,T ),X) = Lq((0,T ),X ) 1 1 where p + q = 1, 1 < p < ∞. Returning to our arguments we thus have that the solution to the equa- ∗ ∗ tion exists in the L2((0,T ),V ) = L2((0,T ),V ) sense and is given by x˜. To obtainx ˜(0) = x0, we may use the same arguments with arbitrary ∈ C1(0,T ), (T ) = 0, but (0) ∕= 0. To prove uniqueness of the solution, it suffices to argue that the solution corresponding to x0 = 0, f = 0 is identically zero. With these specific values for f and x0, (27) can be written as

⟨x˙(t),'⟩V ∗,V + (x(t),') = 0, (34) x(0) = 0. Let ' = x(t). Then (34) becomes 1 d ∣x(t)∣2 + (x(t), x(t)) = 0. 2 dt H Integrating by parts and using the V -ellipticity of , we obtain Z t 1 2 2 ∣x(t)∣H + c1∣x(s)∣V ds ≤ 0. 2 0 Therefore, x(t) = 0 and thus the solution is unique. To estsblish continuous dependence of the solution, deﬁne

∗ \ x(⋅; x0, f):(x0, f) ∈ H ×L2((0,T ),V ) → x ∈ L2((0,T ),V ) C((0,T ),H).

∗ Therefore, x ∈ L2((0,T ),V ). Taking the limits in (31) and using the property that the norms are weakly lower semi-continuous, we obtain the following relation: Z t Z t 1 2 2 1 2 1 2 ∣x(t)∣H + (c1 − ) ∣x(s)∣V ds ≤ ∣x0∣ + ∣f(s)∣V ∗ ds. 2 0 2 4 0

41 and hence Z T Z T 1 2 2 1 2 1 2 sup ∣x(t)∣H + (c1 − ) ∣x(s)∣V ds ≤ ∣x0∣H + ∣f(s)∣V ∗ ds. t∈[0,T ] 2 0 2 4 0

∗ Since the map (x0, f) → x is a linear map on H × L2((0,T ),V ), we have ∗ immediately that x is continuous on H × L2((0,T ),V ) into C((0,T ),H), ∗ L2((0,T ),V ) and L2((0,T ),V ). Finally, we need to prove the equivalence between this solution and the mild solution given by (28). From (28) we have that the map (x0, f) → ∗ ∗ x(⋅, x0, f) is continuous from H × L2((0,T ),V ) to L2((0,T ),V ). We have just argued that the variational solution xvar(⋅, x0, f) is continuous in the same sense. Thus xvar and xm are both continuous in the above sense. Recall that if two functions agree on a dense subset of some set, then the solutions must agree on the entire set. Therefore, if there is a dense subset ∗ of H × L2((0,T ),V ) on which xvar and xm agree, then they will agree on the entire set. 1 Choose x0 ∈ DA and f ∈ C ((0,T ),H). Then Theorem 13 guarantees that xm is the unique solution in the H sense. However, if xm is a strong solution in the H sense, then it must also be a variational solution (i.e., a strong solution in the V ∗ sense). However, the mild solution being 1 unique means xm(⋅, x0, f) = xvar(⋅, x0, f) for (x0, f) ∈ DA × C ((0,T ),H). 1 ∗ But DA × C ((0,T ),H) is dense in H × L2((0,T ),V ). Hence, we have the equivalence between the solutions corresponding to data (x0, f) in H × ∗ L2((0,T ),V ).

42 16 Weak Formulations for Second Order Systems

16.1 Model Formulation We return to the general second order systems (18) of Section 14 to illustrate well-posedness ideas in the context of the abstract hyperbolic model with time dependent stiﬀness and damping given by

⟨y¨(t), ⟩V ∗,V + d(t;y ˙(t), ) + a(t; y(t), ) = ⟨f(t), ⟩V ∗,V

∗ ∗ where V ⊂ VD ⊂ H ⊂ VD ⊂ V are Hilbert spaces with continuous and dense injections, where H is identified with its dual and ⟨⋅, ⋅⟩ denotes the associated duality product. We first show under reasonable assumptions on the time-dependent sesquilinear forms a(t; ⋅, ⋅): V × V → C and d(t; ⋅, ⋅): VD × VD → C that this model possesses a unique solution and that the solution depends continuously on the data of the problem. We also consider well-posedness as well as finite element type approximations in associated inverse problems. The model is a weak formulation that includes models in abstract differential operator form that include plate, beam and shell equations with several important kinds of damping. Applications for such systems are abundant and range from systems with periodic or structured pattern time dependence that occurs for example in thermally dependent systems orbiting in space (periodic exposure to sunlight) to earth bound structures (bridges, buildings) and aircraft/space structure components with extreme temperature exposures (winter vs. sum- mer). On a longer time scale, such systems are important in time dependent health of elastic structures where long term (slowly varying) time dependence of parameters may be used in detecting aging/fatiguing as represented by changes in stiffness and/or damping. Moreover, applications may be found in modern “smart material structures” [BSW, RCS] where one “controls” damping and/or elasticity via piezoceramic patches, electrically active polymers, etc. As before let V and H be complex Hilbert spaces forming a “Gelfand ∗ ∗ triple” V ⊂ H = H ⊂ V with duality product ⟨⋅, ⋅⟩V ∗,V . The injections are assumed to be dense and continuous and the spaces are assumed to be separable. Moreover, we assume that there exists a Hilbert space VD (the ∗ ∗ ∗ damping space), such that V ⊂ VD ⊂ H = H ⊂ VD ⊂ V , allowing for a wide class of damping models. Thus the duality products ⟨⋅, ⋅⟩V ∗,V and ⟨⋅, ⋅⟩ ∗ are the natural extensions by continuity of the inner product ⟨⋅, ⋅⟩ VD,VD ∗ ∗ in H to V × V and VD × VD, respectively. The H-norm is denoted by ∣ ⋅ ∣ or ∣ ⋅ ∣H for clarity if needed and the V and VD-norms are denoted ∣ ⋅ ∣V and

43 ∣ ⋅ ∣VD , respectively. We denote ∂ty byy ˙, where y = y(t) is considered as a function of time taking values in one of the occurring Hilbert spaces, that is, y(t, ) = y(t)() where y(t) ∈ H, for example. Our goal is to investigate the system

y¨(t) + D(t)y ˙(t) + A(t)y(t) = f(t), in V ∗, t ∈ (0,T ); (35)

with y(0) = y0 ∈ V andy ˙(0) = y1 ∈ H which is equivalent to

⟨y¨(t), ⟩ ∗ + ⟨D(t)y ˙(t), ⟩ ∗ + ⟨A(t)y(t), ⟩ ∗ = ⟨f(t), ⟩ ∗ , (36) V ,V VD,VD V ,V V ,V

for all ∈ V and t ∈ (0,T ), with y(0) = y0 ∈ V andy ˙(0) = y1 ∈ H. Following standard terminology we will call a function y ∈ L2(0,T ; V ), ∗ withy ˙ ∈ L2(0,T ; H) andy ¨ ∈ L2(0,T ; V ) a weak solution of the initial- value problem (35) if it solves the equation (36) in the strong V ∗ sense of the previous section, or, equivalently, solves the equation (45) below, with y(0) = y0 ∈ V andy ˙(0) = y1 ∈ H given. We will assume that the operators A and D arise (as deﬁned precisely in (42)-(43) below) from time-dependent sesquilinear forms a and d satisfying the following natural ellipticity, coercivity and diﬀerentiability conditions. First, we assume hermitian symmetry, that is

(H1) a(t, , ) = a(t, , ) for all , ∈ V .

(H2) ∣a(t; , )∣ ≤ c1∣∣V ∣ ∣V , , ∈ V where c1 is independent of t.

We assume, further, that

(H3) a(t; , ) for , ∈ V fixed is continuously differentiable with respect to t for t ∈ [0,T ](T finite) and

∣a˙(t; , )∣ ≤ c2∣∣V ∣ ∣V , for all t ∈ [0,T ], (37)

c2 once again independent of t. We also assume that the sesquilinear form a(t; , ) is V -elliptic, so that

(H4) There exist a constant > 0 such that

2 ∣a(t; , )∣ ≥ ∣∣V for all t ∈ [0,T ], and for all u ∈ V. (38)

For the sesquilinear form d we assume similarly

44 (H5) ∣d(t; , )∣ ≤ c3∣∣VD ∣ ∣VD , , ∈ VD where c3 is independent of t.

We assume, further, that

(H6) d(t; , ) for , ∈ VD fixed is continuously differentiable with respect to t for t ∈ [0,T ](T finite) and ˙ ∣d(t; , )∣ ≤ c4∣∣VD ∣ ∣VD , for all t ∈ [0,T ], (39)

c4 once again independent of t.

1 Then t → d(t; , ) and t → a(t; , ) are C [0,T ] for all , ∈ VD, and , ∈ V , respectively, which implies that d(t; , ) and a(t; , ) are suf- ﬁciently well-behaved in order to have existence for (35) or (36). We also assume that the sesquilinear form d(t; , ) is VD-coercive. That is,

(H7) There exist constants d and d > 0, such that

Re d(t; , ) + ∣∣2 ≥ ∣∣2 for all t ∈ [0,T ], and for all ∈ V . d d VD D (40)

We know then from our earlier considerations that there exist representation operators A(t) and D(t)

∗ ∗ A(t): V → V ,D(t): VD → VD, (41)

which for each ﬁxed t are continuous and linear, with

a(t; , ) = ⟨A(t), ⟩V ∗,V , for all , ∈ V, (42)

and d(t; , ) = ⟨D(t), ⟩ ∗ , for all , ∈ V . (43) VD,VD D

We will now consider the following problem: Given ﬁnite T and f ∈ ∗ L2(0,T ; VD) along with initial conditions y0 ∈ V , y1 ∈ H,

∗ we wish to ﬁnd a function y ∈ L2(0,T ; V ), y˙ ∈ L2(0,T ; VD) such that in V we have ( y¨(t) + D(t)y ˙(t) + A(t)y(t) = f(t), t ∈ (0,T ), (44) y(0) = y0,y ˙(0) = y1.

45 ∗ That is, for f ∈ L2(0,T ; VD)

⟨y¨(t), ⟩V ∗,V + d(t;y ˙(t), ) + a(t; y(t), ) = ⟨f(t), ⟩V ∗,V for all ∈ V. (45) ∗ ∗ We remark that (45) makes sense since f(t) ∈ VD ⊂ V . This formulation covers linear beam (i.e., Example 6 discussed earlier), plate and shell models with numerous damping models (Kelvin-Voigt, viscous, square-root, structural and spatial hysteresis) frequently studied in the literature. The formulation above is non-standard in the sense that the damping sesquilinear form is incorporated in the variational model and is time-dependent. The problem without damping (d = 0) and f ∈ L2(0,T ; H) was treated by Lions in [Li], and subsequently in [W]. The less general case 1 without damping and V = H0 (Ω) is treated for example in [Ev]. The model above with d : VD → C independent of time is treated in [BSW]. The following theorem is a time-dependent extension of the previous results.

