Chapter 4 Optimal Control Problems in Infinite Dimensional Function Space

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1 Chapter 4 Optimal Control Problems in Infinite 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 E and dual space E and X is another separable Banach space with norm X and separable dual space X. We further suppose A : DA E E to be the infinitesimal 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[, T ]; E C[, T ] and h : L [, T ]; X L [, T ] be strictly convex, coercive and continuously Fréchet-differentiable mappings with Fréchet-derivatives g and h, respectively. We consider the control constrained optimal control problem: 4.1a minimize Jy, u : gyt + hut dt, 4.1b 4.1c 4.1d subject to ẏt Ayt + But, t [, T ], y y, u L [, T ]; K, K X, where ẏ : dy/dt, y E, and K is assumed to be weakly closed and convex. We note that L [, T ]; X denotes the linear space of functions v with vt X for almost all t [, T ] such that vt X is essentially bounded in t [, T ]. It is a normed space with respect to the norm v L [,T ];X : ess sup vt X. t [,T ] Consequently, the control constraint 4.1d implies ut K for almost all t [, T ]. In addition to the above assumptions, we require the control constraint set L [, T ]; K to be L 1 [, T ]; X-weakly compact. In the sequel, for Banach spaces E and F with norms E and F we refer to LE, F as the normed space of bounded linear operators 97

2 98 Ronald H.W. Hoppe L : E F with norm L : Ly F sup. y E\{} y E 4.2 Evolution equations in function space Let E be a reflexive, separable Banach space with norm E and dual E and A : DA E E be a densely defined, closed linear operator. We consider the following initial-value problem Cauchy problem for an abstract evolution equation 4.2a 4.2b ẏt Ayt, t, y y E. The Cauchy problem is said to be well posed, if for every y DA there exists a solution y of 4.2a satisfying 4.2b, there exists a positive function C Ct, t, that is bounded on bounded subsets of the nonnegative real line such that 4.3 yt E Ct y E, t. If the Cauchy problem is well posed, the solution operator S is given by Sty yt, t. It can be easily extended to a mapping S : [, LE, E according to Stz lim Stz n, t, n where z E and z n DA, n ln, such that lim n z n z. The solution operator S is an operator valued function that is strongly continuous in t lr +, i.e., the mapping t Sty is continuous in the norm of E for every y E. Moreover, it has the properties 4.4a 4.4b S I, St 1 + t 2 St 1 St 2, t i lr +, 1 i 2. Definition 4.1 Strongly continuous and analytic semigroup An operator valued function S : lr + LE, E that satisfies 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 >, it is referred to as an analytic semigroup.

3 Optimization Theory II, Spring 27; Chapter4 99 Given a strongly continuous semigroup S, we define an operator A : DA E E according to 4.5 Ay : lim h 1 Sh I y, y DA, h + where DA E consists of all y E for which the limit exists. Definition 4.2 Infinitesimal generator The operator A : DA E E given by 4.5 is called the infinitesimal generator of the strongly continuous semigroup S. Theorem 4.1 Properties of the infinitesimal generator The infinitesimal generator A : DA E E of a strongly continuous semigroup S is a closed densely defined operator such that the Cauchy problem 4.2a, 4.2b is well posed. Proof: In order to show that A is densely defined, let y E and for ε > define 4.6 y ε : ε 1 Using 4.4b, for h ε we find h Sh 1 I y ε ε 1 h 1 ε Sty dt. ε+h Sty dt h 1 h Sty dt ε ε Sεy 1 y as h +, where we have used the continuity of S y. Hence, y ε DA and Ay ε ε Sεy 1 y. Similarly, one can show y ε y as ε + which proves the denseness of A. For the proof of the closedness of A, let y, z E and {y n } DA such that y n y and Ay n z as n. Then h Sh 1 I y y n lim Ay n h z h, n where Ay n h and z h as in 4.6. The limit process h + reveals y DA and Ay z. lim h 1 Sh I n Theorem 4.2 Hille-Yosida theorem An operator A : DA E E is the infinitesimal generator of a strongly continuous semigroup S with 4.7 St C expωt, t lr +

