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 with norm ∗ k · kE and 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 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([0,T ]; E) → C([0,T ]) and h : L∞([0,T ]; X∗) → L∞([0,T ]) be strictly convex, coercive and continuously Fr´echet-differentiable mappings with Fr´echet- 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 ∈ [0,T ] ,

(4.1c) y(0) = y0 , (4.1d) u ∈ L∞([0,T ]; K) ,K ⊆ X∗,

∗ wherey ˙ := dy/dt, y0 ∈ E, and K is assumed to be weakly closed and convex. We note that L∞([0,T ]; X∗) denotes the linear space of functions v ∗ with v(t) ∈ X for almost all t ∈ [0,T ] such that kv(t)kX∗ is essentially bounded in t ∈ [0,T ]. It is a normed space with respect to the norm

kvkL∞([0,T ];X∗) := ess sup kv(t)kX∗ . t∈[0,T ]

Consequently, the control constraint (4.1d) implies

u(t) ∈ K for almost all t ∈ [0,T ] .

In addition to the above assumptions, we require the control constraint set L∞([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 . y∈E\{0} 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 defined, 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 ∈ E. The Cauchy problem is said to be well posed, if

• for every y0 ∈ D(A) there exists a solution y of (4.2a) satisfying (4.2b), • 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, ∞) → L(E,E) ac- cording to

S(t)z = lim S(t)zn , t ≥ 0 , n→∞ where z ∈ E and zn ∈ D(A), n ∈ lN, such that lim zn = z. n→∞ The solution operator S is an operator valued function that is strongly continuous in t ∈ lR+, i.e., the mapping t 7−→ S(t)y is continuous in the norm of E for every y ∈ E. Moreover, it has the properties (4.4a) S(0) = I,

(4.4b) S(t1 + t2) = S(t1)S(t2) , ti ∈ lR+ , 1 ≤ i ≤ 2 .

Definition 4.1 (Strongly continuous and analytic semigroup) An operator valued function S : lR+ → L(E,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 > 0, it is referred to as an analytic semigroup. Optimization Theory II, Spring 2007; Chapter4 99

Given a strongly continuous semigroup S, we define an operator A : D(A) ⊂ E → E∗ according to ³ ´ (4.5) Ay := lim h−1 S(h) − I y , y ∈ D(A) , h→0+ where D(A) ⊂ E consists of all y ∈ E for which the limit exists. Definition 4.2 (Infinitesimal generator) The operator A : D(A) ⊂ E → E∗ given by (4.5) is called the infini- tesimal generator of the strongly continuous semigroup S. Theorem 4.1 (Properties of the infinitesimal generator) The infinitesimal generator A : D(A) ⊂ E → E∗ of a strongly conti- nuous 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 > 0 define Zε −1 (4.6) yε := ε S(t)y dt . 0 Using (4.4b), for h ≤ ε we find ³ ´ ³ Zε+h Zh ´ −1 −1 −1 −1 h S(h) − I yε = ε h S(t)y dt − h S(t)y dt ³ ε ´ 0 −1 → ε S(ε)y − y as h → 0+ , where we have used the continuity of S(·)y. Hence, yε ∈ D(A) and ³ ´ −1 Ayε = ε S(ε)y − y .

Similarly, one can show yε → y as ε → 0+ which proves the denseness of A. For the proof of the closedness of A, let y, z ∈ E and {yn} ⊂ D(A) such that y → y and Ay → z as n → ∞. Then ³ n ´ n ³ ´ −1 −1 h S(h) − I y = lim h S(h) − I yn = lim (Ayn)h = zh , n→∞ n→∞ where (Ayn)h and zh as in (4.6). The limit process h → 0+ reveals y ∈ D(A) and Ay = z. Theorem 4.2 (Hille-Yosida theorem) An operator A : D(A) ⊂ E → E∗ is the infinitesimal generator of a strongly continuous semigroup S(·) with

(4.7) kS(t)k ≤ C exp(ωt) , t ∈ lR+ 100 Ronald H.W. Hoppe for some C ∈ R+, ω > 0, if and only if A is a closed densely defined operator whose resolvent R(λ; A) := (λI − A)−1 exists for all λ > ω and satisfies (4.8) kR(λ; A)nk ≤ 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 : D(A) ⊂ E → E∗ be the infinitesimal generator of a strongly continuous semigroup S(·) in a reflexive Banach space E. Then, its adjoint A∗ : D(A∗) ⊂ E∗ → E is the infinitesimal 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 ∈ E, where A is assumed to be the infinitesimal generator of a strongly continuous semigroup S(·) in E and f ∈ C([0,T ]; E). Theorem 4.3 (Representation of a solution) The solution y = y(t), t ∈ lR+, of (4.9a),(4.9b) has the representation Zt

(4.10) y(t) = S(t)y0 + S(t − τ)f(τ) dτ . 0 Proof: Let g(τ) := S(t − τ)y(τ). Then, there holds g˙(τ) = −AS(t − τ)y(τ) + S(t − τ)y ˙(τ) = ³ ´ = −AS(t − τ)y(τ) + S(t − τ) Ay(τ) + f(τ) = = S(t − τ)f(τ) , whence Zt

g(t) − g(0) = y(t) − S(t)y0 = S(t − τ)f(τ) dτ . 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 ∞ ∗ u ∈ L ([0,T ]; X ) have a measurable norm. Thus, given y0 ∈ E and ∞ ∗ u ∈ L ([0,T ]; X ), the solution y = y(t), t ∈ lR+, of (4.1a) is given by Zt