0 1 ∗ Theorem 18 Assume that (f, y , y ) ∈ L2(0,T ; VD) × V × H. Then there exists a unique solution y to (45) with (y, y˙) ∈ L2(0,T ; V ) × L2(0,T ; VD), and the mapping (f, y0, y1) → (y, y˙), (46) is continuous and linear on

∗ L2(0,T ; VD) × V × H → L2(0,T ; V ) × L2(0,T ; VD). (47)

As we will see, Theorem 18 can then be extended to stronger smoothness results on solutions.

0 1 ∗ Theorem 19 Assume that (f, y , y ) ∈ L2(0,T ; VD) × V × H. Then there exists (perhaps after modiﬁcations on a set of measure zero) a unique solution y to (45) with (y, y˙) ∈ C(0,T ; V ) × (C(0,T ; H) ∩ L2(0,T ; VD)), and the mapping (f, y0, y1) → (y, y˙), (48) is continuous and linear on

∗ L2(0,T ; VD) × V × H → C(0,T ; V ) × (C(0,T ; H) ∩ L2(0,T ; VD)). (49)

Remark: If we only have that the inequality (40) for the damping sesquilinear form d is satisﬁed with d = 0, the results are still true with modiﬁcations. Then it will be necessary that f ∈ L2(0,T ; H) and one obtains only thaty ˙ ∈ L2(0,T ; H); that is, we have the same results as if

46 there was no damping. (See, e.g., [Li].) As an added comment, we note that J. L. Lions was one of the early and most proliﬁc contributors to functional analytic formulations of partial diﬀerential equations and control theory. His many contributions include the seminal texts [Li, LiMag].

16.2 Discussion of the Model It is well known from the literature that the strong form of the operator formulation (35) of the problem in general causes computational problems due to irregularities stemming from non-smooth terms - typically in the force/moment terms in, for example, elasticity problems. The weak formulation has proven advantageous both for theoretical and practical purposes, specifically in the effort to estimate parameters or for control purposes [BSW]. To give a particular example illustrating and motivating our discussions here, we consider a version of the beam of Example 6 but with both ends fixed. For an Euler-Bernoulli beam of length l, width b, thickness ℎ and linear density , where the parameters b, ℎ, may be functions that depend on time and/or spatial position along the beam, the equation for transverse displacements y = y(t, ) (in strong form [BSW]) is given by

∂2y ∂2 ∂3y ∂2y + CgDI + EIf = f 0 < < l, (50) ∂t2 ∂2 ∂2∂t ∂2 with fixed end boundary conditions ∂y ∂y y(t, 0) = (t, 0) = y(t, l) = (t, l). (51) ∂ ∂ Here we assume Kevin-Voigt structural damping with damping coefficient CgDI = CgDI(t, ) and the possibly time and spatially dependent stiffness coefficient given by EIf = EIf (t, ). For simplicity we assume is constant and scale the system by taking = 1. One can readily compute that EIf = 3 3 Eℎ b/12, CgDI = CDℎ b/12, where the Young’s modulus E, the damping coefficient CD and the geometric parameters ℎ, b may in general all be time and/or spatially dependent. As we have seen, in weak form this can be written ∂y˙2(t) ∂2y(t) ∂2 ⟨y¨(t), ⟩V ∗,V + ⟨CgDI + EIf , ⟩H = ⟨f(t), ⟩V ∗,V , (52) ∂2 ∂2 ∂2

2 for all ∈ V , where H = L2(0, l) and V = VD = H0 (0, l) with

2 2 ′ ′ H0 (0, l) ≡ { ∈ H (0, l)∣ (0) = (0) = (l) = (l) = 0}. (53)

47 ′ ∂ Here we have adopted the usual notation = ∂ . This has the form (36) or (45) with

Z l ′′ ′′ a(t; , ) = ⟨A(t), ⟩V ∗,V = EIf (t, ) () ()d (54) 0 Z l ′′ ′′ d(t; , ) = ⟨D(t), ⟩V ∗,V = CgDI(t, ) () ()d. (55) 0 Models such as this can be generalized to higher dimensions (in Rn for n = 2, 3) to treat more general beams, plates, shells, and solid bodies [BSW, RCS]. Of great interest are a number of useful damping models that can be readily used with these equations and treated using the abstract formulation developed here. We discuss brieﬂy some of these damping models without going into much detail, as the purpose of this section is to establish well- posedness and approximation properties of the abstract model. We consider brieﬂy several damping models of interest in practice.

Time-dependent Kelvin-Voigt damping: Let ! ⊂ [0, l], with 1! denoting the characteristic function of !. Let , > 0 denote material parameters and let t → k(t) denote a suﬃciently smooth function. Then a time−dependent damping sesquilinear form is given by

Z l ′′ ′′ d(t; , ) = ( + k(t)1!() () ()d, (56) 0

2 for , ∈ VD = V = H0 (0, l). This gives a model for a mechanical structure damped by a time-varying actuator, localized somewhere inside the structure. This could be piezoceramic actuators or other “smart” devices, with the possibility of them varying in time.

Time-dependent viscous damping: This is a velocity-proportional damping, given (with the notation from above) by the sesquilinear form

Z l d(t; , ) = k(t, )() ()d, (57) 0

1 with k ∈ C (0,T ; L∞(0, l)) denoting the damping coeﬃcient. One can take VD = L2(0, l) here.

Time-dependent spatial hysteresis damping: This model, without time- dependence, is discussed in [DLR], and, as noted in [BSW], it has been

48 shown to be appropriate for composite material models where graphite ﬁbers are embedded in an epoxy matrix. The time-dependent sesquilinear form that we consider here can now be constructed with the following compact operator K(t) on L2(0, l):

Z l (K(t))() = k(t, , )()d, (58) 0

1 where the nonnegative integral kernel k belongs to C (0,T ; L∞(0, l)×L∞(0, l)). Letting Z l () = (, )d (59) 0 denote some material property, we can deﬁne

Z l d(t; , ) = (() − K(t))′() ′()d, (60) 0

1 with VD = H (0, l).

16.3 Proof of Theorems 18 and 19 As with the first order systems investigated earlier, we will follow a standard Galerkin approximation method (see for example [BSW, LiMag, W]) with necessary, non-trivial modifications as given in [BaPed] due to the presence ∞ of the time-dependent forms. So, let {wj}1 denote a basis (i.e., a linearly independent total set) in V that is also a basis in V . This is possible since V is dense in H. For a fixed m we denote by Vm the finite dimensional m 0 1 subspace spanned by {wj}1 , and we let um and ym be chosen in Vm such that 0 0 1 1 ym → y in V , ym → y in H, for m → ∞. (61)

We now deﬁne the approximate solution ym(t) of order m of our problem in the following way: m X ym(t) = gjm(t)wj, (62) j=1 where the gjm(t) are determined uniquely from the m-dimensional linear system:

(¨ym(t), wj)+d(t;y ˙m(t), wj)+a(t; ym(t), wj) = ⟨f(t), wj⟩V ∗,V , j = 1, 2, ..., m; (63)

49 0 1 ˙ with ym(0) = ym andy ˙m(0) = ym. Multiplying (63) with gjm(t) and summing over j yields

(¨ym(t), y˙m(t)) + d(t;y ˙m(t), y˙m(t)) + a(t; ym(t), y˙m(t)) = ⟨f(t), y˙m(t)⟩V ∗,V . (64) Now, since d a(t; y (t), y (t)) = 2 Re a(t; y (t), y˙ (t)) +a ˙(t; y (t), y (t)), (65) dt m m m m m m we see that d {∣y˙ (t)∣2 + a(t; y (t), y (t))} + 2 Re d(t;y ˙ (t), y˙ (t)) = dt m m m m m a˙(t; ym(t), ym(t)) + 2 Re⟨f(t), y˙m(t)⟩V ∗,V and by integrating this equality we ﬁnd Z t 2 ∣y˙m(t)∣ + a(t; ym(t), ym(t)) + 2 Re d(t;y ˙m(s), y˙m(s))ds = 0 Z t Z t 2 0 0 ∣y˙m(0)∣ + a(0; ym, ym) + a˙(s; ym(s), ym(s))ds + 2 Re⟨f(s), y˙m(s)⟩V ∗,V ds. 0 0 Using the coercivity conditions for a and d, together with the inequality ∗ (recall that f(s) ∈ VD)

1 2 2 ∣⟨f(s), y˙m(s)⟩V ∗,V ∣ ≤ ∣f(s)∣V ∗ + ∣y˙m(s)∣V (66) 4 D D we obtain, for all > 0 Z t ∣y˙ (t)∣2 + ∣y (t)∣2 + 2( − )∣y˙ (s)∣2 ds ≤ ∣y1 ∣2 + c ∣y0 ∣2 + m m V d m VD m 1 m V 0 Z t Z t Z t 2 2 1 2 c2 ∣ym(s)∣V ds + 2d ∣y˙m(s)∣ ds + ∣f(s)∣V ∗ ds. 0 0 0 2 D 0 0 1 1 ∗ Since ym → y in V , ym → y in H and f ∈ L2(0,T ; VD), we have that, for > 0 ﬁxed and m large, there exist a constant C > 0, such that Z t 1 2 0 2 1 2 ∣ym∣ + c1∣ym∣V + ∣f(s)∣V ∗ ds ≤ C, (67) 0 2 D hence Z t ∣y˙ (t)∣2 + ∣y (t)∣2 + 2( − )∣y˙ (s)∣2 ds ≤ m m V d m VD 0 Z t Z t 2 2 C + c2 ∣ym(s)∣V ds + 2d ∣y˙m(s)∣ ds. 0 0

50 Then, in particular 2 2 ∣y˙m(t)∣ + ∣ym(t)∣V ≤ Z t Z t 2 2 C + c2 ∣ym(s)∣V ds + 2d ∣y˙m(s)∣ ds. (68) 0 0

By Gronwall’s inequality we then see that the sequence {y˙m} is bounded in C(0,T ; H) and that the sequence {ym} is bounded in C(0,T ; V ). From this fact together with the inequality (68) we conclude that {y˙m} is also bounded in L2(0,T ; VD). Then it is possible to extract a subsequence {ymk } ⊂ {ym} and functions y ∈ L2(0,T ; V ) andy ˜ ∈ L2(0,T ; VD), such that ymk ⇀ y, weakly in L2(0,T ; V ) andy ˙mk ⇀ y˜, weakly in L2(0,T ; VD). But for 0 ≤ t < T we have in V , hence in VD and H, that Z t

ymk (t) = ymk (0) + y˙mk (s)ds. (69) 0 0 R t But ymk (0) → y in V and hence in VD, while, for t fixed, 0 y˙mk (s)ds ⇀ R t 0 y˜(s)ds, weakly in VD. So, by taking the weak limit in VD in (69), we obtain in VD the equality Z t y(t) = y0 + y˜(s)ds, (70) 0 0 from which we conclude thaty ˙(t) is in VD a.e., withy ˙ =y ˜ and y(0) = y . We need now to show that y is actually a solution to the problem (45), withy ˙(0) = y1. To see this, take a function ' ∈ C1([0,T ]), satisfying '(T ) = 0, and define, for j < m, the function 'j by 'j(t) = '(t)wj, where m {wj}1 was the basis spanning Vm. Now, for a fixed j < m, we multiply (63) with '(t) and integrate to obtain Z T ((ÿm(s),'j(s))+d(s;y ˙m(s),'j(s)) + a(s; ym(s),'j(s)))ds = 0 Z T ⟨f(s),' (s)⟩ ∗ ds. j VD,VD 0 ∗ Noticing that, for each t, we have that d(t; ⋅,'j(t)) ∈ VD and a(t; ⋅,'j(t)) ∈ ∗ V , we find, using the weak convergence above that for m = mk → ∞ and integration by parts in the first term, that Z T (−(y ˙(s), '˙ j(s)) + d(s;y ˙(s),'j(s)) + a(s; y(s),'j(s)))ds = 0 Z T 1 ⟨f(s),' (s)⟩ ∗ ds + (y ,' (0)), (71) j VD,VD j 0

