Optimal distributed control of a stochastic Cahn-Hilliard equation
aa r X i v : . [ m a t h . O C ] J u l Optimal distributed controlof a stochastic Cahn-Hilliard equation ∗ Luca Scarpa
E-mail: [email protected]
Faculty of Mathematics, University of ViennaOskar-Morgenstern-Platz 1, 1090 Vienna, Austria
Abstract
We study an optimal distributed control problem associated to a stochastic Cahn-Hilliard equation with a classical double-well potential and Wiener multiplicativenoise, where the control is represented by a source-term in the definition of thechemical potential. By means of probabilistic and analytical compactness argu-ments, existence of an optimal control is proved. Then the linearized system and thecorresponding backward adjoint system are analysed through monotonicity and com-pactness arguments, and first-order necessary conditions for optimality are proved.
AMS Subject Classification:
Key words and phrases: stochastic Cahn-Hilliard equation, phase separation,optimal control, linearized state system, adjoint state system, first-order optimalityconditions.
The pure Cahn-Hilliard equation on a smooth bounded domain D ⊂ R N , N = 2 , , canbe written in its simplest form as ∂ t y − ∆ w = 0 , w = − ∆ y + Ψ ′ ( y ) − u in (0 , T ) × D , where
T > is a fixed final time, y and w denote the order parameter and the chemicalpotential of the system, respectively, and u represents a given distributed source term.Furthermore, Ψ ′ is the derivative of a so-called double-well potential Ψ , which may be seen ∗ This paper was funded by Vienna Science and Technology Fund (WWTF) through Project MA14-009.
Optimal control of a stochastic CH equation as the sum of a convex function and a concave quadratic perturbation: typical examples of Ψ which are relevant in applications are discussed in [18]. Usually, in order to ensure theconservation of the mean on D , the equation is complemented by homogenous Neumannconditions for both y and w , and a given initial value, namely ∂ n y = ∂ n w = 0 in (0 , T ) × ∂D , y (0) = y in D , where n denotes the outward normal unit vector on ∂D .The Cahn-Hilliard equation was originally introduced in [7] (see also [31, 32, 49]) tocapture the spinodal decomposition phenomenon occurring in a phase-separation of abinary metallic alloy. The mathematical literature on the deterministic Cahn-Hilliardequation has been widely developed in the last years, especially in much more generalsettings as the presence of viscosity terms and dynamic boundary conditions: in thisdirection we mention, among all, the contributions (as well as the references therein) [4,8–10, 13, 14, 18, 37, 48, 57] on the well-posedness of the system, and [15, 21, 38] on asymptoticbehaviour of the solutions. Optimal distributed and boundary control problems havebeen studied in the context of Allen-Cahn and Cahn-Hilliard equations in the works[11, 12, 16, 17, 19, 20, 42, 54].More recently, in order to account also for the random vibrational movements at amicroscopic level in the system, which may be of magnetic, electronic or configurationalnature, the equation has been modified by adding a cylindrical Wiener process W (see[23, 46]). This has resulted in the well-accepted version of the stochastic Cahn-Hilliardequation dy − ∆ w dt = B ( y ) dW in (0 , T ) × D =: Q , (1.1) w = − ∆ y + Ψ ′ ( y ) − u in (0 , T ) × D , (1.2) ∂ n y = ∂ n w = 0 in (0 , T ) × ∂D =: Σ , (1.3) y (0) = y in D , (1.4)where B is a stochastically integrable operator with respect to W . The mathematicalliterature on the stochastic Cahn-Hilliard and Allen-Cahn equations is significantly lessdeveloped. Let us mention the works [24, 25, 30, 55] dealing with existence, uniquenessand regularity for the pure equation, and [41, 45, 55] for an analysis of the viscous casein terms of well-posedness, regularity and vanishing viscosity limit. We point out forcompleteness also the contributions [1] for a study of a stochastic Cahn-Hilliard equationwith unbounded noise, and [26,27,39] dealing with stochastic Cahn-Hilliard equations withreflections. The reader can refer also to [3, 51] for the context of stochastic Allen-Cahnequations, and [33] for a study of a diffuse interface model with termal fluctuations.While the literature on stochastic optimal control problems is widely developed, weare not aware of any result dealing with controllability of the stochastic Cahn-Hilliardequation. The main novelty of the present contribution is to provide a first study inthis direction, and represents a starting point for the study of optimal control problemsassociated to the wide class of more general phase-field models with stochastic pertur-bation. Optimal control problems have been studied in the stochastic case especially inconnection with the stochastic maximum principle: the reader can refer to [61] for a gen-eral treatment on the subject. Let us mention the works [36, 50] dealing with stochastic uca Scarpa u in the definition of the chemical potential and the cost functional is ofstandard quadratic tracking-type. More precisely, we want to minimize J ( y, u ) := α E Z Q | y − x Q | + α E Z D | y ( T ) − x T | + α E Z Q | u | , (1.5)subject to the state equation (1.1)–(1.4) and a control constraint on u ∈ U , where U isa suitable convex closed subset of L (Ω × Q ) which will be specified in Section 2 below.Here, α , α , α are nonnegative constants, x Q and x T are given functions in L (Ω × Q ) and L (Ω × D ) , respectively. The main results of this work are the existence of a relaxedoptimal control and the proof of first-order necessary conditions for optimality.The first step of our analysis consists in studying the control-to-state mapping. Inparticular, we show that for every admissible control u ∈ U , the state system (1.1)–(1.4)admits a unique solution y , and the map S : u y is Lipschitz-continuous in somesuitable spaces. Consequently, the cost functional J can be expressed in a reduced formonly in terms of the control u , i.e. introducing the reduced cost functional ˜ J as ˜ J ( u ) := J ( S ( u ) , u ) , u ∈ U . At this point, in the deterministic setting the most natural necessary condition foroptimality of ¯ u ∈ U would read D ˜ J (¯ u )( v − ¯ u ) ≥ ∀ v ∈ U , where D ˜ J represents the derivative of ˜ J at least in the sense of Gâteaux. In this direction,the classical approach consists in showing that the map S is Fréchet-differentiable, henceso is ˜ S by the usual chain rule for Fréchet-differentiable functions, and to characterize thederivative DS (¯ u ) as the solution of a suitable linearized system. In the context of Cahn-Hilliard equations with possibly degenerate potentials (for example if Ψ is the double-well logarithmic potential), the Fréchet differentiability of the control-to-state mapping isusually obtained by requiring sufficient conditions in the box constraint for u , ensuringat least that Ψ ′′ (¯ y ) ∈ L ∞ ( Q ) , where ¯ y := S (¯ u ) and Ψ ′′ is the second derivative of Ψ (forexample that U is contained in a closed ball in L ∞ ( Q ) ).However, if we add a stochastic perturbation in the equation, under reasonable as-sumptions on the data it is not possible to prove that Ψ ′′ (¯ y ) is uniformly bounded in L ∞ (Ω × Q ) , even if we add a constraint on the L ∞ -norm in the definition of the admis-sible controls. This behaviour gives rise to several nontrivial difficulties: among all, it isnot true a priori that the control-to-state map S is Fréchet-differentiable in some space. Optimal control of a stochastic CH equation
This issue is usually overcome in the stochastic setting using specific time-variations onthe control (the so-called “spike-variation” technique). In our case, however, we are ableto avoid such procedure by analysing explicitly the linearized system. More specifically,we prove that the linearized system admits a unique variational solution by means of com-pactness and monotonicity arguments. Then, we show that the control-to-state mappingis Gâteaux differentiable in a suitable weak sense, and that the (weak) Gâteaux derivativeof S can still be identified as the unique solution z to the linearized system. Performingusual first-order variations around a fixed optimal control ¯ u , we then prove that the weakGâteaux-differentiability is enough to ensure first-order necessary conditions for optimal-ity. The second main issue that we tackle in this work consist in removing the dependenceon z in the first-order necessary conditions by studying the adjoint problem. As it is well-known, in the stochastic framework the adjoint problem becomes a backward stochasticpartial differential equation (BSPDE) of the form ˜ p = − ∆ p in Q , (1.6) − dp − ∆˜ p dt + Ψ ′′ ( y )˜ p dt = α ( y − x Q ) dt + DB ( y ) ∗ q dt − q dW in Q , (1.7) ∂ n p = ∂ n ˜ p = 0 in Σ , (1.8) p ( T ) = α (¯ y ( T ) − x T ) in D , (1.9)where the unknown is the triple ( p, ˜ p, q ) . Since Ψ ′′ ( y ) does not belong to L ∞ (Ω × Q ) , as wehave pointed out above, the adjoint problem cannot be framed in any available existencetheory for BSPDEs, and is absolutely nontrivial and interesting on its own. Through asuitable approximation involving a truncation on Ψ ′′ and a passage to the limit, we showexistence and uniqueness of a solution to the adjoint problem. Furthermore, we provea suitable duality relation between z and ˜ p , which allows us to express the first-orderoptimality conditions only in terms of ˜ p and ¯ u in a much more simplified form.The paper is organized as follows. In Section 2 we fix the assumptions, the generalsetting of the work and the main results. Section 3 contains the proof of well-posedness ofthe state system. In Section 4 we prove that a relaxed optimal control always exists, usingProkhorov and Skorokhod theorems and natural lower semicontinuity results. In Section 5we study the control-to-state map: we show that it is well-defined and differentiable ina certain weak sense, and we identify its (weak) derivative as the unique solution to thelinearized problem. Finally, in Section 6 we study the adjoint problem and prove thefirst-order necessary conditions for optimality. Throughout the paper (Ω , F , ( F t ) t ∈ [0 ,T ] , P ) denotes a filtered probability space satisfyingthe usual conditions, with T > fixed, and W is a cylindrical Wiener process on aseparable Hilbert space U . The progressive σ -algebra on Ω × [0 , T ] is denoted by P .Furthermore, D ⊂ R N , with N = 2 , , is a smooth bounded connected domain, and weuse the notation Q := (0 , T ) × D and Q t := (0 , t ) × D for every t ∈ (0 , T ) .For every Hilbert spaces E and E we denote by L ( E , E ) and L ( E , E ) thespaces of linear continuous and Hilbert-Schmidt operators from E to E , respectively. uca Scarpa k·k and h· , ·i , respectively, with a sub-scriptindicating the specific spaces in consideration. We shall use the symbols → , ⇀ and ∗ ⇀ to denote strong, weak, and weak* convergences, respectively. For any Banach space E and p ∈ [1 , + ∞ ] we shall use the symbols L p (Ω; E ) and L p (0 , T ; E ) for the usual spacesof Bochner-integrable functions, and the symbols C ([0 , T ]; E ) and C w ([0 , T ]; E ) for thespaces of continuous functions from [0 , T ] to E endowed with the norm topology or weaktolopogy, respectively. If p, q ∈ [1 , + ∞ ) we shall denote by L p P (Ω; L q (0 , T ; E )) the space of E -valued progressively measurable processes X such that E (cid:16)R T k X ( s ) k qE ds (cid:17) p/q < + ∞ .We define the functional spaces H := L ( D ) , V := H ( D ) , Z := (cid:8) ϕ ∈ H ( D ) : ∂ n ϕ = 0 a.e. on ∂D (cid:9) , endowed with their natural norms. In the sequel H is identified to H ∗ , so that ( V, H, V ∗ ) is a Hilbert triplet, with dense, continuous and compact inclusions. The Laplace operatorwith Neumann homogeneous conditions will be intended in the usual variational way asthe operator − ∆ : V → V ∗ , h− ∆ x, ϕ i V := Z D ∇ x · ∇ ϕ , x, ϕ ∈ V , or − ∆ : H → Z ∗ , h− ∆ x, ϕ i Z := − Z D x ∆ ϕ , x ∈ H , ϕ ∈ Z .
