Chapter 4 Optimal Control Problems in In¯nite Dimensional Function Space 4.1 Introduction In this chapter, we will consider optimal control problems in function space where we will restrict ourselves to state equations that are both linear in the state and linear in the control: We assume that E is a reflexive, separable Banach space with norm ¤ k ¢ kE and dual space E and X is another separable Banach space ¤ with norm k ¢ kX and separable dual space X . We further suppose A : D(A) ⊆ E ! E¤ to be the in¯nitesimal generator of a compact, analytic semigroup S(¢) in E (cf. section 4.2 below) and B : X¤ ! E to be a bounded linear operator. Finally, let g : C([0;T ]; E) ! C([0;T ]) and h : L1([0;T ]; X¤) ! L1([0;T ]) be strictly convex, coercive and continuously Fr¶echet-di®erentiable mappings with Fr¶echet-derivatives g0(¢) and h0(¢), respectively. We consider the control constrained optimal control problem: ZT ³ ´ (4.1a) minimize J(y; u) := g(y(t)) + h(u(t)) dt ; 0 (4.1b) subject toy _(t) = Ay(t) + Bu(t) ; t 2 [0;T ] ; (4.1c) y(0) = y0 ; (4.1d) u 2 L1([0;T ]; K) ;K ⊆ X¤; ¤ wherey _ := dy=dt; y0 2 E, and K is assumed to be weakly closed and convex. We note that L1([0;T ]; X¤) denotes the linear space of functions v ¤ with v(t) 2 X for almost all t 2 [0;T ] such that kv(t)kX¤ is essentially bounded in t 2 [0;T ]. It is a normed space with respect to the norm kvkL1([0;T ];X¤) := ess sup kv(t)kX¤ : t2[0;T ] Consequently, the control constraint (4.1d) implies u(t) 2 K for almost all t 2 [0;T ] : In addition to the above assumptions, we require the control constraint set L1([0;T ]; K) to be L1([0;T ]; X)-weakly compact. In the sequel, for Banach spaces E and F with norms k¢kE and k¢kF we refer to L(E; F ) as the normed space of bounded linear operators 97 98 Ronald H.W. Hoppe L : E ! F with norm kLyk kLk := sup F : y2Enf0g kykE 4.2 Evolution equations in function space Let E be a reflexive, separable Banach space with norm k ¢ kE and dual E¤ and A : D(A) ⊆ E ! E¤ be a densely de¯ned, closed linear operator. We consider the following initial-value problem (Cauchy problem) for an abstract evolution equation (4.2a) y_(t) = Ay(t) ; t ¸ 0 ; (4.2b) y(0) = y0 2 E: The Cauchy problem is said to be well posed, if ² for every y0 2 D(A) there exists a solution y of (4.2a) satisfying (4.2b), ² there exists a positive function C = C(t); t ¸ 0; that is bounded on bounded subsets of the nonnegative real line such that (4.3) ky(t)kE · C(t) ky0kE ; t ¸ 0 : If the Cauchy problem is well posed, the solution operator S(¢) is given by S(t)y0 = y(t) ; t ¸ 0 : It can be easily extended to a mapping S(¢) : [0; 1) !L(E; E) ac- cording to S(t)z = lim S(t)zn ; t ¸ 0 ; n!1 where z 2 E and zn 2 D(A); n 2 lN, such that lim zn = z. n!1 The solution operator S is an operator valued function that is strongly continuous in t 2 lR+, i.e., the mapping t 7¡! S(t)y is continuous in the norm of E for every y 2 E. Moreover, it has the properties (4.4a) S(0) = I; (4.4b) S(t1 + t2) = S(t1)S(t2) ; ti 2 lR+ ; 1 · i · 2 : De¯nition 4.1 (Strongly continuous and analytic semigroup) An operator valued function S : lR+ !L(E; E) that satis¯es (4.4a) and (4.4b) is called a semigroup. If it is strongly continuous, it is said to be a strongly continuous semigroup. Moreover, if it is analytic in t > 0, it is referred to as an analytic semigroup. Optimization Theory II, Spring 2007; Chapter4 99 Given a strongly continuous semigroup S, we de¯ne an operator A : D(A) ½ E ! E¤ according to ³ ´ (4.5) Ay := lim h¡1 S(h) ¡ I y ; y 2 D(A) ; h!0+ where D(A) ½ E consists of all y 2 E for which the limit exists. De¯nition 4.2 (In¯nitesimal generator) The operator A : D(A) ½ E ! E¤ given by (4.5) is called the in¯ni- tesimal generator of the strongly continuous semigroup S. Theorem 4.1 (Properties of the in¯nitesimal generator) The in¯nitesimal generator A : D(A) ½ E ! E¤ of a strongly conti- nuous semigroup S is a closed densely de¯ned operator such that the Cauchy problem (4.2a), (4.2b) is well posed. Proof: In order to show that A is densely de¯ned, let y 2 E and for " > 0 de¯ne Z" ¡1 (4.6) y" := " S(t)y dt : 0 Using (4.4b), for h · " we ¯nd ³ ´ ³ Z"+h Zh ´ ¡1 ¡1 ¡1 ¡1 h S(h) ¡ I y" = " h S(t)y dt ¡ h S(t)y dt ³ " ´ 0 ¡1 ! " S(")y ¡ y as h ! 