51 ∞ for every j. Now further restrict ' to also satisfy ' ∈ C0 (0,T ) and write (71) as Z T '˙ (s)(−y˙(s), wj)+ 0 Z T '(s)(d(s;y ˙(s), w ) + a(s; y(s), w ) − ⟨f(s), w ⟩ ∗ )ds = 0, (72) j j j VD,VD 0 for each j. But by (72), we have d (y ˙(t), wj) + d(t;y ˙(t), wj) + a(t; y(t), wj) = ⟨f(t), wj⟩V ∗ ,V (73) dt D D ∞ ∗ for all wj. By of density of ∪m Vm in V we conclude thaty ¨ ∈ L2(0,T ; V ) and that for all ∈ V

(ÿ(t), ) + d(t;y ˙(t), ) + a(t; y(t), ) = ⟨f(t), ⟩ ∗ , (74) VD,VD which was (45). Hence the y we have constructed is indeed a solution to the equation and by (70) we have that y(0) = y0. In order to verify that y˙(0) = y1 we integrate by parts in (71), and by application of (73) we find that, for all j: s=T 1 −(y ˙(s),'j(s))∣s=0 = (y ,'j(0)), (75) or, equivalently 1 (y ˙(0), wj)'(0) = (y , wj)'(0). (76) Hencey ˙(0) = y1. In order to prove uniqueness, let y be a solution of our problem (45) 0 1 corresponding to (y , y , f) = (0, 0, 0), and define for a fixed t1 ∈ (0,T ) (arbitrarily chosen) the function by ( − R t1 y(s)ds for t < t , (t) = t 1 (77) 0 for t ≥ t1, so (T ) = 0. Obviously (t) ∈ V for all t, so we can take (t) = in (45) which yields

⟨y¨(t), (t)⟩V ∗,V + d(t;y ˙(t), (t)) + a(t; y(t), (t)) = ⟨f(t), (t)⟩V ∗,V . (78)

Because ˙(t) = y(t) for t < t1 (a.e), we have that

Z t1 (⟨y¨(t), (t)⟩V ∗,V + ⟨y˙(t), y(t)⟩V ∗,V )dt = 0 Z t1 d (⟨y˙(t), (t)⟩V ∗,V )dt = 0, (79) 0 dt

52 due to (t1) = 0 and the initial conditions. Using this and by integration of (78) we ﬁnd

Z t1 (⟨y˙(t), y(t)⟩V ∗,V − d(t;y ˙(t), (t)) − a(t; y(t), (t)))dt = 0; (80) 0 hence Z t1 d (∣y(t)∣2 − a(t; (t), (t)))dt = 0 dt Z t1 2 (˙a(t; (t), (t)) + Re d(t;y ˙(t), (t)))dt. (81) 0

0 Because (t1) = 0 and y(0) = y = 0 this yields

Z t1 2 ∣y(t1)∣ + a(0; (0), (0)) = 2 (˙a(t; (t), (t)) + Re d(t;y ˙(t), (t)))dt. 0 (82) From the assumptions on a anda ˙ we arrive at

Z t1 2 2 2 ∣y(t1)∣ + ∣ (0)∣V ≤ 2 (c2∣ (t)∣V + Re d(t;y ˙(t), (t)))dt. (83) 0 Now notice that d d(t;y ˙(t), (t)) = (d(t; y(t), (t))) − d˙(t; y(t), (t)) − d(t; y(t), y(t)), (84) dt so (from the initial conditions)

Z t1 Z t1 d(t;y ˙(t), (t))dt = (−d˙(t; y(t), (t)) − d(t; y(t), y(t)))dt. (85) 0 0 Because − Re d(t; y(t), y(t)) ≤ ∣y(t)∣2 − ∣y(t)∣2 (86) D d VD we have that

2 2 ∣y(t1)∣ + ∣ (0)∣V ≤ Z t1 2 (c ∣ (t)∣2 + ∣y(t)∣2 − ∣y(t)∣2 + Re d˙(t; y(t), (t)))dt. (87) 2 V D d VD 0 R t Now we introduce the function !(t) = 0 y(s)ds and use 2 2 2 2 ∣ (t)∣V = ∣!(t) − !(t1)∣V ≤ 2∣!(t)∣V + 2∣!(t1)∣V (88)

53 to obtain

2 2 ∣y(t1)∣ + ( − 4c2t1)∣!(t1)∣V ≤ Z t1 2 (2c ∣!(t)∣2 + ∣y(t)∣2 − ∣y(t)∣2 + Re d˙(t; y(t), (t)))dt. (89) 2 V D d VD 0 Finally we use the diﬀerentiability of the damping sesquilinear form d: ˙ ∣d(t; y(t), (t))∣ ≤ c4∣y(t)∣VD ∣ (t)∣VD c 1 ≤ 4 (∣y(t)∣2 + ∣ (t)∣2 ) 2 VD VD c4 2 2 2 2 2 ≤ (∣y(t)∣ + ∣!(t)∣ + ∣!(t1)∣ ) 2 VD VD VD for all > 0. We now have an inequality

2 2 2c4t1 2 ∣y(t1)∣ + ( − 4c2t1)∣!(t1)∣ − ∣!(t1)∣ ≤ V VD Z t1 c 2 2 (2c ∣!(t)∣2 + ∣y(t)∣2 − ∣y(t)∣2 + 4 (∣y(t)∣2 + ∣!(t)∣2 ))dt 2 V D d VD VD VD 0 2 (90) and using here that

2 2 2 ( − 4c2t1)∣!(t1)∣ ≥ ( − 2c2t1)(∣!(t1)∣ + ∣!(t1)∣ ), (91) V 2 V VD we ﬁnally arrive at the formidable inequality

2 2 2c4t1 2 ∣y(t1)∣ + ( − 2c2t1)∣!(t1)∣ + ( − 2c2t1 − )∣!(t1)∣ ≤ 2 V 2 VD Z t1 c c 2 (2c ∣!(t)∣2 + ∣y(t)∣2 + ( 4 − )∣y(t)∣2 + 4 ∣!(t)∣2 )dt. 2 V D d VD VD 0 2

c4 Now ﬁx > 0 such that ( 2 − d) < 0 and ﬁx t1 = c4 such that 8(c2+ ) 2c4t1 ( 2 − 2c2t1 − ) = 4 . Then also ( 2 − 2c2t1) > 0 and the inequality above implies that for some constant M > 0:

Z t1 ∣y(t )∣2 + ∣!(t )∣2 + ∣!(t )∣2 ≤ M (∣y(t)∣2 + ∣!(t)∣2 + ∣!(t)∣2 )dt, (92) 1 1 V 1 VD V VD 0 and from Gronwall’s inequality we see that y = 0 in the interval [0, t1]. Since the length of t1 is independent of the choice of origin, we conclude that y = 0 on [t1, 2t1], etc. Hence y = 0 and uniqueness is proved. That the

54 solution depends continuously on the data is obvious from the inequalities used to show existence; indeed, from (67) and (68) and the weak lower semicontinuity of norms we conclude that the constructed solution satisﬁes

Z t ∣y(t)∣2 + ∣y˙(t)∣2 + ∣y˙(t)∣2 ≤ V VD 0 Z t 0 2 1 2 2 K(∣y ∣V + ∣y ∣ + ∣f(s)∣V ∗ ds) (93) 0 D for some positive constants and K. Integrating from 0 to T yields the desired result (since (y0, y1, f) → (y, y˙) is linear). This completes the proof of Theorem 18.

The proof of Theorem 19 follows from the inequality (92) and the original proof in the case d = 0 from of [LiMag, p. 275–279], because we do not gain any additional regularity from the form d in this case.

55 17 Inverse or Parameter Estimation Problems

In the generic abstract parameter estimation problem, we consider a dynamic model of the form (35) where the operators A and D and possibly the input f depend on some unknown (i.e., to be estimated) functional 1 parameters q in an admissible family Q ⊂ C (0,T ; L∞(Ω; Q)) of parameters. Here Ω is the underlying set on which the functions of H,V,VD are deﬁned (e.g., the spatial set Ω = (0, l) in the heat, transport, and beam examples). We assume that the time dependence of the operators A and D are through the time dependence of the parameters q(t) ∈ L∞(Ω; Q) where Q ⊂ Rp is a given constraint set for the values of the parameters. That is, A(t) = A1(q(t)),D(t) = A2(q(t)) so that we have

∗ y¨(t) + A2(q(t))y ˙(t) + A1(q(t))y(t) = f(t, q) in V (94) y(0) = y0, y˙(0) = y1.