We recall also that in the context of Cahn-Hilliard equations it is useful to introduce theoperator N as the inverse of − ∆ restricted to the subspace of null-mean elements in V .More specifically, if we denote x D := | D | h x, i V for any x ∈ V ∗ , by the Poincaré inequalitywe know that − ∆ : { x ∈ V : x D = 0 } → { x ∈ V ∗ : x D = 0 } is an isomorphism, hence its inverse N is well-defined. Furthermore, it is well-known(see [18, pp. 979-980]) that x
7→ k x k ∗ := k∇N ( x − x D ) k H + | y D | , x ∈ V ∗ , defines a norm on V ∗ , equivalent to the usual one, such that ∀ σ > , ∃ C σ > k x k H ≤ σ k∇ x k H + C σ k x k ∗ ∀ x ∈ V , (2.1)and h ∂ t x ( t ) , N x ( t ) i V = 12 ddt k∇N x ( t ) k H for a.e. t ∈ (0 , T ) for every x ∈ H (0 , T ; V ∗ ) with x D = 0 almost everywhere in (0 , T ) . We shall denote forsimplicity H := { x ∈ H : x D = 0 } . The following assumptions on the data of the problem will be in force throughout:(A1) Ψ ∈ C ( R , R + ) ; Optimal control of a stochastic CH equation (A2) there exist c , c > such that, for every r ∈ R , Ψ ′′ ( r ) ≥ − c , | Ψ ′′ ( r ) | ≤ c (1 + | r | ) , | Ψ ′ ( r ) | ≤ c (1 + Ψ( r )) . (A3) y ∈ L (Ω , F ; H ) ∩ L (Ω , F ; V ) and Ψ( y ) ∈ L (Ω; L ( D )) ;(A4) B : [0 , T ] × H → L ( U, V ) is measurable and there exists a constant L B > suchthat, for every t ∈ [0 , T ] , k B ( t, x ) − B ( t, x ) k L ( U,H ) ≤ L B k x − x k H ∀ x , x ∈ H , k B ( t, x ) k L ( U,H ) ≤ L B (1 + k x k H ) ∀ x ∈ H , k B ( t, x ) k L ( U,V ) ≤ L B (1 + k x k V ) ∀ x ∈ V . If B is genuinely of multiplicative type, i.e. if it is not constant in the last variable,we further assume that the image of B is contained in L ( U, H ) .(A5) for every t ∈ [0 , T ] , the operator B ( t, · ) : H → L ( U, H ) is of class C . Remark 2.1.
Let us comment on assumptions (A1)–(A5). In order for the state system(1.1)–(1.4) to be well-posed, one can require less stringent assumptions on the data (seefor example [55, 56]). However, in order to study the linearized and the adjoint problems,one needs some further regularity on the solution y to the state equation, and for thisreason (A1)–(A5) are in order. Let us point out that by the hypotheses (A1)–(A2) we candecompose Ψ ′ as the sum of a continuous increasing function and a Lipschitz-continuousfunction as Ψ ′ ( r ) = (Ψ ′ ( r ) + c r ) − c r , r ∈ R . Furthermore, note that (A3)–(A4) aretrivially satisfied for example when y ∈ V is nonnradom with Ψ( y ) ∈ L ( D ) and B ∈ L ( U, V ) is time-independent and of additive type. The reason why we assume existenceof higher moments on y might not be intuitive at this level and will be clarified later: letus mention that these assumptions will be needed to solve the linearized system and theadjoint problem, and that the hypothesis on the moment of order is “optimal” in thissense. In case of multiplicative noise, assumption (A4) corresponds to usual boundednessand Lipschitz-continuity conditions on B , and the differentiability assumption (A5) isneeded in order to analyse the linearized system. In particular, (A4)–(A5) imply that k DB ( t, x ) k L ( H ; L ( U,H )) ≤ L B ∀ ( t, x ) ∈ [0 , T ] × H .
We define the set of admissible controls U as U := (cid:8) u ∈ L P (Ω; L (0 , T ; H )) ∩ L P (Ω; L (0 , T ; V )) : k u k L (Ω; L (0 ,T ; H )) ∩ L (Ω; L (0 ,T ; V )) ≤ C o , where C > and s ∈ (0 , / are fixed constants. It will be useful to introduce also thebigger set U ′ := (cid:8) u ∈ L P (Ω; L (0 , T ; H )) ∩ L P (Ω; L (0 , T ; V )) : k u k L (Ω; L (0 ,T ; H )) ∩ L (Ω; L (0 ,T ; V )) < C o , uca Scarpa L P (Ω; L (0 , T ; H )) ∩ L P (Ω; L (0 , T ; V )) , and U ⊂ U ′ .Moreover, we define the cost functional J : L P (Ω; C ([0 , T ]; H )) × L P (Ω; L (0 , T ; H )) → R + ,J ( y, u ) := α E Z Q | y − x Q | + α E Z D | y ( T ) − x T | + α E Z Q | u | , where α , α , α ≥ are fixed constants and α x Q ∈ L P (Ω; L (0 , T ; H )) , α x T ∈ L (Ω , F T ; V ) . Remark 2.2.
The choices α x T ∈ L (Ω; V ) and α x Q ∈ L (Ω; L (0 , T ; H )) might lookunnnatural to the reader at this level, due to the form of the cost functional. However,this will be necessary in order to solve the adjoint system. Let us mention that conditionsof this type are not new in literature of optimal control problems: see for example [22]for an analogous assumption in the context of the Allen-Cahn equation.As we have anticipated in Section 1, we are interested in minimizing J ( y, u ) subjectto the constraint u ∈ U and the state system (1.1)–(1.4). We shall call optimal pair anycouple ( y, u ) with u ∈ U satisfying (1.1)–(1.4) and minimizing the cost functional J .Under the hypotheses (A1)–(A4), we can prove that the state system is well-posed forevery admissible control, and that the map u y is well-defined and Lipschitz-continuous.These results are summarized in the following theorem. Theorem 2.1.
Assume (A1)–(A4). Then for every u ∈ U ′ there exists a unique pair ( y, w ) with y ∈ L P (cid:0) Ω; C ([0 , T ]; H ) ∩ L (0 , T ; Z ) (cid:1) , (2.2) y ∈ L P (Ω; L ∞ (0 , T ; V )) ∩ L (Ω; L (0 , T ; H ( D ))) , (2.3) y − Z · B ( s, y ( s )) dW ( s ) ∈ L (Ω; H (0 , T ; V ∗ )) , (2.4) w ∈ L P (Ω; L (0 , T ; V ))) , Ψ ′ ( y ) ∈ L P (Ω; L (0 , T ; V )) , (2.5) such that y (0) = y and, for every ϕ ∈ V , for almost every t ∈ (0 , T ) , P -almost surely, (cid:28) ∂ t (cid:18) y − Z · B ( s, y ( s )) dW ( s ) (cid:19) ( t ) , ϕ (cid:29) V + Z D ∇ w ( t ) · ∇ ϕ = 0 , (2.6) Z D w ( t ) ϕ = Z D ∇ y ( t ) · ∇ ϕ + Z D Ψ ′ ( y ( t )) ϕ − Z D u ( t ) ϕ . (2.7) Moreover, there exists a constant M ′ > , only depending on y , C , c , c , L B and Q ,such that, for every u ∈ U ′ and for any respective state ( y, w ) satisfying (2.2) – (2.7) , k y k L (Ω; C ([0 ,T ]; H ) ∩ L (0 ,T ; Z )) ≤ M ′ , (2.8) k y k L (Ω; L ∞ (0 ,T ; V )) ∩ L (Ω; L (0 ,T ; H ( D )) ≤ M ′ , (2.9) k w k L (Ω; L (0 ,T ; V )) + k Ψ ′ ( y ) k L (Ω; L (0 ,T ; V )) ≤ M ′ . (2.10) Optimal control of a stochastic CH equation
Finally, there exists a constant
M > , only depending on y , C , c , c , L B and Q ,such that, for any u , u ∈ U ′ and for any respective pairs ( y , w ) , ( y , w ) satisfying (2.2) – (2.7) , it holds k y − y k L (Ω; C ([0 ,T ]; V ∗ )) ∩ L (Ω; L (0 ,T ; V )) ≤ M k u − u k L (Ω; L (0 ,T ; V ∗ )) , (2.11) k y − y k L (Ω; C ([0 ,T ]; H ) ∩ L (0 ,T ; Z )) ≤ M k u − u k L (Ω; L (0 ,T ; H )) . (2.12)By Theorem 2.1, it is clear that uniqueness of y holds for the state system. Conse-quently, it is well-defined the control-to-state map S : U ′ → L (cid:0) Ω; C ([0 , T ]; H ) ∩ L ∞ (0 , T ; V ) ∩ L (0 , T ; Z ) (cid:1) , which is Lipschitz-continuous in the sense specified in (2.11)–(2.12). This allows us tointroduce the reduced cost functional as ˜ J : U ′ → R + , ˜ J ( u ) := J ( S ( u ) , u ) , u ∈ U ′ . The optimal control problem is thus equivalent to minimizing ˜ J over U ⊂ U ′ . Thefollowing definitions of optimal control are very natural. Definition 2.3.