4 1 Ronald H.W. Hoppe for some C R +, ω >, if and only if A is a closed densely defined operator whose resolvent Rλ; A : λi A 1 exists for all λ > ω and satisfies 4.8 Rλ; A n C λ ω n, n 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 : DA E E be the infinitesimal generator of a strongly continuous semigroup S in a reflexive Banach space E. Then, its adjoint A : DA E E is the infinitesimal generator of the adjoint semigroup S S in E. Proof: We refer to Theorem in [2]. Instead of the Cauchy problem 4.2a,4.2b let us consider the slightly more general problem 4.9a 4.9b ẏt Ayt + ft, t, y y E, where A is assumed to be the infinitesimal generator of a strongly continuous semigroup S in E and f C[, T ]; E. Theorem 4.3 Representation of a solution The solution y yt, t lr +, of 4.9a,4.9b has the representation 4.1 yt Sty + St τfτ dτ. Proof: Let gτ : St τyτ. Then, there holds whence ġτ ASt τyτ + St τẏτ ASt τyτ + St τ Ayτ + fτ St τfτ, gt g yt Sty St τfτ dτ.

5 Optimization Theory II, Spring 27; Chapter 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 u L [, T ]; X have a measurable norm. Thus, given y E and u L [, T ]; X, the solution y yt, t lr +, of 4.1a is given by 4.11 yt Sty + St τbuτ dτ. Indeed, for z E, we have z, Buτ B z, uτ 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 defines a mapping 4.12 G : L [, T ]; X C[, T ]; E Gut : Sty + St τbuτ dτ, 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 semigroup 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 4.13 inf u L [,T ];K J red u : ggut + hut dt. 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 C[, T ]; E and u L [, T ]; K. Proof: Let {u n } ln, u n L [, T ]; K, n ln, be a minimizing sequence, i.e., lim J redu n inf J redu. n u L [,T ];K Due to the coerciveness assumption on the functionals g and h, the sequence {u n } ln is bounded in L [, T ]; K. Since L [, T ]; K is

6 12 Ronald H.W. Hoppe L 1 [, T ]; X-weakly compact, there exist a subsequence ln ln and an element u L [, T ]; K such that for all v L 1 [, T ]; X u n t, vt X,X dt u t, vt X,X dt n ln, n. Due to the compactness of the state-to-control map G, we have Gu n t Gu t n N, n uniformly in t [, T ]. Due to its convexity, the reduced functional J red is lower semicontinuous with respect to the L 1 [, T ]; X-weak topology of L [, T ]; X, and hence lim inf n J redu n J red u, which allows to conclude with y Gu. 4.4 Optimality conditions The necessary optimality conditions for the reduced optimal control problem 4.13 are given by the variational inequality 4.14 J redu, u u, u L [, T ]; K. Evaluating the Gâteaux derivative J red u allows to characterize the optimal solution y, u by means of an adjoint state p p t satisfying a terminal value problem for an associated adjoint equation. We note that in this case the optimality conditions are also sufficient due to the convexity of the objective functional. Theorem 4.5 Optimality conditions Let y, u C[, T ]; E L [, T ]; K be the optimal solution of the control constrained optimal control problem 4.1a-4.1d. Then, there exists an adjoint state p C[, T ]; E such that the following optimality conditions are satisfied 4.15a 4.15b 4.15c 4.15d ẏ t Ay t Bu t, t [, T ], y y, ṗ t + A p t g y t, t [, T ], p T, u L [, T ]; K, u t ut, B p t X,X + + h u tu t ut dt, u L [, T ]; K.

7 Optimization Theory II, Spring 27; Chapter4 13 Proof: Observing the definition 4.13 of the reduced objective functional J red, the variational inequality 4.14 reads as follows 4.16 g Gu t, GBu t ut + + h u tu t ut dt, u L [, T ]; K. For the first part in 4.16 we obtain 4.17 g Gu t, GBu t ut dt u t ut, B G g Gu t dt. We know that Gu t yt Sty + St τbuτdτ. For the evaluation of G, assume v C[, T ]; E. Then, there holds St τbuτ dτ, vt dt τ Sτ tbut, vτ dtdτ τ uτ, B S τ tvt dt dτ ut, B S t τvτ dτ dt. Since S is an analytic semigroup with infinitesimal generator A, we may interpret zt : S t τvτ dτ as the solution of the initial value problem żt A zt + vt, t [, T ], z.

8 14 Ronald H.W. Hoppe We set p t : T t S τ T tg y T τ dτ T and employ the transformation ˆτ T τ which gives 4.18 p t S t ˆτg y ˆτ dˆτ. It follows that p satisfies the terminal value problem 4.15b. Substituting G g y t in 4.17 by p t readily gives 4.15d. References [1] N. Dunford and T. Schwartz; Linear Operators. Part I. Interscience, New York, 1958 [2] H.O. Fattorini; Infinite Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge, 1999