Thus we introduce the sesquilinear forms 1, 2 by

a(t; , ) ≡ 1(q(t))(, ) = ⟨A1(q(t)), ⟩V ∗,V , (95) d(t; , ) ≡ (q(t))(, ) = ⟨A (q(t)), ⟩ ∗ . (96) 2 2 VD,VD It will be convenient in subsequent arguments to use the notation d ˙ (q)(, ) ≡ (q)(, ) (97) i dt i which in the event that i is linear in q becomes ˙ i(q)(, ) = i(q ˙)(, ). For example, in the Euler Bernoulli example of (52), we would have

Z l ′′ ′′ 1(q(t))(, ) = EIf (t, ) () ()d (98) 0 Z l ′′ ′′ 2(q(t))(, ) = CgDI(t, ) () ()d, (99) 0

1 ∞ 2 where q = (EI,f CgDI) ∈ C (0,T ; L (0, l; R+)). Note that in this case we do have ˙ i(q)(, ) = i(q ˙)(, ). In terms of parameter dependent sesquilinear forms we thus will write (94) as

⟨y¨(t), ⟩ + 2(q(t))(y ˙(t), ) + 1(q(t))(y(t), ) = ⟨f(t, q), ⟩ (100) y(0) = y0, y˙(0) = y1

56 for all ∈ V. As in (36), ⟨⋅, ⋅⟩ denotes the duality product ⟨⋅, ⋅⟩V ∗,V . In some problems the initial data y0, y1 may also depend on parametersq ˜ to be estimated, i.e., y0 = y0(˜q), y1 = y1(˜q). We shall not discuss such problems here, although the ideas we present can be used to effectively treat such problems. We instead refer readers to [13] for discussions of general estimation problems where not only the initial data but even the underlying spaces V and H themselves may depend on unknown parameters. It is assumed that the parameter-dependent sesquilinear forms 1(q), 2(q) of (100) satisfy the continuity and ellipticity conditions (H2)-(H7) of Sec- tion 1 uniformly in q ∈ Q; that is, the constants c1, c2, , c3, c4, d, d of (H2)-(H7) can be found independently of q ∈ Q. In general inverse problems, one must estimate the functional parameters q from dynamic observations of the system (94) or (100). A fundamental consideration in problem formulation involves what will be measured in the dynamic experiments producing the observations. To discuss these measurements in a specific setting, we consider the transverse vibrations (for example, equation (52)) of our beam example where y(t) = y(t, ⋅). Measure- ments, of course, depend on the sensors available. If one considers a truly smart material structure as in [BSW, RCS], it contains both sensors and actuators which may or may not rely on the same physical device or material. In usual mechanical experiments, there are several popular measurement devices [BSW, BT], some of which could possibly be used in a smart material configuration. If one uses an accelerometer placed at the point ¯ ∈ (0, ℓ) along the beam, then one obtains observationsy ¨(t, ¯) of beam acceleration. A laser vibrometer will yield data of velocity y˙(t, ¯) while proximity probes including displacement solenoids produce measurements of displacement y(t, ¯). In the case of a beam (or structure) with piezoceramic patches, the patches may be used as sensors as well as actuators. In this case one obtains observations of voltages which are proportional to the accumulated strain; this is discussed fully in [BSW, RCS]. Whatever the measuring devices, the resulting observations can be used in a maximum likelihood or one of several least squares formulations of the parameter estimation problem, depending on assumptions about the statistical model [7, BT] for errors in the observation process. In the least squares formulations, the problems are stated in terms of finding parameters which give the best fit of the parameter-dependent solutions of the partial differential equation to dynamic system response data collected after various excitations, while the maximum likelihood estimator results from maximizing a given (assumed) likelihood function for the parameters, given the data.

57 In the beam example, the parameters to be estimated include the stiﬀ- ness coeﬃcient EIf (t, ), the Kelvin-Voigt damping parameter cgDI(t, ), and any control related parameters that arise in the actuator input f. Details regarding the estimation of these parameters in the time independent case for an experimental beam are given in Section 5.4 of [BSW], while similar experimental results for a plate are summarized in Section 5.5 of the same reference. The general ordinary least squares parameter estimation problem can be formulated as follows. For a given discrete set of measured observations z = Nt {zi}i=1 corresponding to model observations zob(ti) at times ti as obtained in most practical cases, we consider the problem of minimizing over q ∈ Q the least squares output functional

n o 2 ˜ ˜ J(q, z) = C2 C1{y(ti, ⋅; q)} − {zi} , (101) where {y(ti, ⋅; q)} are the parameter dependent solutions of (94) or (100) evaluated at each time ti, i = 1, 2,...,Nt and ∣ ⋅ ∣ is an appropriately chosen Euclidean norm. Here the operators C˜1 and C˜2 are observation operators that depend on the type of observed or measured data available. The operator C˜1 may have several forms depending on the type of sensors being used. When the collected data zi consists of time domain displacement, velocity, or acceleration values at a point ¯ on the beam as discussed above, the functional takes the form

Nt 2 X ∂ w ¯ J(q, z) = (ti, ; q) − zi , (102) ∂t i=1 for = 0, 1, 2, respectively. In this case the operator C˜1 involves diﬀerenti- ation (either = 0, 1 or 2 times, respectively) with respect to time followed by pointwise evaluation in t and .¯ We shall in our presentation of the next section adopt the ordinary least squares functional (101) to formulate and develop our results.

17.1 Approximation and Convergence In this section we present a corrected version (Theorem 5.2 of [BSW] contains error in statement and proof) and extension (to treat time dependent coeﬃ- cients) of arguments for approximation and convergence in inverse problems found in Section 5.2 of [BSW]. For more details on general inverse problem methodology in the context of abstract structural systems, the reader may

58 consult [BSW]. The Banks-Kunisch book [13] contains a general treatment of inverse problems for partial differential equations in a functional analytic setting. The minimization in our general abstract parameter estimation problems for (101) involves an infinite dimensional state space H and an infinite dimensional admissible parameter set Q (of functions). To obtain computa- tionally tractable methods, we thus consider Galerkin type approximations in the context of the variational formulation (100). Let HN be a sequence of finite dimensional subspaces of H, and QM be a sequence of finite dimensional sets approximating the parameter set Q. We denote by P N the orthogonal projections of H onto HN . Then a family of approximating estimation problems with finite dimensional state spaces and parameter sets can be formulated by seeking q ∈ QM which minimizes

n o 2 N ˜ ˜ N J (q, z) = C2 C1{y (ti, ⋅; q)} − {zi} , (103)

where yN (t; q) ∈ HN is the solution to the ﬁnite dimensional approximation of (100)) given by

N N N ⟨y¨ (t), ⟩ + 2(q(t))(y ˙ (t), ) + 1(q(t))(y (t), ) = ⟨f(t, q), ⟩ (104) yN (0) = P N y0, y˙N (0) = P N y1,

for ∈ HN . For the parameter sets Q and QM , and state spaces HN , we make the following hypotheses.

(A1M) The sets Q and QM lie in a metric space Q˜ with metric d. It is assumed that Q and QM are compact in this metric and there is a mapping iM : Q → QM so that QM = iM (Q). Furthermore, for each q ∈ Q, iM (q) → q in Q˜ with the convergence uniform in q ∈ Q.

(A2N) The ﬁnite dimensional subspaces HN satisfy HN ⊂ V as well as the approximation properties of the next two statements.

N (A3N) For each ∈ V, ∣ − P ∣V → 0 as N → ∞.

N (A4N) For each ∈ VD, ∣ − P ∣VD → 0 as N → ∞.

The reader is referred to Chapter 4 of [BSW] for a complete discussion N motivating the spaces H and VD. We also need some regularity with respect to the parameters q in the parameter dependent sesquilinear forms 1, 2. In addition to (uniform in Q)

59 ellipticity/coercivity and continuity conditions (H2)-(H7), the sesquilinear forms 1 = 1(q), 2 = 2(q) and ˙ 1 = ˙ 1(q) are assumed to be deﬁned on Q and satisfy the continuity-with-respect-to-parameter conditions

(H8) ∣1(q)(, ) − 1(˜q)(, )∣ ≤ 1d(q, q˜)∣∣V ∣ ∣V , for , ∈ V

(H9) ∣˙ 1(q)(, ) − ˙ 1(˜q)(, )∣ ≤ 3d(q, q˜)∣∣V ∣ ∣V , for , ∈ V

(H10) ∣2(q)(, ) − 2(˜q)(, ) ≤ 2d(q, q˜)∣∣VD ∣∣VD , for , ∈ VD

for q, q˜ ∈ Q where the constants 1, 2, 3 depend only on Q. Solving the approximate estimation problems involving (103),(104), we obtain a sequence of parameter estimates {q¯N,M }. It is of paramount importance to establish conditions under which {q¯N,M } (or some subsequence) converges to a solution for the original inﬁnite dimensional estimation problem involving (100),(101). Toward this goal we have the following results.

Theorem 20 To obtain convergence of at least a subsequence of {q¯N,M } to a solution q¯ of minimizing (101) subject to (100), it suﬃces, under assumption (A1M), to argue that for arbitrary sequences {qN,M } in QM with qN,M → q in Q, we have

N N,M C˜2C˜1y (t; q ) → C˜2C˜1y(t; q). (105)

Proof: Under the assumptions (A1M), let {q¯N,M } be solutions minimizing (103) subject to the ﬁnite dimensional system (104) and letq ˆN,M ∈ Q be such that iM (ˆqN,M ) =q ¯N,M . From the compactness of Q, we may select sub- sequences, again denoted by {qˆN,M } and {q¯N,M }, so thatq ˆN,M → q¯ ∈ Q and q¯N,M → q¯ (the latter follows the last statement of (A1M)). The optimality of {q¯N,M } guarantees that for every q ∈ Q

J N (¯qN,M , z) ≤ J N (iM (q), z). (106)

Using (105), the last statement of (A1M) and taking the limit as N,M → ∞ in the inequality (106), we obtain J(¯q, z) ≤ J(q, z) for every q ∈ Q, or thatq ¯ is a solution of the problem for (100),(101). We note that under uniqueness assumptions on the problems (a situation that we hasten to add is not often realized in practice), one can actually guarantee convergence of the entire sequence {q¯N,M } in place of subsequential convergence to solutions. We note that the essential aspects in the arguments given above involve compactness assumptions on the sets QM and Q. Such compactness ideas play a fundamental role in other theoretical and computational aspects of

60 these problems. For example, one can formulate distinct concepts of problem stability and method stability as in [13] involving some type of continuous dependence of solutions on the observations z, and use conditions similar to those of (105) and (A1M), with compactness again playing a critical role, to guarantee stability. We illustrate with a simple form of method stability (other stronger forms are also amenable to this approach–see [13]). We might say that an approximation method, such as that formulated above involving QM ,HN and (103), is stable if

dist(˜qN,M (zk), q˜(z∗)) → 0 as N, M, k → ∞ for any zk → z∗ (in this case in the appropriate Euclidean space), whereq ˜(z) denotes the set of all solutions of the problem for (101) andq ˜N,M (z) denotes the set of all solutions of the problem for (103). Here “dist” represents the usual distance set function. Under (105) and (A1M), one can use arguments very similar to those sketched above to establish that one has this method stability. If the sets QM are not deﬁned through a mapping iM as supposed above, one can still obtain this method stability if one replaces the last statement of (A1M) by the assumptions:

(i) If {qM } is any sequence with qM ∈ QM , then there exist q∗ in Q and subsequence {qMk } with qMk → q∗ in the Q˜ topology.

(ii) For any q ∈ Q, there exists a sequence {qM } with qM ∈ QM such that qM → q in Q˜.

Similar ideas may be employed to discuss the question of problem stability for the problem of minimizing (101) over Q (i.e., the original problem) and again compactness of the admissible parameter set plays a critical role. Compactness of parameter sets also plays an important role in computational considerations. In certain problems, the formulation outlined above (involving QM = iM (Q)) results in a computational framework wherein the QM and Q all lie in some uniform set possessing compactness properties. The compactness criteria can then be reduced to uniform constraints on the derivatives of the admissible parameter functions. There are numerical examples (for example, see [BI86]) which demonstrate that imposition of these constraints is necessary (and suﬃcient) for convergence of the resulting algorithms. (This oﬀers a possible explanation for some of the numerical failures [YY] of such methods reported in the engineering literature.) Thus we have that compactness of admissible parameter sets play a fundamental role in a number of aspects, both theoretical and computational,

61 in parameter estimation problems. This compactness may be assumed (and imposed) explicitly as we have outlined here, or it may be included implicitly in the problem formulation through Tikhonov regularization as discussed for example by Kravaris and Seinfeld [KS], Vogel [Vog] and widely by many others. In the regularization approach one restricts consideration to a subset Q1 of parameters which has compact embedding in Q and modiﬁes the least-squares criterion to include a term which insures that minimizing sequences will be Q1 bounded and hence compact in the original parameter set Q. After this short digression on general inverse problem concepts, we return to the condition (105). To demonstrate that this condition can be readily established in many problems of interest to us here, we give the following general convergence results.