An optimal control is an element u ∈ U such that ˜ J ( u ) ≤ ˜ J ( v ) ∀ v ∈ U . A relaxed optimal control is a family ((Ω ∗ , F ∗ , ( F ∗ t ) t ∈ [0 ,T ] , P ∗ ) , W ∗ , x ∗ Q , x ∗ T , y ∗ , u ∗ , y ∗ , w ∗ ) , where (Ω ∗ , F ∗ , ( F ∗ t ) t ∈ [0 ,T ] , P ∗ ) is a filtered probability space satisfying the usual conditions, W ∗ is a ( F ∗ t ) t -cylindrical Wiener process with values in U , X ∗ Q is a ( F ∗ t ) t -progressivelymeasurable L (0 , T ; H ) -valued process with the same law of x Q , x ∗ T is a F ∗ T -measurable H -valued random variable with the same law of x T , y ∗ is a F ∗ -measurable random variablewith the same law of y , u ∗ is a process in the set U ∗ (defined as U replacing Ω with Ω ∗ ), ( y ∗ , w ∗ ) is the unique solution to the system (2.2) – (2.7) on Ω ∗ with respect to the data ( W ∗ , y ∗ , u ∗ ) , and such that ˜ J ∗ ( u ∗ ) := α E ∗ Z Q | y ∗ − x ∗ Q | + α E ∗ Z D | y ∗ ( T ) − x ∗ T | + α E ∗ Z Q | u ∗ | ≤ inf v ∈U ˜ J ( v ) . The first main result that we prove concerns with the existence of a relaxed optimalcontrol. Note however that existence of (strong) optimal controls is nontrivial, since theminimization problem in not convex, hence uniqueness of optimal controls may fail. Incase of uniqueness of optimal controls, existence of a strong optimal control can be provedfor example by a well-known criterion on convergence in probability due to Gyöngy–Krylov(see [40, Lem. 1.1], [43, Prop. 4.16], and [2, Def. 2.4 and Thm. 2.5]).
Theorem 2.2.
Assume (A1)–(A4). Then there exists a relaxed optimal control.
We focus now on the necessary conditions for optimality. As we have anticipated,the first step consists in showing that S is Gâteaux-differentiable in certain weak-senseand to characterize its weak derivative as the unique solution of a linearized system.The following proposition ensures that the linearized system is well-posed in a suitablevariational sense. uca Scarpa Proposition 2.4.
Assume (A1)–(A5). Then, for every u ∈ U ′ and for every h ∈ L P (Ω; L (0 , T ; H )) , setting y := S ( u ) , there exists a unique pair ( z h , µ h ) with z h ∈ L P (cid:0) Ω; C ([0 , T ]; H ) ∩ L (0 , T ; Z ) (cid:1) , (2.13) z h − Z · DB ( s, y ( s )) z h ( s ) dW ( s ) ∈ L (Ω; H (0 , T ; Z ∗ )) , (2.14) µ h ∈ L P (Ω; L (0 , T ; H )) , (2.15) such that z h (0) = 0 and, for every ϕ ∈ Z , for almost every t ∈ (0 , T ) , P -almost surely, (cid:28) ∂ t (cid:18) z h ( t ) − Z t DB ( s, y ( s )) z h ( s ) dW ( s ) (cid:19) , ϕ (cid:29) Z − Z D µ h ( t )∆ ϕ = 0 , (2.16) Z D µ h ( t ) ϕ = Z D ∇ z h ( t ) · ∇ ϕ + Z D Ψ ′′ ( y ( t )) z h ( t ) ϕ − Z D h ( t ) ϕ . (2.17) Remark 2.5.
Let us point out that (2.13)–(2.17) is the weak formulation of the linearizedsystem, which can be obtained formally differentiating the state system (1.1)–(1.4) withrespect to u in the direction h , i.e. dz h − ∆ µ h dt = DB ( y ) z h dW in (0 , T ) × D ,µ h = − ∆ z h + Ψ ′′ ( y ) z h − h in (0 , T ) × D ,∂ n z h = ∂ n µ h = 0 in (0 , T ) × ∂D ,z h (0) = 0 in D .
We are now able to give a characterization of the weak Gâteaux derivative of S interms of the unique solution to the linearized system (2.13)–(2.17). Theorem 2.3.
Assume (A1)–(A5). Then the control-to-state map S is weakly Gâteaux-differentiable from U ′ to L (Ω; C ([0 , T ]; H )) ∩ L (Ω; L (0 , T ; Z )) in the following sense:for every u, h ∈ U ′ , as ε ց , S ( u + εh ) − S ( u ) ε → z h in L p (Ω; L (0 , T ; V )) ∀ p ∈ [1 , ,S ( u + εh ) − S ( u ) ε ⇀ z h in L (cid:0) Ω; L (0 , T ; Z ) (cid:1) ,S ( u + εh )( t ) − S ( u )( t ) ε ⇀ z h ( t ) in L (Ω; H ) ∀ t ∈ [0 , T ] , where z h is the unique solution to the linearized system (2.13) – (2.17) . The first natural necessary optimality condition is collected in the following result.
Theorem 2.4.
Assume (A1)–(A5), let ¯ u ∈ U be an optimal control and ¯ y := S (¯ u ) be therespective optimal state. Then α E Z Q (¯ y − x Q ) z v − ¯ u + α E Z D (¯ y ( T ) − x T ) z v − ¯ u ( T ) + α E Z Q ¯ u ( v − ¯ u ) ≥ ∀ v ∈ U , where z v − ¯ u is the unique solution to (2.13) – (2.17) with respect to the choice h := v − ¯ u . Optimal control of a stochastic CH equation
The last result that we present is an alternative formulation of the first-order necessaryconditions for optimality which does not involve the solution z to the linearized problem,but the solution to the corresponding adjoint problem. In this sense, the advantage isthat the resulting variational inequality that we obtain is much simpler to interpret. Thefollowing proposition states that the adjoint problem is well-posed in a suitable variationalsense. Proposition 2.6.
Assume (A1)–(A5). Then for every u ∈ U ′ , setting y := S ( u ) , thereexists a triple of processes ( p, ˜ p, q ) , with p ∈ C w ([0 , T ]; L (Ω; V )) ∩ L P (Ω; L (0 , T ; Z ∩ H ( D ))) , (2.18) ˜ p ∈ C w ([0 , T ]; L (Ω; V ∗ )) ∩ L P (Ω; L (0 , T ; V )) , (2.19) q ∈ L P (Ω; L (0 , T ; L ( U, V ))) , (2.20) such that, for every ϕ ∈ Z , P -almost surely and for every t ∈ [0 , T ] , Z D ˜ p ( t ) ϕ = Z D ∇ p ( t ) · ∇ ϕ , (2.21) Z D p ( t ) ϕ − Z Tt Z D ˜ p ( s )∆ ϕ ds + Z Tt Z D Ψ ′′ ( y )˜ p ( s ) ϕ ds = α Z D (¯ y ( T ) − x T ) ϕ + α Z Tt Z D ( y − x Q )( s ) ϕ ds + Z Tt ( DB ( s, y ( s )) ∗ q ( s ) , ϕ ) H ds − Z D (cid:18)Z Tt q ( s ) dW ( s ) (cid:19) ϕ . (2.22) Moreover, if ( p , ˜ p , q ) and ( p , ˜ p , q ) are two solutions to (2.18) – (2.22) , then p − ( p ) D = p − ( p ) D , ˜ p = ˜ p . Our last result is a simplified version of the first-order necessary optimality conditionswhich do not involve the solution z to the linearized system, but the unique solution ˜ p tothe adjoint problem instead. Theorem 2.5.
Assume (A1)–(A5), let ¯ u ∈ U be an optimal control and let ¯ y := S (¯ u ) bethe respective optimal state. Then the following variational inequality holds: E Z Q ( ˜ p + α ¯ u ) ( v − ¯ u ) ≥ ∀ v ∈ U , where ˜ p is the unique second solution component satisfying (2.18) – (2.22) with respect to (¯ u, ¯ y ) . In particular, if α > , then ¯ u is the orthogonal projection of the point − ˜ pα onthe closed convex set U in the Hilbert space L (Ω; L (0 , T ; H )) . Remark 2.7.