(4.11) y(t) = S(t)y0 + S(t − τ)Bu(τ) dτ . 0 Indeed, for z ∈ E∗, we have hz, Bu(τ)i = hB∗z, u(τ)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 is a Lebesgue-. Moreover, the integral is continuous so that the solution y is a continuous E-valued function. It follows that (4.11) defines a mapping G : L∞([0,T ]; X∗) → C([0,T ]; E)(4.12) Zt

Gu(t) := S(t)y0 + S(t − τ)Bu(τ) dτ , 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 . u∈L∞([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∗ ∈ C([0,T ]; E) and u∗ ∈ L∞([0,T ]; K). ∞ Proof: Let {un}lN, un ∈ L ([0,T ]; K), n ∈ lN, be a minimizing se- quence, i.e., lim Jred(un) = inf Jred(u) . n→∞ u∈L∞([0,T ];K) Due to the coerciveness assumption on the functionals g and h, the ∞ ∞ sequence {un}lN is bounded in L ([0,T ]; K). Since L ([0,T ]; K) is 102 Ronald H.W. Hoppe

L1([0,T ]; X)-weakly compact, there exist a subsequence lN0 ⊂ lN and an element u∗ ∈ L∞([0,T ]; K) such that for all v ∈ L1([0,T ]; X) ZT ZT ∗ 0 hun(t), v(t)iX∗,X dt → hu (t), v(t)iX∗,X dt (n ∈ lN , n → ∞) . 0 0 Due to the compactness of the state-to-control map G, we have

∗ 0 Gun(t) → Gu (t)(n ∈ N , n → ∞) uniformly in t ∈ [0,T ] .

Due to its convexity, the reduced functional Jred is lower semicontinuous with respect to the L1([0,T ]; X)- of L∞([0,T ]; X∗), and hence ∗ lim inf Jred(un) ≤ Jred(u ) , n→∞ 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

0 ∗ ∗ ∞ (4.14) hJred(u ), u − ui ≤ 0 , u ∈ L ([0,T ]; K) . 0 ∗ Evaluating the Gˆateauxderivative Jred(u ) allows to characterize the optimal solution (y∗, u∗) by means of an adjoint state p∗ = p∗(t) satis- fying 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([0,T ]; E) × L∞([0,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([0,T ]; E∗) such that the following optimality conditions are satisfied

∗ ∗ ∗ ∗ (4.15a) y˙ (t) − Ay (t) − Bu (t) = 0 , t ∈ [0,T ] , y (0) = y0 , (4.15b) p˙∗(t) + A∗p∗(t) − g0(y∗(t)) = 0 , t ∈ [0,T ] , p∗(T ) = 0 , (4.15c) u∗ ∈ L∞([0,T ]; K) , ZT ³ ∗ ∗ ∗ hu (t) − u(t),B p (t)iX∗,X +(4.15d) 0 ´ + h0(u∗(t))(u∗(t) − u(t)) dt ≤ 0 , u ∈ L∞([0,T ]; K) . Optimization Theory II, Spring 2007; Chapter4 103

Proof: Observing the definition (4.13) of the reduced objective func- tional Jred, the variational inequality (4.14) reads as follows

ZT ³ hg0(Gu∗(t)), GB(u∗(t) − u(t))i +(4.16) 0 ´ + h0(u∗(t))(u∗(t) − u(t)) dt ≤ 0 , u ∈ L∞([0,T ]; K) .

For the first part in (4.16) we obtain

ZT ³ hg0(Gu∗(t)), GB(u∗(t) − u(t))i dt =(4.17)

0 ZT = hu∗(t) − u(t),B∗G∗g0(Gu∗(t))i dt .

0

Rt ∗ We know that Gu (t) = y(t) = S(t)y0 + S(t − τ)Bu(τ)dτ. For the 0 evaluation of G∗, assume v ∈ C([0,T ]; E). Then, there holds

ZT Zt h S(t − τ)Bu(τ) dτ, v(t)i dt =

0 0 ZT Zτ = hS(τ − t)Bu(t), v(τ)i dtdτ =

0 0 ZT Zτ = hu(τ), B∗S∗(τ − t)v(t) dti dτ =

0 0 ZT Zt = hu(t),B∗ S∗(t − τ)v(τ) dτi dt .

0 0

∗ ∗ Since S (·) is an analyticR semigroup with infinitesimal generator A , we t ∗ may interpret z(t) := 0 S (t − τ)v(τ) dτ as the solution of the initial value problem

z˙(t) = A∗z(t) + v(t) , t ∈ [0,T ] , z(0) = 0 . 104 Ronald H.W. Hoppe

We set ZT −t p∗(t) := S∗(τ − (T − t))g0(y∗(T − τ)) dτ

T and employ the transformationτ ˆ = T − τ which gives Zt (4.18) p∗(t) = − S∗(t − τˆ)g0(y∗(ˆτ)) dτˆ .

0 It follows that p∗ satisfies the terminal value problem (4.15b). Substi- tuting G∗g0(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. Cam- bridge University Press, Cambridge, 1999