Theorem 21 Suppose that HN satisfies (A2N),(A3N),(A4N) and assume that the sesquilinear forms 1(q), ˙ 1(q) and 2(q) satisfy (H8),(H9),(H10), respectively, as well as (H1)-(H7) of Section 1 (uniformly in q ∈ Q). Fur- thermore, assume that ∗ q → f(⋅; q) is continuous from Q to L2(0,T ; VD). (107) Let qN be arbitrary in Q such that qN → q in Q. Then if in addition y˙ ∈ L2(0,T ; V ), we have as N → ∞, yN (t; qN ) → y(t; q) in V norm for each t > 0 N N y˙ (t; q ) → y˙(t; q) in L2(0,T ; VD) ∩ C(0,T ; H), where (yN , y˙N ) are the solutions to (104) and (y, y˙) are the solutions to (100). Proof: From Theorem 19 of Section 1 we find that the solution of (100) satisfies y(t) ∈ V for each t,y ˙(t) ∈ VD for almost every t > 0. Because N N N N N N ∣y (t; q ) − y(t; q)∣V ≤ ∣y (t; q ) − P y(t; q)∣V + ∣P y(t; q) − y(t; q)∣V , and (A3N) implies that second term on the right side converges to 0 as N → ∞, it suffices for the first convergence statement to show that N N N ∣y (t; q ) − P y(t; q)∣V → 0 as N → ∞. Similarly, we note that this same inequality with yN , y replaced byy ˙N , y˙ and the V -norm replaced by the VD-norm along with (A4N) permits us to claim that the convergence N N N ∣y˙ (t; q ) − P y˙(t; q)∣VD → 0 as N → ∞,

62 is suﬃcient to establish the second convergence statement of the theorem. We shall, in fact, establish the convergence ofy ˙N − P N y˙ in the stronger V norm. Let yN = yN (t; qN ), y = y(t; q), and ΔN = ΔN (t) ≡ yN (t; qN ) − P N y(t; q). Then d Δ˙ N =y ˙N − P N y =y ˙N − P N y˙ dt and d2 Δ¨ N =y ¨N − P N y dt2 ∗ becausey ˙ ∈ L2((0,T ),VD), y¨ ∈ L2((0,T ),V ). We suppress the dependence on t in the arguments below when no confusion will result. From (100) and (104), we have for ∈ HN

2 N N d N ⟨Δ¨ , ⟩ ∗ = ⟨y¨ − y¨ +y ¨ − P y, ⟩ ∗ V ,V dt2 V ,V N N N N N = ⟨f(q ), ⟩ ∗ − (q )(y ˙ , ) − (q )(y , ) VD,VD 2 1 − ⟨f(q), ⟩ ∗ + (q)(y, ˙ ) + (q)(y, ) VD,VD 2 1 2 d N + ⟨y¨ − P y, ⟩ ∗ . dt2 V ,V This can be written as

N N N ⟨Δ¨ , ⟩V ∗,V + 1(q )(Δ , ) 2 d N N = ⟨y¨ − P y, ⟩V ∗,V − ⟨f(q) − f(q ), ⟩V ∗ ,V dt2 D D N N N (108) + 2(q )(y ˙ − P y,˙ ) + 2(q)(y, ˙ ) − 2(q )(y, ˙ ) N N N + 1(q )(y − P y, ) + 1(q)(y, ) − 1(q )(y, ) N N − 2(q )(Δ˙ , ).

Choosing Δ˙ N as the test function in (108) and employing the equality ¨ N ˙ N 1 d ˙ N 2 ⟨Δ , Δ ⟩V ∗,V = 2 dt ∣Δ ∣H (this follows using deﬁnitions of the duality mapping - see [?] and the hypothesis (A2N)), we have using the symmetry

63 of 1 1 d n o ∣Δ˙ N ∣2 + (qN )(ΔN , ΔN ) 2 dt H 1 2 n d N N N N =Re ⟨y¨ − P y, Δ˙ ⟩V ∗,V − ⟨f(q) − f(q ), Δ˙ ⟩V ∗ ,V dt2 D D N N N N + 2(q )(y ˙ − P y,˙ Δ˙ ) + 2(q)(y, ˙ Δ˙ ) N N N N N − 2(q )(y, ˙ Δ˙ ) + 1(q )(y − P y, Δ˙ ) N N N N N N + 1(q)(y, Δ˙ ) − 1(q )(y, Δ˙ ) − 2(q )(Δ˙ , Δ˙ ) N N N o + ˙ 1(q )(Δ , Δ ) . (109)

d2 N N N ˙ ∗ We observe that ⟨y¨ − dt2 P y, Δ ⟩V ,V ≡ 0 because P is an orthogonal projection. Thus, we ﬁnd 1 d n o ∣Δ˙ N ∣2 + (qN )(ΔN , ΔN ) 2 dt H 1 n d = Re −T N + (qN )(y ˙ − P N y,˙ Δ˙ N ) + T N + ( (qN )(y − P N y, ΔN )) 3 2 2 dt 1 d − (qN )( (y − P N y), ΔN ) − ˙ (qN )(y − P N y, ΔN ) 1 dt 1 N N ˙ N ˙ N N N N o + T1 + 2(q )(Δ , Δ ) + ˙ 1(q )(Δ , Δ ) , (110) where N N ˙ N ˙ N N ˙ N T1 = Δ1 (y, Δ ) ≡ 1(q)(y, Δ ) − 1(q )(y, Δ ) N N ˙ N ˙ N N ˙ N T2 = Δ2 (y, ˙ Δ ) ≡ 2(q)(y, ˙ Δ ) − 2(q )(y, ˙ Δ ) (111) N N N N N T = ⟨Δf , Δ˙ ⟩ ∗ ≡ ⟨f(q) − f(q ), Δ˙ ⟩ ∗ . 3 VD,VD VD,VD

Note that here we have used thaty ˙ ∈ L2(0,T ; V ). Integrating the terms in (110) from 0 to t and using the initial conditions

ΔN (0) = yN (0) − P N y(0) = yN (0) − P N y0 = 0 Δ˙ N (0) =y ˙N (0) − P N y˙(0) =y ˙N (0) − P N y1 = 0, we obtain (here we do include arguments (t) and (s) in our calculations and

64 estimates to avoid confusion)

1 ∣Δ˙ N (t)∣2 + (qN (t))(ΔN (t), ΔN (t)) 2 H 1 Z t n N N N = Re{2(q (s))(y ˙(s) − P y˙(s), Δ˙ (s)) 0 N N N N N N − 1(q (s))(y ˙(s) − P y˙(s), Δ (s)) − ˙ 1(q (s))(y(s) − P y(s), Δ (s)) N ˙ N ˙ N N − 2(q (s))(Δ (s), Δ (s)) + T1 (s) N N N N N o + T2 (s) + T3 (s) + ˙ 1(q (s))(Δ (s), Δ (s))} ds

n N N o + Re 1(q(t))(y(t) − P y(t), Δ (t)) . (112)

N We consider the Ti terms in this equation. We have d T N = ΔN (y, ΔN ) − ΔN (y, ˙ ΔN ) − Δ ˙ N (y, ΔN ) 1 dt 1 1 1 so that

Z t N N N T1 (s)ds = Δ1 (y(t), Δ (t)) 0 Z t n N N N N o − Δ1 (y ˙(s), Δ (s)) − Δ ˙ 1 (y(s), Δ (s)) ds. 0 Using (H8),(H9) we thus obtain

Z t 2 N 1 N 2 2 N 2 Re T1 (s)ds ≤ d(q , q) ∣y(t)∣V + ∣Δ (t)∣V 0 4 Z t 2 n 1 N 2 2 N 2 + d(q , q) ∣y˙(s)∣V + ∣Δ (s)∣V 0 4 2 o + 3 d(qN , q)2∣y(s)∣2 + ∣ΔN (s)∣2 ds. (113) 4 V V Similarly, using (H10) we ﬁnd

Z t N N Re (T2 (s) + T3 (s))ds ≤ 0 Z tn 2 1 o 2 d(qN , q)2∣y˙(s)∣2 + 2∣Δ˙ N (s)∣2 + ∣Δf N (s)∣2 ds. (114) V VD V 0 4 4

65 These can then be used in (112) to obtain the estimate 1 Z t ∣Δ˙ N (t)∣2 + ∣ΔN (t)∣2 + ∣Δ˙ N (s)∣2 ds H V d VD 2 0 Z tn c2 ≤ ∣Δ˙ N (s)∣2 + 3 ∣y˙(s) − P N y˙(s)∣2 + 3∣Δ˙ N (s)∣2 d H VD VD 0 4 c2 c2 + 1 ∣y˙(s) − P N y˙(s)∣2 + 2 ∣y(s) − P N y(s)∣2 + (1 + + c )∣ΔN (s)∣2 2 V 2 V 2 V 2 2 2 + 1 d(qN , q)2∣y˙(s)∣2 + 3 d(qN , q)2∣y(s)∣2 + 2 d(qN , q)2∣y˙(s)∣2 4 V 4 V 4 V 1 o c2 2 + ∣Δf N (s)∣2 ds + 1 ∣y(t) − P N y(t)∣2 + 2∣ΔN (t)∣2 + 1 d(qN , q)2∣y(t)∣2 . 4 V 4 V V 4 V This ﬁnally reduces to

1 Z t ∣Δ˙ N (t)∣2 + ( − 2)∣ΔN (t)∣2 + ( − 3)∣Δ˙ N (s)∣2 ds H V d VD 2 0 Z t n ˙ N 2 N 2 o ≤ d∣Δ (s)∣H + (1 + + c2)∣Δ (s)∣V ds 0 c2 Z tnc2 c2 o + 1 ∣y(t) − P N y(t)∣2 + 3 ∣y˙(s) − P N y˙(s)∣2 + 1 ∣y˙(s) − P N y˙(s)∣2 ds V VD V 4 0 4 2 | {z } N 1 (t) 2 Z t 2 1 N 2 2 n 1 N 2 c2 N 2 o + d(q , q) ∣y(t)∣V + ∣Δf (s)∣V + ∣y(s) − P y(s)∣V ds 4 0 4 2 | {z } N 2 (t) Z t 2 2 2 n 1 N 2 2 3 N 2 2 2 N 2 2 o + d(q , q) ∣y˙(s)∣V + d(q , q) ∣y(s)∣V + d(q , q) ∣y˙(s)∣V + ds . 0 4 4 4 | {z } N 3 (t)

Therefore, under the assumptions we have as N → ∞ and qN → q N N N sup [1 (t) + 2 (t) + 3 (t)] → 0. t∈(0,T ) Applying Gronwall’s inequality, we ﬁnd that Δ˙ N → 0 in C(0,T ; H) ΔN → 0 in C(0,T ; V ) N Δ˙ → 0 in L2(0,T ; VD),

66 and hence the convergence statement of the theorem holds. Remark: The condition “y ˙ ∈ L2(0,T ; V )” routinely occurs if the structure under investigation has suﬃciently strong damping so that VD is equivalent to V .