The form of the cost functional J considered in this paper is of standardquadratic tracking-type and is widely used in optimal control theory. However, let us pointout that this choice is a particular case of the more general class of nonlinear performances J ( y, u ) = E Z T F Q ( t, y ( t ) , u ( t )) dt + E F T ( y ( T )) , uca Scarpa F Q : [0 , T ] × H × H → R and and F T : H → R are B ([0 , T ]) ⊗ B ( H ) ⊗ B ( H ) –measurable and B ( H ) –measurable, respectively. The techniques used here can also beadapted to deal with such more general situations. For example, one can show existenceof relaxed optimal controls under very natural lower semicontinuity assumptions on F Q and F T . Furthermore, requiring that F ( t, · , · ) : H × H → R and F T : H → R areFréchet-differentiable for every t ∈ [0 , T ] , with k D y F Q ( t, y, u ) k H + k D u F Q ( t, y, u ) k H . k y k H + k u k H k D y F T ( y ) k H . k y k H for every ( t, y, u ) ∈ [0 , T ] × H × H , necessary conditions for optimality can be also studied.Note however that in the case of nonlinear performance, the resulting variational inequal-ity in Theorem 2.5 would not give a characterization of the optimal controls in terms oforthogonal projection on U . This section is devoted to the proof of Theorem 2.1, ensuring the the state system iswell-posed.For any λ > , we consider the approximated problem dy λ − ∆ w λ dt = B ( y λ ) dW in (0 , T ) × D ,w λ = − ∆ y λ + Ψ ′ λ ( y λ ) − u in (0 , T ) × D ,∂ n y λ = ∂ n w λ = 0 in (0 , T ) × ∂D ,y λ (0) = y in D , where Ψ ′ λ is a Lipschitz-continuous smooth Yosida-type approximation of Ψ ′ . The classicalvariational theory ensures the existence and uniqueness of an approximated solution y λ ∈ L P (Ω; C ([0 , T ]; H ) ∩ L (0 , T ; Z )) . Arguing as in [55, 56], Itô’s formula for the square ofthe H -norm and the linear growth condition on B in assumption (A4) yields, togetherwith the Gronwall lemma, that k y λ k L (Ω; C ([0 ,T ]; H ) ∩ L (0 ,T ; Z )) ≤ M ′ , (3.1)where the constant M ′ > only depends on y , C , c , c , L B and Q . Furthermore,writing Itô’s formula for the free-energy functional (see again [55]) yields Z D |∇ y λ ( t ) | + Z D Ψ λ ( y λ ( t )) + Z Q t ∇ w λ · ∇ ( w λ + u )= 12 Z D |∇ y | + Z D Ψ λ ( y ) + Z t (( w λ + u )( s ) , B ( s, y λ ( s ))) H dW ( s )+ 12 Z t k∇ B ( s, y λ ( s )) k L ( U,H ) ds + 12 ∞ X k =0 Z Q t Ψ ′′ λ ( y λ ) | B ( · , y λ ) e k | . (3.2)We want now to take power at both sides, and then supremum in time and expectations.Note that the trace term on the right-hand side can be estimated thanks to the Hölder2 Optimal control of a stochastic CH equation inequality, the Sobolev embedding
V ֒ → L ( D ) and the growth conditions (A2) and (A4)on Ψ ′′ and B , as ∞ X k =0 Z Q t Ψ ′′ λ ( y λ ) | B ( · , y λ ) e k | . k B ( · , y λ ) k L (0 ,t ; L ( U,H )) + Z t k y λ ( s ) k L ( D ) k B ( · , y λ ) k L ( U,V ) ds . k y λ k L (0 ,T ; V ) + k y λ k L (0 ,T ; V ) . Since by interpolation we have k y λ k L (0 ,t ; V ) . k y λ k L ∞ (0 ,T ; H ) ∩ L (0 ,T ; Z ) , the right-hand side is uniformly bounded in L (Ω) by the estimate (3.1). Furthermore,for the stochastic integral we note that ( w λ + u, B ) H = ( w λ − ( w λ ) D , B ) H + | D | ( w λ ) D B D + ( u, B ) . The Burkholder-Davis-Gundy, Young, and Poincaré-Wirtinger inequalities imply, togetherwith assumption (A4), that E sup r ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)(cid:12)Z r (( w λ + u )( s ) , B ( s, y λ ( s ))) H dW ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . E (cid:18)Z t (cid:0) k ( w λ − ( w λ ) D )( s ) k H + k u ( s ) k H (cid:1) k B ( s, y λ ( s )) k L ( U,H ) ds (cid:19) / + E (cid:18)Z t | ( w λ ( s )) D | k B D ( s, y λ ( s )) k L ( U, R ) ds (cid:19) / . Now, in case of multiplicative noise we have that B D = 0 by (A4) and the second termon the right-hand side vanishes, so that we can continue the estimate by E h k B ( · , y λ ) k L ∞ (0 ,T ; L ( U,H )) (cid:16) k w λ − ( w λ ) D k L (0 ,t ; H ) + k u k L (0 ,T ; H ) (cid:17)i . δ k∇ w λ k L (Ω; L (0 ,T ; H )) + C δ k y λ k L (Ω; C ([0 ,T ]; H )) + k u k L (Ω; L (0 ,T ; H )) . In case of additive noise, on the right-hand side we obtain the further contribution k B k L ∞ (0 ,T ; L ( U,V ∗ )) k ( w λ ) D k L (Ω; L (0 ,t )) . t / k ( w λ ) D k L (Ω; L ∞ (0 ,t )) . We go back now to (3.2), and take power , supremum in t ∈ [0 , T ] for a certain T ∈ (0 , T ) and expectations. Since w λ = − ∆ y λ + Ψ ′ λ ( y λ ) − u , by (A2) the term involving Ψ λ ( y λ ) on the left-hand side of (3.2) yields a bound from below for ( w λ ) D in L (Ω; L ∞ (0 , T )) :hence, choosing δ > and T small enough, rearranging the terms, and using the Gronwalllemma yields k y λ k L (Ω; L ∞ (0 ,T ; V )) + k Ψ λ ( y λ ) k L (Ω; L ∞ (0 ,T ; L ( D ))) + k∇ w λ k L (Ω; L (0 ,T ; H )) . k y λ k L (Ω; C ([0 ,T ]; H ) ∩ L (0 ,T ; Z )) + k u k L (Ω; L (0 ,T ; V )) , uca Scarpa λ . Taking these remarks into account,noting that k w λ k V . k∇ w λ k H + | ( w λ ) D | , using (3.1) and the fact that u ∈ U ′ , we inferby using a classical patching-in-time technique that k y λ k L (Ω; L ∞ (0 ,T ; V )) + k w λ k L (Ω; L (0 ,T ; V )) ≤ M ′ . (3.3)Now, by comparison in the equation for the chemical potential we deduce that k Ψ ′ λ ( y λ ) k L (Ω; L (0 ,T ; H )) ≤ M ′ , while by (A2), the embedding V ֒ → L ( D ) , and interpolation we have k∇ Ψ ′ ( y λ ) k L (0 ,T ; H ) = k Ψ ′′ λ ( y λ ) ∇ y λ k . k y λ k L ∞ (0 ,T ; V ) k y λ k L ∞ (0 ,T ; H ) ∩ L (0 ,T ; Z ) , so that by (3.1)–(3.3) k Ψ ′ λ ( y λ ) k L (Ω; L (0 ,T ; V )) ≤ M ′ . By elliptic regularity we infer then also k Ψ ′ ( y λ ) k L (Ω; L (0 ,T ; V )) + k y λ k L (Ω; L (0 ,T ; H ( D ))) ≤ M ′ . (3.4)It is now a standard matter to pass to the limit as λ ց in the approximated problem,and recover (2.2)–(2.10): for further details we refer to [55, 56].It only remains to prove the continuous dependence property (2.12), as (2.11) hasalready been proved in [56]. To this end, note that d ( y − y ) − ∆( w − w ) dt = ( B ( t, y ) − B ( t, y )) dW ,w − w = − ∆( y − y ) + Ψ ′ ( y ) − Ψ ′ ( y ) − ( u − u ) , so that Itô’s formula for the square of the H -norm yields k ( y − y )( t ) k H + Z Q t | ∆( y − y ) | = Z Q t (Ψ ′ ( y ) − Ψ ′ ( y )) ∆( y − y ) − Z Q t ( u − u )∆( y − y )+ 12 k B ( · , y ) − B ( · , y ) k L (0 ,t ; L ( U,H )) + Z t (( y − y )( s ) , B ( s, y ( s )) − B ( s, y ( s ))) H dW ( s ) . Using the Burkholder-Davis-Gundy and Young inequalities, and employing the Lipschitzcontinuity of B , we have k y − y k L (Ω; C ([0 ,t ]; H )) + k ∆( y − y ) k L (Ω; L (0 ,t ; H )) . E Z Q | u − u | + E Z Q | Ψ ′ ( y ) − Ψ ′ ( y ) | + k y − y k L (Ω; L (0 ,t ; H )) + δ k y − y k L (Ω; C ([0 ,t ]; H )) + C δ k y − y k L (Ω; L (0 ,t ; H )) , Optimal control of a stochastic CH equation for every δ > and a suitable constant C δ > . By the mean-value theorem, assumption(A2), the Hölder inequality and the embedding V ֒ → L ( D ) , E Z Q | Ψ ′ ( y ) − Ψ ′ ( y ) | . E Z Q (cid:0) | y | + | y | (cid:1) | y − y | . E Z T (cid:16) k y ( s ) k L ( D ) + k y ( s ) k L ( D ) (cid:17) k ( y − y )( s ) k L ( D ) ds . E (cid:16) k y k L ∞ (0 ,T ; V ) + k y k L ∞ (0 ,T ; V ) (cid:17) k y − y k L (0 ,T ; V ) . (cid:16) k y k L (Ω; L ∞ (0 ,T ; V )) + k y k L (Ω; L ∞ (0 ,T ; V )) (cid:17) k y − y k L (Ω; L (0 ,T ; V )) . Hence, (2.12) follows rearranging the terms and using (2.11) and (2.9).