0+ ; where we have used the continuity of S(¢)y. Hence, y" 2 D(A) and ³ ´ ¡1 Ay" = " S(")y ¡ y : Similarly, one can show y" ! y as " ! 0+ which proves the denseness of A. For the proof of the closedness of A, let y; z 2 E and fyng ½ D(A) such that y ! y and Ay ! z as n ! 1. Then ³ n ´ n ³ ´ ¡1 ¡1 h S(h) ¡ I y = lim h S(h) ¡ I yn = lim (Ayn)h = zh ; n!1 n!1 where (Ayn)h and zh as in (4.6). The limit process h ! 0+ reveals y 2 D(A) and Ay = z. Theorem 4.2 (Hille-Yosida theorem) An operator A : D(A) ½ E ! E¤ is the in¯nitesimal generator of a strongly continuous semigroup S(¢) with (4.7) kS(t)k · C exp(!t) ; t 2 lR+ 100 Ronald H.W. Hoppe for some C 2 R+; ! > 0, if and only if A is a closed densely de¯ned operator whose resolvent R(¸; A) := (¸I ¡ A)¡1 exists for all ¸ > ! and satis¯es (4.8) kR(¸; A)nk · C (¸ ¡ !)¡n ; n 2 lN : Proof: We refer to [1]. For the optimality conditions associated with (4.1a)-(4.1d) we need the following result: Theorem 4.3 (Adjoint semigroup) Let A : D(A) ½ E ! E¤ be the in¯nitesimal generator of a strongly continuous semigroup S(¢) in a reflexive Banach space E. Then, its adjoint A¤ : D(A¤) ½ E¤ ! E is the in¯nitesimal generator of the adjoint semigroup S¤(¢) = S(¢)¤ in E¤. Proof: We refer to Theorem 5.2.6 in [2]. Instead of the Cauchy problem (4.2a),(4.2b) let us consider the slightly more general problem (4.9a) y_(t) = Ay(t) + f(t) ; t ¸ 0 ; (4.9b) y(0) = y0 2 E; where A is assumed to be the in¯nitesimal generator of a strongly continuous semigroup S(¢) in E and f 2 C([0;T ]; E). Theorem 4.3 (Representation of a solution) The solution y = y(t); t 2 lR+, of (4.9a),(4.9b) has the representation Zt (4.10) y(t) = S(t)y0 + S(t ¡ ¿)f(¿) d¿ : 0 Proof: Let g(¿) := S(t ¡ ¿)y(¿). Then, there holds g_(¿) = ¡AS(t ¡ ¿)y(¿) + S(t ¡ ¿)_y(¿) = ³ ´ = ¡AS(t ¡ ¿)y(¿) + S(t ¡ ¿) Ay(¿) + f(¿) = = S(t ¡ ¿)f(¿) ; whence Zt g(t) ¡ g(0) = y(t) ¡ S(t)y0 = S(t ¡ ¿)f(¿) d¿ : 0 Optimization Theory II, Spring 2007; Chapter4 101 4.3 Existence and uniqueness of an optimal solution We consider the control constrained optimal control problem (4.1a)- (4.1d) under the assumptions made in section 4.1. Since X is a separable reflexive Banach space, it follows that functions 1 ¤ u 2 L ([0;T ]; X ) have a measurable norm. Thus, given y0 2 E and 1 ¤ u 2 L ([0;T ]; X ), the solution y = y(t); t 2 lR+, of (4.1a) is given by Zt (4.11) y(t) = S(t)y0 + S(t ¡ ¿)Bu(¿) d¿ : 0 Indeed, for z 2 E¤, we have hz; Bu(¿)i = hB¤z; u(¿)i so that Bu(¢) is an E-valued, E¤-weakly measurable function. Consequently, the integrand in (4.11) is a strongly measurable E-valued function so that the integral is a Lebesgue-Bochner integral. Moreover, the integral is continuous so that the solution y is a continuous E-valued function. It follows that (4.11) de¯nes a mapping G : L1([0;T ]; X¤) ! C([0;T ]; E)(4.12) Zt Gu(t) := S(t)y0 + S(t ¡ ¿)Bu(¿) d¿ ; 0 which we refer to as the control-to-state map. We note that due to the boundedness of the operator B and the compactness of the semi- group S(¢), the control-to-state map G is a compact linear operator. Substituting y = Gu in (4.1a), we get the reduced optimal control problem ZT ³ ´ (4.13) inf Jred(u) := g(Gu(t)) + h(u(t)) dt : u2L1([0;T ];K) 0 Theorem 4.4 (Existence and uniqueness) Under the assumptions on the data, the control constrained optimal control problem (4.1a)-(4.1d) admits a unique solution (y¤; u¤) with y¤ 2 C([0;T ]; E) and u¤ 2 L1([0;T ]; K). 1 Proof: Let funglN; un 2 L ([0;T ]; K); n 2 lN; be a minimizing se- quence, i.e., lim Jred(un) = inf Jred(u) : n!1 u2L1([0;T ];K) Due to the coerciveness assumption on the functionals g and h, the 1 1 sequence funglN is bounded in L ([0;T ]; K).