17.2 Some Further Remarks We have provided sufficient conditions and detailed arguments for existence and uniqueness of solutions to abstract second order non-autonomous hyperbolic systems such as those with time dependent “stiffness”, “damping” and input parameters. In addition, we have considered a class of corresponding inverse problems for these systems and argued convergence results for approximating problems that yield a type of method stability as well as a framework for finite element type computational techniques. The efficacy of such methods have been demonstrated for autonomous systems in earlier efforts [BSW]. The approaches developed here can readily be extended (al- beit with considerable tedium) for higher dimensional spatial systems such as plates, shells, etc.

67 18 “Weak” or “Variational Form”

We consider the origin of the terms “weak or variational form” as opposed to strong or closed form of PDE’s. We use the beam equation to illustrate ideas. Recall Example 6, the cantilever beam. This example, given in classical form (which can be derived in a straight forward manner using force and moment balance–see [BT]) is

∂2y ∂y ∂2 ∂2y ∂3y + + EI + c I = f(t, ) ∂t2 ∂t ∂2 ∂2 D ∂2∂t with boundary conditions

y(t, 0) = 0 ∂y (115) ∂ (t, 0) = 0

∂2y ∂3y EI 2 + cDI 2 ∣=l = 0 ∂ ∂ ∂t (116) ∂ h ∂2y ∂3y i ∂ EI ∂2 + cDI ∂2∂t ∣=l = 0 and initial conditions y(0, ) = Φ() ∂y ∂t (0, ) = Ψ() To facilitate our discussions, we consider an undamped and unforced version (i.e., = cDI = 0, f = 0) of the above system. Rather than force and moment balance, we consider energy formulations for the beam. For a segment of the beam in [, + Δ], one can argue that the kinetic energy (at a given time t) is given by

1 Z +Δ ∂y KE(t) = T (t) = ( (t, ))2d, 2 ∂t and hence the kinetic energy of the entire beam is given by

1 Z l T (t) = y˙2(t, )d. 2 0 Similarly, the potential (or strain) energy U of the beam at any given time t is given by Z l 2 1 ∂ y 2 PE(t) = U(t) = EI( 2 ) d. 2 0 ∂

68 A fundamental tenant of the mechanics of rigid or elastic bodies is Hamil- ton’s “Principle of Stationary Action” (often, in a misnomer, referred to as Hamilton’s principle of “least action”) which postulates that any system undergoing motion during a period [t0, t1] will exhibit motion y(t, ) that provides the “least action” for the system with a stationary value. The “Action” is deﬁned by

A = R t1 (KE − PE)dt t0

= R t1 [T (t) − U(t)]dt. t0 For the beam of Example 6, this means that the motion or vibrations y(t, ) must provide a stationary value to the action

Z t1 Z l 1 1 A[y] = [ y˙2 − EI(y′′)2]ddt t0 0 2 2 integrated over any time interval [t0, t1]. Through the calculus of variations (a field of mathematics that was the precursor to modern control theory), this leads to an equation of motion for the vibrations y that the beam motion must satisfy. To further explore this, we consider y(t, ) as the motion of the beam and consider a family of variations y(t, )+(t, ) where is chosen so that y + is an “admissible variation”, i.e., y + must satisfy the essential boundary conditions (115). 2 2 ′ We define V = HL(0, l) = {' ∈ H (0, l)∣'(0) = ' (0) = 0}. Let ∈ 2 C (t0, t1) with (t0) = (t1) = 0. Then ∈ N ≡ {∣ = ', ' ∈ V } 2 2 satisfies is C in t, H in with (t0, ) = (t1, ) = 0 and (t, 0) = ′(t, 0) = 0. Then by Hamilton’s principle, we must have that A[y + ] for > 0, ∈ N , must have a stationary value at = 0. That is, d A[y + ]∣ = 0. (117) d =0 Since Z t1 Z l 1 1 A[y + ] = [ (y ˙ + ˙)2 − EI(y′′ + ′′)2]ddt, t0 0 2 2 we find from (117) the weak (in time t and space ) form of the equation given by Z t1 Z l 0 = [y˙˙ − EIy′′′′]ddt (118) t0 0

69 for all ∈ N . To explore further, we formally integrate by parts in the ﬁrst term (with respect to t) to obtain

R t1 R l y˙ddt˙ = − R t1 R l yddt¨ + R l yd˙ ∣t=t1 t0 0 t0 0 0 t=t0

= − R t1 R l yddt¨ t0 0 since (t0, ) = (t1, ) = 0. Since has the form = ', equation (118) has the form Z t1 Z l [y'¨ + EIy′′'′′] ddt = 0 (119) t0 0 2 for all ∈ C [t0, t1] with (t0) = (t1) = 0, and all ' ∈ V . Since this holds for arbitrary , we must have in the L2(t0, t1) sense Z l [y'¨ + EIy′′'′′]d = 0 for all ' ∈ V. 0 2 In our former notation of Gelfand triples with V = HL(0, l) and H = L2(0, l), this may be written ′′ ′′ ⟨y,¨ '⟩V ∗,V + ⟨EIy ,' ⟩H = 0 for all ' ∈ V in the L2(t0, t1) sense, which is exactly the “weak” or “variational” form of the beam equation we have encountered previously. Note that in fact the true variational form was given in (118); that is,

Z t1 [−⟨y,˙ '⟩ ˙ + ⟨EIy′′,'′′⟩ ]dt = 0 t0 2 for all ' ∈ V and ∈ C [t0, t1] with (t0) = (t1) = 0. (See the proofs and ∗ ∼ ∗ our remarks concerning solutions in the L2(t0, t1; V ) = L2(t0, t1; V ) sense in the well-posedness (existence) results for second order systems discussed earlier in Section 16. We note that if the variational solution y has additional smoothness so that y ∈ V T H4(0, l) (more precisely EIy′′ ∈ H2(0, l)), then we can integrate by parts twice (with respect to ) in the second term of (119) to obtain in place of (119):

Z t1 Z l Z t1 ′′ ′′ ′′ ′ =l [y'¨ + (EIy ) '] ddt + −(EIy )' ∣=0 dt t0 0 t0 Z t1 ′′ ′ =l + (EIy ) '∣=0 dt = 0 t0

70 2 for ' ∈ V, ∈ C [t0, t1] with (t0) = (t1) = 0. This can be written

Z t1 Z l ′′ ′′ ′′ ′ ′′ ′ [y'¨ + (EIy ) ']d + −EIy ' ∣=l + (EIy ) '∣=l dt = 0 t0 0 for arbitrary ' ∈ V . We note once again that this results in the strong or classical form of the equations

∂2y ∂2 ∂2y + (EI ) = 0 ∂t2 ∂2 ∂2 with the essential boundary conditions

y(t, 0) = y′(t, 0) = 0 as well as the natural boundary conditions

∂2y ∂ ∂2y EI (t, l) = (EI )(t, l) = 0 ∂2 ∂ ∂2 holding.

We remark on an additional perspective of the weak or variational or energy formulation of dynamical systems such as the beam equation. The weak or variational form may be thought of as Euler’s equations in the calculus of variations. If Z J(y) = F (t, y, y˙)dt = A[y], then the condition of stationarity d J(y + )∣ = 0 d =0 implies Z (Fy + Fy˙ ˙)dt = 0 which is the “true” Euler’s equation. We may then invoke the duBois Rey- mond lemma below to obtain

∂F d ∂F = dy dt dy˙ in a distributional sense. Speciﬁcally, we have

71 Lemma 1 (duBois Raymond’s Lemma) If Z (G1 + G2˙) = 0 for all , then d G = G dt 2 1 in a weak sense. In other words, G1 is the distributional derivative of G2. If we assume enough smoothness and integrate by parts, we obtain the strong form of Euler’s equation: d − F (t, y, y˙) + F (t, y, y˙) = 0. dt y˙ y

In the derivation above, we considered the undamped and unforced version of the equation. In the case of the forced beam, we can add a conservative force term W = fy in our derivation, and we will obtain the desired ⟨f, '⟩ term in our result. However, there is no known way to derive the weak form with damping. In other words, Hamilton’s principle is essentially valid for conservative forces, but it doesn’t conveniently handle nonconservative (dissipative) forces (damping).

72 19 Finite Element Approximations

We will now consider finite element approximations or Galerkin approximations for general first order systems for which parabolic systems are special cases. Consider x˙(t) = Ax + F in V ∗ (H if possible) (120) x(0) = x0 where V,→ H,→ V ∗ is the usual Gelfand triple. We can write the above system in the weak or variational form (i.e., in V ∗ ) as ⟨x˙(t),'⟩V ∗,V + (x(t),') = ⟨F (t),'⟩V ∗,V x(0) = x0 for ' ∈ V . If is V continuous and V -elliptic, and S(t) ∼ eAt (i.e., A is the infinitesimal generator of a C0 semigroup S(t)), we can write Z t x(t) = S(t)x0 + S(t − )F ()d (121) 0 T ∗ where x ∈ L2(0,T ; V ) C(0,T ; H) andx ˙ ∈ L2(0,T ; V ). We can use this formulation to give a nice treatment of finite element approximations of Galerkin type. In general, this is an infinite dimensional space; therefore, we want to project the system into a finite dimensional space in which we can compute. N N N N Let H = span{B1 ,B2 , ..., BN } ⊂ V be the approximation of H. Typical N choices for the Bj are piecewise linear or piecewise cubic splines (e.g., see [13]). The idea is to replace (120) by

x˙ N (t) = AN xN (t) + F N (t) in HN N N x (0) = x0 , or equivalently, replace (121) by

Z t N N N N N x (t) = S (t)x0 + S (t − )F ()d 0 where SN (t) ∼ eAN t. One of the key constructs we need is P N : H → HN which is called the orthogonal projection of H onto HN . In other words, P N is deﬁned by

⟨P N ' − ', ⟩ = 0 for all ∈ HN

73 or N ∣P ' − '∣H = inf ∣ − '∣H . ∈HN N N N N N We would like F → F and x0 → x0, so we take x0 = P x0 and F (t) = P N F (t). We also want AN ∈ ℒ(HN ) and AN ≈ A. However, we have defined SN (t) = eAN t and S(t) ∼ eAt; therefore, if we had SN (t) → S(t), then we would be able to argue xN (t) → x(t). This is considered in the Trotter-Kato theorem which will be discussed later. Now to relate this to the computational aspects of finite elements, we first restrict the equations