This section is devoted to the proof of Theorem 2.2: we show that a relaxed optimalcontrol always exists.Let ( u n ) n ⊂ U be a minimizing sequence for the reduced cost functional ˜ J , and set ( y n , w n ) as the respective solution to (2.2)–(2.7). Then, by definition of U and the uniformestimates (2.8)–(2.10), we deduce that there exists a positive constant c , independent of n , such that k u n k L (Ω; L (0 ,T ; H )) ∩ L (Ω; L (0 ,T ; V )) ≤ C , k y n k L (Ω; C ([0 ,T ]; H ) ∩ L (0 ,T ; Z )) ∩ L (Ω; L ∞ (0 ,T ; V )) ∩ L (Ω; L (0 ,T ; H ( D ))) ≤ c , k w n k L (Ω; L (0 ,T ; V )) + k Ψ ′ ( y n ) k L (Ω; L (0 ,T ; V )) ≤ c . Recalling hypothesis (A4), we also deduce that k B ( · , y n ) k L (Ω; L ∞ (0 ,T ; L ( U,V ))) ≤ c . Hence, by [34, Lem. 2.1], for every s ∈ (0 , / , there exists c s > , independent of n ,such that k B ( · , y n ) · W k L (Ω; W s, (0 ,T ; V )) ≤ c s , where we have used the classical notation · for the stochastic integral. Since − >s − , we have that H (0 , T ; V ∗ ) ֒ → W s, (0 , T ; V ∗ ) by the Sobolev embeddings, and bycomparison in (2.6) we infer that k y n k L (Ω; W s, (0 ,T ; V ∗ )) ≤ c s . Let us define now π y n as the law of y n on C ([0 , T ]; H ) ∩ L (0 , T ; Z ) and show that ( π y n ) n is tight. Fixing now s ∈ (1 / , / so that s > , by [58, Sec. 8, Cor. 4–5] we have thecompact inclusions L (0 , T ; H ( D ) ∩ Z ) ∩ W s, (0 , T ; V ∗ ) c ֒ → L (0 , T ; Z ) ,L ∞ (0 , T ; V ) ∩ W s, (0 , T ; V ∗ ) c ֒ → C ([0 , T ]; H ) . uca Scarpa W := L ∞ (0 , T ; V ) ∩ L (0 , T ; H ( D ) ∩ Z ) ∩ W s, (0 , T ; V ∗ ) , we deduce that W c ֒ → C ([0 , T ]; H ) ∩ L (0 , T ; Z ) compactly and also the estimate k y n k L (Ω; W ) ≤ c . This ensures by a standard argument that ( π y n ) n is tight on C ([0 , T ]; H ) ∩ L (0 , T ; Z ) .Indeed, if B R denotes the closed ball of radius R > in W , for any R > , we have that B R is compact in C ([0 , T ]; H ) ∩ L (0 , T ; Z ) , and by Markov’s inequality π y n ( B cR ) = P (cid:8) k y n k W > R (cid:9) ≤ R k y n k L (Ω; W ) ≤ c R ∀ n ∈ N , from which the tightness of ( π y n ) n . Similarly, by [58, Sec. 8, Cor. 4–5] we also have thecompact inclusion W s, (0 , T ; V ) c ֒ → C ([0 , T ]; H ) , so that an entirely analogous argumentyields that the laws of ( B ( · , y n ) · W ) n on the space C ([0 , T ]; H ) are tight. Moreover,denoting by L w (0 , T ; V ) the space L (0 , T ; V ) endowed with its weak topology, it is clearthat the laws of ( u n ) n on L w (0 , T ; V ) are tight.Now, taking into account the remarks above, we deduce in particular that the septuple ( y n , u n , B ( · , y n ) · W, W, y , x Q , x T ) n is tight on the space C ([0 , T ]; H ) × L w (0 , T ; V ) × C ([0 , T ]; H ) × C ([0 , T ]; U ) × V × L (0 , T ; H ) × H .
By Skorokhod theorem (see [44, Thm. 2.7] and [59, Thm. 1.10.4, Add. 1.10.5]) andthe Jakubowski-Skorokhod version (see [5, Thm. 2.7.1]), there is a probability space (Ω ∗ , F ∗ , P ∗ ) , a sequence of maps ( φ n ) n , where φ n : (Ω ∗ , F ∗ ) → (Ω , F ) are measur-able and satisfy P = P ∗ ◦ φ − n for every n ∈ N , and measurable random variables ( y ∗ , u ∗ , I ∗ , W ∗ , y ∗ , x ∗ Q , x ∗ T ) defined on (Ω ∗ , F ∗ ) with values in C ([0 , T ]; H ) × L (0 , T ; V ) × C ([0 , T ]; H ) × C ([0 , T ]; U ) × V × L (0 , T ; H ) × H , such that y ∗ n := y n ◦ φ n → y ∗ in C ([0 , T ]; H ) P ∗ -a.s. ,u ∗ n := u n ◦ φ n ⇀ u ∗ in L (0 , T ; V ) P ∗ -a.s. ,I ∗ n := ( B ( · , y n ) · W ) ◦ φ n → I ∗ in C ([0 , T ]; H ) P ∗ -a.s. ,W ∗ n := W ◦ φ n → W ∗ in C ([0 , T ]; U ) P ∗ -a.s. ,y ∗ ,n := y ◦ φ n → y ∗ in V P ∗ -a.s. ,x ∗ Q,n := x Q ◦ φ n → x ∗ Q in L (0 , T ; H ) P ∗ -a.s. ,x ∗ T,n := x T ◦ φ n → x ∗ T in H P ∗ -a.s. . Since the sequence ( y , x Q , x T ) n is constant, it is clear that the laws of ( y ∗ , x ∗ Q , x ∗ T ) and ( y , x Q , x T ) coincide. Moreover, setting w ∗ n := w n ◦ w n , since the maps ( φ n ) n preserve thelaws, we readily deduce that k y ∗ n k L (Ω ∗ ; C ([0 ,T ]; H ) ∩ L (0 ,T ; Z )) ∩ L (Ω ∗ ; L ∞ (0 ,T ; V )) ∩ L (Ω ∗ ; L (0 ,T ; H ( D ))) ≤ c , k u ∗ n k L (Ω ∗ ; L (0 ,T ; H )) ∩ L (Ω ∗ ; L (0 ,T ; V )) ≤ C , k I ∗ n k L (Ω ∗ ; W s, (0 ,T ; V )) ≤ c s , k w ∗ n k L (Ω ∗ ; L (0 ,T ; V )) + k Ψ ′ ( y ∗ n ) k L (Ω ∗ ; L (0 ,T ; V )) ≤ c , Optimal control of a stochastic CH equation hence in particular that y ∗ ∈ L (Ω ∗ ; C ([0 , T ]; H ) ∩ L (0 , T ; Z )) ,y ∗ ∈ L (Ω ∗ ; L ∞ (0 , T ; V ) ∩ L (Ω ∗ ; L (0 , T ; H ( D )) ,u ∗ ∈ U ∗ ,I ∗ ∈ L (Ω ∗ ; W s, (0 , T ; V )) and y ∗ n → y ∗ in L p (Ω ∗ ; C ([0 , T ]; H )) ∀ p ∈ [1 , ,y ∗ n ⇀ y ∗ in L (Ω ∗ ; L (0 , T ; Z )) ∩ L (Ω ∗ ; L (0 , T ; H ( D ))) ,u ∗ n ⇀ u ∗ in L (Ω ∗ ; L (0 , T ; H )) ∩ L (Ω ∗ ; L (0 , T ; V )) ,w ∗ n ⇀ w ∗ in L (Ω ∗ ; L (0 , T ; V )) , Ψ ′ ( y ∗ n ) ⇀ ξ ∗ in L (Ω ∗ ; L (0 , T ; V )) , for some w ∗ ∈ L (Ω ∗ ; L (0 , T ; V ) , ξ ∗ ∈ L (Ω ∗ ; L (0 , T ; V )) . The strong-weak closure of the maximal monotone operator r Ψ ′ ( r ) + c r , r ∈ R ,ensures that ξ ∗ = Ψ ′ ( y ∗ ) almost everywhere. Moreover, by (A4) we have that B ( · , y ∗ n ) → B ( · , y ∗ ) in L p (Ω ∗ ; L (0 , T ; L ( U, H ))) ∀ p ∈ [1 , . Now, defining the filtrations ( F ∗ n,t ) t ∈ [0 ,T ] and ( F ∗ t ) t ∈ [0 ,T ] as F ∗ n,t := σ ( W ∗ n ( s )) s ∈ [0 ,t ] , F ∗ t := σ ( y ∗ ( s ) , u ∗ ( s ) , I ∗ ( s ) , W ∗ ( s )) s ∈ [0 ,T ] , t ∈ [0 , T ] , using classical representation theorems for martingales (see for example the argumentsin [60, § 4]) it is possible to show that W ∗ n is a ( F ∗ n,t ) t -cylindrical Wiener process on U , W ∗ is a ( F ∗ t ) t -cylindrical Wiener process on U , and that I ∗ n = Z · B ( s, y ∗ n ( s )) dW ∗ n ( s ) , I ∗ = Z · B ( s, y ∗ ( s )) dW ∗ ( s ) . Since ( y ∗ n , w ∗ n ) satisfies the variational formulation (2.6)–(2.7) on the space Ω ∗ with respectto ( y ∗ ,n , u ∗ n ) , passing to the weak limit it follows that ( y ∗ , w ∗ ) is the unique solution to(2.2)–(2.7) on the probability space (Ω ∗ , F ∗ , P ∗ ) corresponding to ( y ∗ , u ∗ ) . Using theweak lower semicontinuity of J , the fact that φ n preserves the law for every n , and thedefinition of the minimizing sequence ( u n ) n , we deduce that ˜ J ∗ ( u ∗ ) = α E ∗ Z Q | y ∗ − x ∗ Q | + α E ∗ Z D | y ∗ ( T ) − x ∗ T | + α E ∗ Z Q | u ∗ | ≤ lim inf n →∞ α E ∗ Z Q | y ∗ n − x ∗ Q,n | + α E ∗ Z D | y ∗ n ( T ) − x ∗ T,n | + α E ∗ Z Q | u ∗ n | = lim inf n →∞ ˜ J ( u n ) = lim n →∞ ˜ J ( u n ) = inf v ∈U ˜ J ( v ) , so that u ∗ is a relaxed optimal control. uca Scarpa In this section we study the Gâteaux differentiability of the control-to-state map and weprove the first version of first-order necessary conditions for optimality.
We prove here Proposition 2.4. Let u ∈ U ′ be given, set y := S ( u ) , and let h ∈ L P (Ω; L (0 , T ; H )) . We show that the linearized system (2.13)–(2.15) admits a uniquesolution z h . Uniqueness.