⟨x˙(t),'⟩ + 1(x(t),') = ⟨F (t),'⟩ for all ' ∈ V (122) to HN × HN . In other words, let

N N X N N x (t) = wj (t)Bj j=1 be a trial solution with

N N X N N x (0) = w0jBj . j=1

Substituting this into (122), we have

N N X N N X N N ⟨ w˙ j (t)Bj ,'⟩ + 1( wj (t)Bj ,') = ⟨F (t),'⟩ (123) j=1 j=1

N N N N for ' ∈ H . Successively choose ' = B1 ,B2 , ...BN in (123). From this we N N N T obtain an N × N vector system for w (t) = (w1 (y), .., wN (t)) given by

N N X N N N X N N N N w˙ j (t)⟨Bj ,Bi ⟩ + wj (t)(Bj ,Bi ) = ⟨F (t),Bi ⟩ (124) j=1 j=1 for i = 1, 2, ..., N. Using standard engineering and applied mathematics terminology, we N N N N deﬁne the mass matrix M = (⟨Bi ,Bj ⟩), the stiﬀness matrix K = N N N N ((Bi ,Bj )), and the column vector F (t) = (⟨F (t),Bi ⟩). Then (124) can be written M N w˙ N (t) + KN wN (t) = F N (t) N N (125) w (0) = w0

74 or w˙ N (t) = −(M N )−1KN wN (t) + (M N )−1F N (t) N N w (0) = w0 . N N N N N Considering w0 , we have x (0) = P x0 which implies ⟨P x0−x0,Bi ⟩ = N PN N N 0 for i = 1, ..., N. However, x0 = j=1 w0jBj . Therefore,

N X N N N ⟨ w0jBj − x0,Bi ⟩ = 0 j=1 for i = 1, ..., N which gives

N X N N N N w0j⟨Bj ,Bi ⟩ = ⟨x0,Bi ⟩. j=1

N N N Deﬁning w0 = col(w01, ..., w0N ), we have

N N −1 N w0 = (M ) col(⟨x0,Bi ⟩).

From this, our system for w becomes

w˙ N (t) = −(M N )−1KN wN (t) + (M N )−1F N (t) N N −1 N w0 = (M ) col(⟨x0,Bi ⟩).

However, we normally do not solve the system in this form. If ⟨Bi,Bj⟩ = 0 for i ∕= j, then M N is diagonal and the system of the form (125) is an easier system with which to work. More generally, the (ﬁnite element) system is solved in the form

M N w˙ N (t) = −KN wN (t) + F N (t) N N N M w0 = col(⟨x0,Bi ⟩), rather than inverting the matrix M N which is, if not diagonal, usually a banded matrix (tri-banded for piecewise linear splines, seven banded for cubic splines, etc, –see [13]), lending itself to fast algebraic solvers (e.g., those based on LU decompositions).

75 20 Trotter-Kato Approximation Theorem

The Trotter-Kato Approximation Theorem is the functional analysis version of the Lax Equivalence Principle used in ﬁnite diﬀerence approximation for PDE’s which dates back to the 1960’s. The fundamental ideas underlying the Lax Equivalence Principle is that “consistency” and “stability” are achieved if and only if we have “convergence” of our system. If we have a PDE

utt = Au and an approximation N N N utt = A u then consistency refers to AN → A in some sense. Stability refers to ∣eAN t∣ ≤ Me!t, and convergence means eAN t → eAt in some sense. For relevant and much more detailed discussions, see [RM]. There are two diﬀerent versions of the Trotter-Kato theorem which we will discuss here. We will ﬁrst consider the operator convergence form of the Trotter-Kato theorem.

Theorem 22 Let X and XN be Hilbert spaces such that XN ⊂ X. Let P N : X → XN be an orthogonal projection of X onto XN . Assume P N x → N x for all x ∈ X. Let A ,A be inﬁnitesimal generators of C0 semigroups SN (t),S(t) on XN ,X respectively satisfying

(i) there exists M,! such that ∣SN (t)∣ ≤ Me!t for each N

(ii) there exists D dense in X such that for some , (I − A)D is dense in X and AN P N x → Ax for all x ∈ D.

Then for each x ∈ X, SN (t)P N x → S(t)x uniformly in t on compact intervals [0,T ]. The arguments for this result are given in [Pa, Chapter 3, Theorem 4.5 ]. We next will examine theresolvent convergence version of the Trotter-Kato theorem. This form is a modiﬁcation of the previous version.

Theorem 23 Replace (ii) in the above theorem by (ii˜) deﬁned as

∞ T N N N (ii˜) There exists ∈ (A) (A ) with Re() > ! so that R(A )P x N=1 → R(A)x for each x ∈ X.

76 Then under this and the remaining conditions in the above theorem, the conclusions also hold. For proofs, see [Pa, Chap. 3, Theorems 4.2,4.3,4.4]. See also [13, Theo- rem 1.14]. Convergence rates are given in [13, Theorem 1.16]. For certain problems, it is not necessary for HN ⊂ dom(A) which carries both the essential and natural boundary conditions. We may need to only choose an appropriate approximation HN such that HN ⊂ V which carries just the essential boundary conditions. If we restrict ourselves to ﬁrst order systems in the context of the Gelfand triple V,→ H,→ V ∗ with HN approximating V as N → ∞ then we have a special case of the Trotter-Kato theorem. Let the condition (C1) be denoted by N N N (C1) For each z ∈ V , there existsz ˆ ∈ H such that ∣z − zˆ ∣V → 0 as N → ∞. 2 Suppose is V -elliptic, i.e., Re(', ') ≥ ∣'∣V for > 0. Also assume N N is V continuous, i.e., ∣(', )∣ ≤ ∣'∣V ∣ ∣V . Let P : H → H be an orthogonal projection. Then

N N N N ∣P z − z∣ = inf{∣z − z∣H ∣ z ∈ H }.

N N N Under (C1), we have ∣P z − z∣H ≤ ∣zˆ − z∣H ≤ ∣zˆ − z∣V → 0 as N → ∞. Therefore, under (C1), P N z → z for z ∈ H. N Next we define A . We have (', ) = ⟨−A', ⟩H for ' ∈ dom(A) = { ∈ V ∣A ∈ H}, ∈ V where A is an infinitesimal generator of a C0 semigroup of contractions on H, i.e., ∣eAt∣ ≤ 1. We define AN through the restriction of to HN × HN . Therefore, AN : HN → HN is defined by

N N N N N N N N (' , ) = ⟨−A ' , ⟩H ' , ∈ H N N = ⟨−A' , ⟩V ∗,V .

By V -ellipticity, we obtain AN is an inﬁnitesimal generator of a contraction semigroup on HN , i.e., ∣eAN t∣ ≤ 1.

Theorem 24 If is V -elliptic, V continuous and (C1) holds, then

N N R(A )P z → R(A)z in the V norm for z ∈ H and = 0. See [BI] for additional discussion.

77 Proof

N N N Let z ∈ H and take = 0. Now deﬁne w = R(A )P z and w = R(A)z where (', ) = ⟨−A', ⟩H ,' ∈ dom(A), ∈ V . By deﬁnition, we have w ∈ dom(A). N N N By (C1), there existsw ˆ ∈ H such that ∣wˆ − w∣V → 0 as N → ∞. Let zN = wN − wˆN . We need to show zN → 0 in V . −1 Since R(A) = (I − A) , we have N N (w, z ) = ⟨−AR(A)z, z ⟩H = ⟨z, zN ⟩ while N N N N N (w , z ) = ⟨−A R(A )z, z ⟩H = ⟨z, zN ⟩, and hence these are equal. Thus

N 2 N N ∣z ∣V ≤ (z , z ) = (wN , zN ) − (w ˆN , zN ) = (w, zN ) − (w ˆN , zN ) = (w − wˆN , zN ) N N ≤ ∣w − wˆ ∣V ∣z ∣V .

N N N Therefore, we have ∣z ∣V ≤ ∣w − wˆ ∣V . However, ∣wˆ − w∣V → 0 implies N N N N N ∣z ∣V → 0 and thus ∣w − w ∣V ≤ ∣w − wˆ ∣V + ∣wˆ − w ∣V → 0. Remark: Theorem 2.2 of [BI] is a parameter dependent version of this, i.e., A = A(q),AN = AN (qN ), q, qN ∈ Q, which can be used in inverse problems.

Theorem 25 Suppose is V -elliptic, V continuous, and (C1) holds. Then SN (t)P N z → S(t)z in the V norm for each z ∈ H uniformly in t on compact intervals.

Proof The arguments involve using Tanabe type estimates after considering an appropriate setting and using the Trotter-Kato, ﬁrst in H and then in V . Let X = H,XN = HN ,P N : H → HN be an orthogonal projection. Then P N z → z for all z ∈ H by (C1). Convergence in H norm follows immediately from application of the Trotter-Kato in H. To obtain V convergence is somewhat more work and more delicate; for these details we refer the reader to [BI, Theorem 2.3].

78 References

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[Aubin] J.-P. Aubin, Optima and Equilbria: An Introduction to Nonlinear Analysis, Spring-Verlag, Berlin Heidelberg, 1993.

[BJR] A. Bamberger, P. Joly and J. E. Roberts, Second-order absorbing boundary conditions for the wave equation: a solution for the corner problem, SIAM J. Numer. Anal., 27 (1990), 323–352.

[BBDS1] H.T. Banks, J.E. Banks, L.K. Dick, and J.D. Stark, Estimation of dynamic rate parameters in insect populations undergoing sublethal exposure to pesticides, CRSC-TR05-22, NCSU, May, 2005; Bulletin of Mathematical Biology, 69 (2007), 2139–2180.

[BBDS2] H.T. Banks, J.E. Banks, L.K. Dick, and J.D. Stark, Time-varying vital rates in ecotoxicology: selective pesticides and aphid population dynamics, Ecological Modelling, 210 (2008), 155–160.

[BBJ] H. T. Banks, J. E. Banks and S. L. Joyner, Estimation in time- delay modeling of insecticide-induced mortality, CRSC-TR08-15, Octo- ber, 2008; J. Inverse and Ill-posed Problems, 17 (2009), 101–125.

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[BBo] H.T. Banks and V.A. Bokil, Parameter identiﬁcation for dispersive dielectrics using pulsed microwave interrogating signals and acoustic wave induced reﬂections in two and three dimensions, CRSC-TR04-27, July, 2004; Revised version appeared as A computational and statistical framework for multidimensional domain acoustoptic material interrogation, Quart. Applied Math., 63 (2005), 156–200.

[Shrimp] H.T.Banks, V.A. Bokil, S. Hu, F.C.T. Allnutt, R. Bullis, A.K. Dhar and C.L. Browdy, Shrimp biomass and viral infection for produc- tion of biological countermeasures, CRSC-TR05-45, NCSU, December, 2005; Mathematical Biosciences and Engineering, 3 (2006), 635–660.

[BBH] H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variability into the mathematical modeling of viral delays in HIV infection dynamics, Math. Biosciences, 183 (2003), 63–91.