For i = 1 , , let ( z ih , µ ih ) ∈ L P (cid:0) Ω; C ([0 , T ]; H ) ∩ L (0 , T ; Z ) (cid:1) × L P (Ω; L (0 , T ; H )) , such that ( z ih , µ ih ) satisfy (2.16)–(2.17). Then we have, in the variational sense in thetriple ( Z, H, Z ∗ ) , d ( z h − z h ) − ∆( µ h − µ h ) dt = DB ( y ) (cid:0) z h − z h (cid:1) dW in (0 , T ) × D ,µ h − µ h = − ∆( z h − z h ) + Ψ ′′ ( y )( z h − z h ) in (0 , T ) × D ,∂ n ( z h − z h ) = ∂ n ( µ h − µ h ) = 0 in (0 , T ) × ∂D , ( z h − z h )(0) = 0 in D .
Integrating on D the first equation, it follows from (A4) that ( z h − z h ) D = 0 . Hence, Itô’sformula for the square of the V ∗ -norm yields (cid:13)(cid:13) ∇N ( z h − z h )( t ) (cid:13)(cid:13) H + Z Q t |∇ ( z h − z h ) | + Z Q t Ψ ′′ ( y ) | ( z h − z h ) | = 12 Z t Tr (cid:0) DB ( s, y ( s )) (cid:0) z h − z h (cid:1) ( s ) ∗ ◦ N ◦ DB ( s, y ( s )) (cid:0) z h − z h (cid:1) ( s ) (cid:1) ds + Z t (cid:0) N ( z h − z h )( s ) , DB ( s, y ( s )) (cid:0) z h − z h (cid:1) ( s ) (cid:1) H dW ( s ) . Now, by the uniform boundedness of DB , the first term on the right-hand side is boundedby (cid:13)(cid:13) DB ( · , y )( z h − z h ) (cid:13)(cid:13) L (0 ,t ; L ( U,V ∗ )) . L B (cid:13)(cid:13) z h − z h (cid:13)(cid:13) L (0 ,t ; H ) ≤ δ (cid:13)(cid:13) ∇ ( z h − z h ) (cid:13)(cid:13) L (0 ,t ; H ) + C δ (cid:13)(cid:13) z h − z h (cid:13)(cid:13) L (0 ,t ; V ∗ ) for every δ > and a certain C δ > , while the second term on the right-hand side canbe estimated using the Burkholder-Davis-Gundy and Young inequalities as E sup r ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)(cid:12)Z r (cid:0) N ( z h − z h )( s ) , DB ( s, y ( s )) (cid:0) z h − z h (cid:1) ( s ) (cid:1) H dW ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . δ (cid:13)(cid:13) z h − z h (cid:13)(cid:13) L (Ω; C ([0 ,t ]; V ∗ )) + C δ (cid:13)(cid:13) DB ( · , y ) (cid:0) z h − z h (cid:1)(cid:13)(cid:13) L (Ω; L (0 ,t ; L ( U,H ))) . δ (cid:13)(cid:13) z h − z h (cid:13)(cid:13) L (Ω; C ([0 ,t ]; V ∗ )) + δ (cid:13)(cid:13) ∇ ( z h − z h ) (cid:13)(cid:13) L (Ω; L (0 ,t ; H )) + C δ (cid:13)(cid:13) z h − z h (cid:13)(cid:13) L (Ω; L (0 ,t ; V ∗ )) . Optimal control of a stochastic CH equation
Furthermore, since Ψ ′′ ≥ − c , choosing δ sufficiently small and rearranging the termsyields, the Young inequality, (cid:13)(cid:13) z h − z h (cid:13)(cid:13) L (Ω; C ([0 ,t ]; H )) + E Z Q t |∇ ( z h − z h ) | . c E Z Q t | z h − z h | . σ E Z Q t |∇ ( z h − z h ) | + ˜ c σ E Z t (cid:13)(cid:13) ∇N ( z h − z h )( s ) (cid:13)(cid:13) H ds for every σ > and a certain ˜ c σ > . Taking σ small enough, the Gronwall lemma yieldsthen z h = z h , hence also µ h = µ h by comparison in the system, from which uniqueness. Approximation.
Let us focus on existence. To this end, we consider the approximation dz nh − ∆ µ nh dt = DB ( y ) z nh dW in (0 , T ) × D ,µ nh = − ∆ z nh + Ψ ′′ n ( y ) z h − h in (0 , T ) × D ,∂ n z nh = ∂ n µ h = 0 in (0 , T ) × ∂D ,z nh (0) = 0 in D , where Ψ ′′ n := T n ◦ Ψ ′′ and T n : R → R is the truncation operator at level n . i.e. T n ( r ) := n if r > n ,r if | r | ≤ n , − n if r < − n , r ∈ R . Since Ψ ′′ n ( y ) ∈ L ∞ (Ω × Q ) , it is not difficult to check that such approximated problemadmits a unique solution ( z nh , µ nh ) satisfying (2.13)–(2.17) with Ψ ′′ n instead of Ψ ′′ . Indeed,one can reformulate the problem in the variational triple ( Z, H, Z ∗ ) as dz nh + A n z nh dt = DB ( t, y ) z nh dW , z nh (0) = 0 , where A n : Ω × [0 , T ] × Z → Z ∗ is given by h A n ( ω, t, x ) , ϕ i Z := Z D ∆ x ∆ ϕ − Z D Ψ ′′ n ( y ( ω, t )) x ∆ ϕ + Z D h ( ω, t )∆ ϕ , for ( ω, t ) ∈ Ω × [0 , T ] and x, ϕ ∈ Z . Since Ψ ′′ n ( y ) ∈ L ∞ (Ω × Q ) , it is not difficult tocheck that A n is progressively measurable, weakly monotone, weakly coercive and linearlybounded. Moreover, it is clear that the operator x DB ( t, y ( ω, t )) x , x ∈ H , is Lipschitz-continuous and linearly bounded from H to L ( U, H ) , uniformly on Ω × [0 , T ] Hence, the approximated problem admits a unique solution z nh such that, setting µ nh := − ∆ z nh + Ψ ′′ n ( y ) z nh − h , conditions (2.13)–(2.17) are satisfied with Ψ ′′ n . Uniform estimates.
Let us now prove uniform estimates independently of n and passto the limit as n → ∞ . Noting that ( z nh ) D = 0 by (A4), Itô’s formula for the square of uca Scarpa V ∗ -norm yields k∇N ( z nh )( t ) k H + Z Q t |∇ z nh | + Z Q t Ψ ′′ ( y ) | z nh | = Z Q t hz nh + 12 Z t Tr ( DB ( s, y ( s )) z nh ( s ) ∗ ◦ N ◦ DB ( s, y ( s )) z nh ( s )) ds + Z t ( N ( z nh )( s ) , DB ( s, y ( s )) z nh ( s )) H dW ( s ) . for every t ∈ [0 , T ] , P -almost surely. Since Ψ ′′ ≥ − c implies that Ψ ′′ n ≥ − c for every n ∈ N , by the Young inequality we have k∇N ( z nh )( t ) k H + Z Q t |∇ z nh | ≤ Z Q t | h | + (cid:18)
12 + c (cid:19) Z Q t | z nh | + 12 Z t Tr ( DB ( s, y ( s )) z nh ( s ) ∗ ◦ N ◦ DB ( s, y ( s )) z nh ( s )) ds + Z t ( N ( z nh )( s ) , DB ( s, y ( s )) z nh ( s )) H dW ( s ) ., where, by the properties of N , Z Q t | z nh | ≤ δ Z Q t |∇ z nh | + C δ Z t k∇N z nh ( s ) k H ds for every δ > and a positive constant C δ > . Hence, choosing δ sufficiently small,taking power , supremum in time and then expectations, arguing on the right-hand sideexactly as in the proof of uniqueness in Section 5.1, we deduce that k z nh k L (Ω; C ([0 ,T ]; V ∗ ) ∩ L (0 ,T ; V )) . k h k L (Ω; L (0 ,T ; H )) ∀ n ∈ N . (5.1)Now we write Itô’s formula for the square of the H -norm, getting k z nh ( t ) k H + Z Q t | ∆ z nh | = Z Q t Ψ ′′ n ( y ) z nh ∆ z nh − Z Q t h ∆ z nh + Z t k DB ( s, y ( s )) z nh ( s ) k L ( U,H ) ds + Z t ( z nh ( s ) , DB ( s, y ( s )) z nh ( s )) H dW ( s ) for every t ∈ [0 , T ] , P -almost surely. We proceed now similarly to the previous estimate,taking supremum in time and expectations. Using the boundedness of DB together withthe Burkholder-Davis-Gundy and Young inequalities on the right-hand side we have inparticular that E Z t k DB ( s, y ( s )) z nh ( s ) k L ( U,H ) ds + E sup r ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)(cid:12)Z r ( z nh ( s ) , DB ( s, y ( s )) z nh ( s )) dW ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . δ E k z nh k C ([0 ,T ]; H ) + C δ E k z nh k L (0 ,t ; H ) for every δ > and a certain C δ > . Choosing δ sufficiently small we infer that k z nh k L (Ω; C ([0 ,T ]; H )) + k ∆ z nh k L (Ω; L (0 ,T ; H )) . k h k L (Ω; L (0 ,T ; H )) + E Z Q | Ψ ′′ n ( y ) z nh | + k z nh k L (Ω; L (0 ,t ; H )) , Optimal control of a stochastic CH equation where by (A2), the Hölder inequality and the Sobolev embedding
V ֒ → L ( D ) and (5.1), E Z Q | Ψ ′′ n ( y ) z nh | . c E Z Q (1 + | y | ) | z nh | ≤ E Z T (cid:13)(cid:13) | y ( s ) | (cid:13)(cid:13) L / ( D ) (cid:13)(cid:13) | z nh ( s ) | (cid:13)(cid:13) L ( D ) ds . E (1 + k y k L ∞ (0 ,T ; V ) ) k z nh k L (0 ,T ; V ) ≤ (1 + k y k L (Ω; L ∞ (0 ,T ; V ) )) k h k L (Ω; L (0 ,T ; H )) . The estimate (2.9) yields then, thanks to the Gronwall lemma, k z nh k L (Ω; C ([0 ,T ]; H ) ∩ L (0 ,T ; Z )) . k h k L (Ω; L (0 ,T ; H )) ∀ n ∈ N . (5.2)By comparison in the equations we also deduce that k µ nh k L (Ω; L (0 ,T ; H )) . k h k L (Ω; L (0 ,T ; H )) ∀ n ∈ N . (5.3) Passage to the limit.