79 [BBPP] H.T. Banks, D.M. Bortz, G.A. Pinter and L.K. Potter, Modeling and imaging techniques with potential for application in bioterrorism, CRSC-TR03-02, January, 2003; Chapter 6 in Bioterrorism: Mathe- matical Modeling Applications in Homeland Security, (H.T. Banks and C. Castillo-Chavez, eds.), Frontiers in Applied Math, FR28, SIAM, Philadelphia, PA, 2003, 129–154. [BBKW] H.T. Banks, L.W. Botsford, F. Kappel, and C. Wang, Modeling and estimation in size structured population models, LCDS-CCS Re- port 87-13, Brown University; Proceedings 2nd Course on Mathematical Ecology, (Trieste, December 8-12, 1986) World Press (1988), Singapore, 521–541. [BBL] H.T. Banks, M.W. Buksas, T. Lin. Electromagnetic Material Inter- rogation Using Conductive Interfaces and Acoustic Wavefronts. SIAM FR 21, Philadelphia, 2002. [BBu1] H.T. Banks and J.A. Burns, An abstract framework for approximate solutions to optimal control problems governed by hereditary systems, International Conference on Differential Equations (H. An- tosiewicz, Ed.), Academic Press (1975), 10–25. [BBu2] H.T. Banks and J.A. Burns, Hereditary control problems: numerical methods based on averaging approximations, SIAM J. Control & Opt., 16 (1978), 169–208. [BDSS] H.T. Banks, M. Davidian, J.R. Samuels Jr., and K. L. Sutton, An inverse problem statistical methodology summary, Center for Research in Scientific Computation Report, CRSC-TR08-01, January, 2008, NCSU, Raleigh; Chapter XX in Statistical Estimation Approaches in Epidemi- ology, (edited by Gerardo Chowell, Mac Hyman, Nick Hengartner, Luis M.A Bettencourt and Carlos Castillo-Chavez), Springer, Berlin Heidel- berg New York, to appear. [BD] H.T. Banks and J.L. Davis, A comparison of approximation methods for the estimation of probability distributions on parameters, CRSC- TR05-38, NCSU, October, 2005; Applied Numerical Mathematics, 57 (2007), 753–777. [BDTR] H.T. Banks and J.L. Davis, Quantifying uncertainty in the estimation of probability distributions with confidence bands, CRSC-TR07- 21, NCSU, December, 2007; Mathematical Biosciences and Engineer- ing, 5 (2008), 647–667.

80 [GRD-FP] H.T. Banks, J.L. Davis, S.L. Ernstberger, S. Hu, E. Artimovich, A.K. Dhar and C.L. Browdy, A comparison of probabilistic and stochastic formulations in modeling growth uncertainty and variability, CRSC- TR08-03, NCSU, February, 2008; Journal of Biological Dynamics, 3 (2009) 130–148.

[BDEHAD] H.T. Banks, J.L. Davis, S.L. Ernstberger, S. Hu, E. Artimovich and A.K. Dhar, Experimental design and estimation of growth rate distributions in size-structured shrimp populations, CRSC-TR08-20, NCSU, November, 2008; Inverse Problems, to appear.

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[BDEK] H.T. Banks, S. Dediu, S. L. Ernstberger and F. Kappel, Gener- alized sensitivities and optimal experimental design, CRSC-TR08-12, September, 2008, Revised Nov, 2009; J Inverse and Ill-Posed Problems, submitted.

[BF] H.T. Banks and B.G. Fitzpatrick, Estimation of growth rate distributions in size-structured population models, CAMS Tech. Rep. 90-2, University of Southern California, January, 1990; Quarterly of Applied Mathematics, 49 (1991), 215–235.

[BFPZ] H.T. Banks, B.G. Fitzpatrick, L.K. Potter, and Y. Zhang, Es- timation of probability distributions for individual parameters using aggregate population data, CRSC-TR98-6, NCSU, January, 1998; In Stochastic Analysis, Control, Optimization and Applications, (Edited by W. McEneaney, G. Yin and Q. Zhang), Birkha¨user,Boston, 1989.

[BG1] H.T. Banks and N.L. Gibson. Well-posedness in Maxwell systems with distributions of polarization relaxation parameters. CRSC, Tech Report TR04-01, North Carolina State University, January, 2004; Ap- plied Math. Letters, 18 (2005), 423–430.

[BG2] H.T. Banks and N.L. Gibson, Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters, CRSC- TR05-29, August, 2005; Quarterly of Applied Mathematics, 64 (2006), 749–795.

81 [BGIT] H.T. Banks, S.L. Grove, K. Ito and J.A. Toivanen, Static two-player evasion-interrogation games with uncertainty, CRSC-TR06-16, June, 2006; Comp. and Applied Math, 25 (2006), 289–306.

[BI86] H.T. Banks and D.W. Iles, On compactness of admissible parameter sets: convergence and stability in inverse problems for distributed parameter systems, ICASE Report #86-38, NASA Langley Res. Ctr., Hampton VA 1986; Proc. Conf. on Control Systems Governed by PDE’s, February, 1986, Gainesville, FL, Springer Lecture Notes in Con- trol & Inf. Science, 97 (1987), 130-142.

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[BI2] H.T. Banks and K. Ito, Approximation in LQR problems for inﬁnite dimensional systems with unbounded input operators, CRSC-TR94-22, N.C. State University, November, 1994; in J. Math. Systems, Estima- tion and Control, 7 (1997), 119–122.

[BIKT] H. T. Banks, K. Ito, G. M. Kepler and J. A. Toivanen, Material surface design to counter electromagnetic interrogation of targets, Tech. Report CRSC-TR04-37, Center for Research in Scientiﬁc Computation, N.C. State University, Raleigh, NC, November, 2004; SIAM J. Appl. Math., 66 (2006), 1027–1049.

[BIT] H. T. Banks, K. Ito and J. A. Toivanen, Determination of interrogating frequencies to maximize electromagnetic backscatter from objects with material coatings, Tech. Report CRSC-TR05-30, Center for Re- search in Scientiﬁc Computation, N.C. State University, Raleigh, NC, August, 2005; Communications in Comp. Physics, 1 (2006), 357–377.

[BKap] H.T. Banks and F. Kappel, Transformation semigroups and L1 approximation for size structured population models (with ), LCDS/CCS Rep. 88-20, July, 1988; Semigroup Forum, 38 (1989), 141–155.

[BKW1] H.T. Banks, F. Kappel and C. Wang, Weak solutions and diﬀer- entiability for size structured population models (with ), CAMS Tech. Rep. 90-11, August, 1990, University of Southern California; in DPS Control and Applications, Birkh¨auser, Intl. Ser. Num. Math., Vol.100 (1991), 35–50.

82 [BKW2] H.T. Banks, F. Kappel and C. Wang, A semigroup formulation of nonlinear size-structured distributed rate population model, CRSC- TR94-4, March 1994; in Control and Estimation of Distributed Param- eter Systems: Nonlinear Phenomena, Birkh¨auser ISNM, 118 (1994), 1–19.

[BKa] H.T. Banks and P. Kareiva, Parameter estimation techniques for transport equations with application to population dispersal and tis- sue bulk ﬂow models, LCDS Report #82-13, Brown University, July 1982; J. Math. Biol., 17 (1983), 253–273.

[BK] H.T. Banks and K. Kunisch. Estimation Techniques for Distributed Parameter Systems. Birkhausen, Bosten, 1989.

[BKcontrol] H.T. Banks and K. Kunisch, An approximation theory for nonlinear partial diﬀerential equations with applications to identiﬁcation and control, LCDS Technical Report #81-7, Brown University, April 1981; SIAM J. Control & Opt., 20 (1982), 815–849.

[BKurW1] H. T. Banks, A. J. Kurdila and G. Webb, Identiﬁcation of hys- teretic control inﬂuence operators representing smart actuators, Part I: Formulation, Mathematical Problems in Engineering 3 (1997), 287–328.

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[BaPed] H.T. Banks and M. Pedersen, Well-posedness of inverse problems for systems with time dependent parameters, CRSC-TR08-10, August, 2008; Arabian Journal of Science and Engineering: Mathematics, 1 (2009), 39–58.

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83 [BRS] H.T. Banks, K.L. Rehm and K.L. Sutton, Dynamic social network models incorporating stochasticity and delays, CRSC-TR09-11, May, 2009; Quarterly Applied Math., to appear.

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[BSTBRSM] H.T. Banks, Karyn L. Sutton, W. Clayton Thompson, Gen- nady Bocharov, Dirk Roose, Tim Schenkel, and Andreas Meyerhans, Estimation of cell proliferation dynamics using CFSE data, CRSC- TR09-17, August, 2009; Bull. Math. Biology, submitted.

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[6] H. T. Banks, D. M. Bortz, G. A. Pinter and L. K. Potter, Model- ing and imaging techniques with potential for application in bioterrorism, Chapter 6 in Bioterrorism: Mathematical Modeling Applications in Homeland Security, H.T. Banks and C. Castillo-Chavez, eds., Frontiers in Applied Mathematics, SIAM, Philadelphia, 2003, pp. 129–154.

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[9] H. T. Banks and J. L. Davis, Quantifying uncertainty in the estimation of probability distributions with conﬁdence bands, CRSC-TR07- 21, NCSU, December, 2007; Math. Biosci. Engr. , 5 (2008), 647–667.

[10] H. T. Banks, S. Dediu and H. K. Nguyen, Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space, Math. Biosci. and Engineering, 4 (2007), 403–430.

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[12] J. E. Banks, L. K. Dick, H. T. Banks and J. D. Stark, Time-varying vital rates in ecotoxicology: selective pesticides and aphid population dynamics, Ecological Modelling,210 (2008), 155–160.

[13] H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhauser, Boston, 1989.

[14] H. T. Banks and I. G. Rosen, Spline approximations for linear nonau- tonomous delay systems, J. Mathematical Analysis and Applications, 96 (1983), 226–268.

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[29] H.T. Banks and J. Burns, Introduction to Control of Distributed Pa- rameter Systems, CRSC Lecture Notes, 1996, to appear.

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90 Systems Governed by PDE’s, Feb., 1986, Gainesville, FL, Springer Lec- ture Notes in Control and Inf. Science, 97, 1987, pp. 120-142.

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[32] H.T. Banks and K. Ito, Approximation in LQR problems for inﬁnite dimensional systems with unbounded input operators, CRSC Techni- cal Report CRSC-TR94-22, November 1994; Journal of Mathematical Systems, Estimation and Control, to appear.

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[34] H.T. Banks and F. Kappel, Spline approximation for functional diﬀer- ential equations, Journal of Diﬀerential Equations, 34, 1979, pp. 496- 522.

[35] H.T. Banks and K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. Control and Optimization, 22, 1984, pp. 684-698.

[36] H.T. Banks, I.G. Rosen and K. Ito, A spline-based technique for com- puting Riccati operators and feedback controls in regulator problems for delay systems, SIAM J. Sci. Stat. Comp., 5, 1984, pp. 830-855.

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