By the estimates (5.1)–(5.3), we deduce that there are z h ∈ L (Ω; C ([0 , T ]; H ) ∩ L (0 , T ; Z )) , µ h ∈ L (Ω; L (0 , T ; H )) , such that, as n → ∞ , z nh ⇀ z h in L (Ω; L (0 , T ; Z )) , µ nh ⇀ µ h in L (Ω; L (0 , T ; H )) . Moreover, since DB ( · , y ) ∈ L ( H ; L ( U, H )) and DB ( · , y ) ∗ ∈ L ( L ( U, H ); H ) , theboundedness of DB ensures that for every ϕ ∈ L (Ω; L (0 , T ; L ( U, H ))) we have that DB ( · , y ) ∗ ϕ ∈ L (Ω; L (0 , T ; H )) : hence, the weak convergence of ( z nh ) n readily impliesthat E Z T ( DB ( s, y ( s )) z nh ( s ) , ϕ ( s )) L ( U,H ) ds = E Z T ( z nh ( s ) , DB ( s, y ( s )) ∗ ϕ ( s )) H ds → E Z T ( z h ( s ) , DB ( s, y ( s )) ∗ ϕ ( s )) H ds = E Z T ( DB ( s, y ( s )) z h ( s ) , ϕ ( s )) L ( U,H ) ds . Since ϕ is arbitrary we infer that DB ( · , y ) z nh ⇀ B ( · , y ) z h in L (Ω; L (0 , T ; L ( U, H ))) , hence also, by the linearity and continuity of the stochastic intragral, Z · DB ( s, y ( s )) z nh ( s ) dW ( s ) ⇀ Z · DB ( s, y ( s )) z nh ( s ) dW ( s ) in L (Ω; L (0 , T ; H )) . It is clear then that these convergences are enough to pass to the limit in the variationalformulation (2.16)–(2.17), except for the term Ψ ′′ n ( y ) z nh : let us analyse it explicitly. Tothis end, note that since Ψ ′′ has quadratic growth and y ∈ L (Ω; L ∞ (0 , T ; L ( D ))) , wehave in particular that Ψ ′′ ( y ) ∈ L (Ω × (0 , T ) × D ) , hence also Ψ ′′ n ( y ) → Ψ ′′ ( y ) in L (Ω × (0 , T ) × D ) . The weak convergence of ( z nh ) n implies then that Ψ ′′ n ( y ) z nh ⇀ Ψ ′′ ( y ) z h in L / (Ω × (0 , T ) × D ) . Hence, letting n → ∞ in (2.16)–(2.17) we deduce that ( z h , µ h ) satisfies the variationalformulation of the linearized system. uca Scarpa We show here that the map S is weakly Gâteaux-differentiable in the sense specified byTheorem 2.3, and that its weak derivative is the unique solution z h to (2.13)–(2.17).Let u, h ∈ U ′ and fix ε > sufficiently small such that u + εh ∈ U ′ for all ε ∈ [ − ε , ε ] .Set also y := S ( u ) and y εh := S ( u + εh ) for any ε ∈ [ − ε , ε ] \ { } , and let z h be the uniquesolution to the linearized system given by Proposition 2.4. Then we have, in the variationaltriple ( Z, H, Z ∗ ) , d (cid:18) y εh − yε (cid:19) − ∆ (cid:18) w εh − wε (cid:19) dt = B ( y εh ) − B ( y ) ε dW ,w εh − wε = − ∆ (cid:18) y εh − yε (cid:19) + Ψ ′ ( y εh ) − Ψ ′ ( y ) ε − h . By the continuous dependence properties (2.11)–(2.12) we have that (cid:13)(cid:13)(cid:13)(cid:13) y εh − yε (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; C ([0 ,T ]; V ∗ ) ∩ L (0 ,T ; V )) . k h k L (Ω; L (0 ,T ; V ∗ )) (5.4)and (cid:13)(cid:13)(cid:13)(cid:13) y εh − yε (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; C ([0 ,T ]; H ) ∩ L (0 ,T ; Z )) . k h k L (Ω; L (0 ,T ; H )) . (5.5)Furthermore, the mean-value theorem and the fact that Ψ ′′ has quadratic growth implies,by the Hölder inequality and the continuous embedding V ֒ → L ( D ) , E Z Q (cid:12)(cid:12)(cid:12)(cid:12) Ψ ′ ( y εh ) − Ψ ′ ( y ) ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ E Z Q Z | Ψ ′′ ( y + σ ( y εh − y )) | (cid:12)(cid:12)(cid:12)(cid:12) y εh − yε (cid:12)(cid:12)(cid:12)(cid:12) dσ . E Z Q (cid:0) | y | + | y εh | (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) y εh − yε (cid:12)(cid:12)(cid:12)(cid:12) . (cid:16) k y k L (Ω; L ∞ (0 ,T ; V )) + k y εh k L (Ω; L ∞ (0 ,T ; V )) (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) y εh − yε (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; L (0 ,T ; V )) , so that (2.9) and (5.4) imply that (cid:13)(cid:13)(cid:13)(cid:13) Ψ ′ ( y εh ) − Ψ ′ ( y ) ε (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; L (0 ,T ; H )) . k h k L (Ω; L (0 ,T ; H )) . (5.6)Moreover, the Lipschitz-continuity of B and (5.5) ensures that (cid:13)(cid:13)(cid:13)(cid:13) B ( · , y εh ) − B ( · , y ) ε (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; C ([0 ,T ]; L ( U,H ))) . k h k L (Ω; L (0 ,T ; H )) , from which (cid:13)(cid:13)(cid:13)(cid:13)Z · B ( s, y εh ( s )) − B ( s, y ( s )) ε dW ( s ) (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; C ([0 ,T ]; H )) . k h k L (Ω; L (0 ,T ; H )) . (5.7)2 Optimal control of a stochastic CH equation
Hence by comparison in the equation we also have that (cid:13)(cid:13)(cid:13)(cid:13) w εh − wε (cid:13)(cid:13)(cid:13)(cid:13) L (Ω; L (0 ,T ; H )) . k h k L (Ω; L (0 ,T ; V )) . (5.8)Let us pass to the limit as ε ց . From the estimates (5.4)–(5.8) we deduce that thereare z h ∈ L (cid:0) Ω; L ∞ (0 , T ; H ) ∩ L (0 , T ; Z ) (cid:1) , µ h ∈ L P (Ω; L (0 , T ; H )) such that, as ε ց , y εh − yε ⇀ z h in L (cid:0) Ω; L p (0 , T ; H ) ∩ L (0 , T ; Z ) (cid:1) ∀ p ∈ [1 , + ∞ ) , (5.9) w εh − wε ⇀ µ h in L (cid:0) Ω; L (0 , T ; H ) (cid:1) . (5.10)Moreover, note that Ψ ′ ( y εh ) − Ψ ′ ( y ) ε − Ψ ′′ ( y ) z h = Ψ ′ ( y εh ) − Ψ ′ ( y ) − Ψ ′′ ( y )( y εh − y ) ε + Ψ ′′ ( y ) (cid:18) y εh − yε − z h (cid:19) = Z (Ψ ′′ ( y + r ( y εh − y )) − Ψ ′′ ( y )) y εh − yε dr + Ψ ′′ ( y ) (cid:18) y εh − yε − z h (cid:19) . Since Ψ ′′ ( y ) ∈ L (Ω × (0 , T ) × D ) , the weak convergences proved above imply that Ψ ′′ ( y ) (cid:18) y εh − yε − z h (cid:19) ⇀ in L / (Ω × (0 , T ) × D ) . Let us show that also the first term goes to . To this end, note that since k y εh − y k L (Ω; C ([0 ,T ]; H ) ∩ L (0 ,T ; Z )) . ε k h k L (Ω; L (0 ,T ; H )) → , by continuity of Ψ ′′ , we have, along a subsequence, Ψ ′′ ( y + r ( y εh − y )) − Ψ ′′ ( y ) → ∀ r ∈ [0 , , a.e. in Ω × (0 , T ) × D .
Moreover, since Ψ ′′ has quadratic growth, we deduce that (cid:12)(cid:12)(cid:12)(cid:12)Z (Ψ ′′ ( y + r ( y εh − y )) − Ψ ′′ ( y )) dr (cid:12)(cid:12)(cid:12)(cid:12) . | y εh | + | y | , where the right hand side is bounded in L (Ω × (0 , T ) × D ) because y and ( y εh ) ε arebounded in L (Ω; L ∞ (0 , T ; V )) by Theorem 2.1 and V ֒ → L ( D ) . Consequently, Z (Ψ ′′ ( y + r ( y εh − y )) − Ψ ′′ ( y )) dr → in L p (Ω × (0 , T ) × D ) ∀ p ∈ [2 , . In particular, we deduce that Z (Ψ ′′ ( y + r ( y εh − y )) − Ψ ′′ ( y )) y εh − yε dr ⇀ in L p (Ω × (0 , T ) × D ) uca Scarpa p ∈ [1 , / , from which Ψ ′ ( y εh ) − Ψ ′ ( y ) ε ⇀ Ψ ′′ ( y ) z h in L p (Ω × (0 , T ) × D ) ∀ p ∈ [1 , / . (5.11)Let us show the convergence of the stochastic integrals. To this end, note that B ( · , y εh ) − B ( · , y ) ε − DB ( · , y ) z h = B ( · , y εh ) − B ( · , y ) − DB ( · , y )( y ε − y ) ε + DB ( · , y ) (cid:18) y εh − yε − z h (cid:19) = Z ( DB ( · , y + r ( y εh − y )) − DB ( · , y )) y εh − yε dr + DB ( · , y ) (cid:18) y εh − yε − z h (cid:19) . The weak convergences proved above, the linearity of DB ( · , y ) , the boundedness of DB and the dominated convergence theorem yields DB ( · , y ) (cid:18) y εh − yε − z h (cid:19) ⇀ in L (Ω; L (0 , T ; L ( U, H ))) . Moreover, since DB ( t, · ) ∈ C ( H ; L ( H, L ( U, H ))) by assumption (A5), recalling alsothat y εh → y in L (Ω; C ([0 , T ]; H )) , by the dominated convergence theorem we have that Z ( DB ( · , y + r ( y εh − y )) − DB ( · , y )) dr → in L p (Ω; L p (0 , T ; L ( H ; L ( U ; H )))) for every p ∈ [2 , ∞ ) . Since y εh − yε ⇀ z h in L (Ω; L p (0 , T ; H )) , we deduce in particular that Z ( DB ( · , y + r ( y εh − y )) − DB ( · , y )) y εh − yε dr ⇀ in L p (Ω; L (0 , T ; L ( U, H ))) , Consequently, taking this information into account, we have B ( · , y εh ) − B ( · , y ) ε ⇀ DB ( · , y ) z h in L p (Ω; L (0 , T ; L ( U ; H ))) ∀ p ∈ [1 , , from which Z · B ( s, y εh ( s )) − B ( s, y ( s )) ε dW ( s ) ⇀ Z · DB ( s, y ( s )) z h ( s ) dW ( s ) (5.12)in L p (Ω; L (0 , T ; H )) for all p ∈ [1 , .Hence, letting ε → in the variational formulation we deduce that ( z h , µ h ) satisfy thelinearized system (2.13)–(2.17). Since we have already proved uniqueness for such systemin the previous section, we deduce that ( z h , µ h ) is the unique solution to (2.13)–(2.17). We prove here the version of first-order necessary optimality conditions contained inTheorem 2.4.4
Optimal control of a stochastic CH equation
Let ¯ u ∈ U be an optimal control and set ¯ u := S (¯ u ) . For every v ∈ U let us define h := v − u , and y εh := S (¯ u + εh ) for every ε > . Since U is convex, we have that u + ε ( v − u ) ∈ U for all ε ∈ [0 , : hence, by definition of optimal control we have that ˜ J (¯ u ) ≤ ˜ J (¯ u + εh ) , which may be rewritten J (¯ y, ¯ u ) ≤ α E Z Q | y εh − x Q | + α Z D | y εh ( T ) − x T | + α E Z Q | ¯ u + εh | . Using the definition of J (¯ y, ¯ u ) and rearranging the terms we have ≤ α E Z Q (cid:0) | y εh | − | ¯ y | − y εh − ¯ y ) x Q (cid:1) + α E Z D (cid:0) | y εh ( T ) | − | ¯ y ( T ) | − y εh − ¯ y )( T ) x T (cid:1) + α E Z Q (cid:0) ε | h | + 2 ε ¯ uh (cid:1) . Since the functions x E R Q | x | and x E R D | x | are Fréchet-differentiable in L (Ω × Q ) and L (Ω × D ) , respectively, dividing by ε we get ≤ α E Z Q (cid:18)Z (¯ y + σ ( y εh − ¯ y )) dσ − x Q (cid:19) y εh − ¯ yε + α E Z D (cid:18)Z (¯ y + σ ( y εh − ¯ y ))( T ) dσ − x T (cid:19) y εh − ¯ yε ( T )+ α E Z Q ¯ uh + α ε k h k L (Ω × Q ) . Since ¯ u + εh → ¯ u in L (Ω; L (0 , T ; V )) as ε ց , we deduce from (2.11)–(2.12), thedefinition of U and the dominated convergence theorem that Z (¯ y + σ ( y εh − ¯ y )) dσ − x Q → ¯ y − x Q in L (Ω; L (0 , T ; H )) , Z (¯ y + σ ( y εh − ¯ y ))( T ) dσ − x T → ¯ y ( T ) − x T in L (Ω; H ) . Furthermore, by Theorem 2.3 we know that y εh − ¯ yε → z h in L / (Ω; L (0 , T ; H )) ,y εh − ¯ yε ( T ) ⇀ z h ( T ) in L (Ω; H ) , so that letting ε ց in the last inequality Theorem 2.4 is proved. In this section we study the adjoint problem (1.6)–(1.9) in terms of existence and unique-ness of solutions. Moreover, we prove the refined version of first-order necessary optimalityconditions contained in Theorem 2.5. uca Scarpa We prove here Proposition 2.6. Let u ∈ U ′ and y := S ( u ) . Uniqueness.
First of all we prove uniqueness of solutions. Let ( p i , ˜ p i , q i ) satisfy (2.18)–(2.22) for i = 1 , : taking the difference of the respective equations we have, setting p := p − p , ˜ p := ˜ p − ˜ p , and q := q − q , − dp − ∆˜ p dt + Ψ ′′ ( y )˜ p dt = DB ( y ) ∗ q dt − q dW , ˜ p = − ∆ p . Itô’s formula for k∇ p k H yields then E k∇ p ( t ) k H + E Z Tt Z D |∇ ˜ p ( s ) | ds + E Z Tt Z D Ψ ′′ ( y ( s )) | ˜ p ( s ) | + 12 E Z Tt k∇ q ( s ) k L ( U,H ) ds = E Z Tt ( q ( s ) , DB ( s, y ( s ))˜ p ( s )) L ( U,H ) ds . Recalling assumption (A4), we have that DB ( · , y )˜ p ∈ L ( U, H ) , so that ( q, DB ( · , y )˜ p ) L ( U,H ) = ( q − q D , DB ( · , y )˜ p ) L ( U,H ) . Taking into account (A2) and noting that ˜ p D = 0 , we get, by the Young and Poincaréinequalities and (2.1), E k∇ p ( t ) k H + E Z Tt Z D |∇ ˜ p ( s ) | ds + 12 E Z Tt k∇ q ( s ) k L ( U,H ) ds ≤ c E Z Tt Z D | ˜ p ( s ) | ds + L B E Z Tt k ( q − q D )( s ) k L ( U,H ) (1 + k ˜ p ( s ) k H ) ds ≤ σ E Z Tt Z D |∇ ˜ p ( s ) | ds + C σ E Z Tt k∇N ˜ p ( s ) k H ds + σ E Z Tt k∇ q ( s ) k L ( U,H ) ds . σ E Z Tt Z D |∇ ˜ p ( s ) | ds + σ E Z Tt k∇ q ( s ) k L ( U,H ) + C σ E Z Tt k∇ p ( s ) k H ds for every σ > for a certain C σ > . Choosing σ sufficiently small and applying theGronwall lemma yields then ∇ ˜ p = 0 , from which ˜ p = 0 since ˜ p D = 0 . Since ˜ p = − ∆ p , weinfer that − ∆ p = 0 , from which p − ( p ) D = p − ( p ) D . Approximation.
Let us prove now existence of solution to the BSPDE (1.6)–(1.9). Weperform the same approximation that we used for the linearized system in Section 5.1,and we consider for every n ∈ N the approximated problem ˜ p n = − ∆ p n in Q , − dp n − ∆˜ p n dt + Ψ ′′ n ( y )˜ p n dt = α ( y − x Q ) dt + DB ( y ) ∗ q n dt − q n dW in Q ,∂ n p n = ∂ n ˜ p n = 0 in Σ ,p n ( T ) = α ( y ( T ) − x T ) in D , where Ψ ′′ n := T n ◦ Ψ ′′ and T n : R → R is the truncation operator at level n . The variational6 Optimal control of a stochastic CH equation formulation of the approximated problem is given by Z D p n ( t ) ϕ + Z Tt Z D ∆ p n ( s )∆ ϕ ds − Z Tt Z D Ψ ′′ n ( y ( s ))∆ p n ( s ) ϕ ds = α Z D ( y ( T ) − x T ) ϕ + α Z Tt Z D ( y − x Q )( s ) ϕ ds + Z Tt ( DB ( s, y ( s )) ∗ q n ( s ) , ϕ ) H ds − Z D (cid:18)Z Tt q n ( s ) dW ( s ) (cid:19) ϕ for every ϕ ∈ Z , P -almost surely, for every t ∈ [0 , T ] . Hence, we introduce the operator A ∗ n : Ω × [0 , T ] × Z → Z ∗ as h A ∗ n ( ω, t, x ) , ϕ i Z := Z D ∆ x ∆ ϕ − Z D Ψ ′′ n ( y ( ω, t ))∆ xϕ and note that since Ψ ′′ n ( y ) ∈ L ∞ (Ω × Q ) , then A ∗ n is progressively measurable, weaklymonotone, weakly coercive and linearly bounded. Moreover, the operator DB ( · , y ) ∗ isuniformly bounded in Ω × [0 , T ] be (A4). Hence, by the classical variational approach toBSPDEs (see [28, Sec. 3]) the approximated problem admits a unique solution ( p n , ˜ p n , q n ) with p n ∈ L P (Ω; C ([0 , T ]; H )) ∩ L P (Ω; L (0 , T ; Z )) , ˜ p n ∈ L P (Ω; C ([0 , T ]; Z ∗ )) ∩ L P (Ω; L (0 , T ; H )) ,q n ∈ L P (Ω; L (0 , T ; L ( U, H ))) . Moreover, by assumption we have α x T ∈ L (Ω , F T , P ; V ) , while by Theorem 2.1 weknow that y ∈ L (Ω; C ([0 , T ]; H ) ∩ L ∞ (0 , T ; V )) , so that y is weakly continuous in V and y ( T ) ∈ L (Ω , F T , P ; V ) . Consequently, we have that α ( y ( T ) − x T ) ∈ L (Ω , F T , P ; V ) ,and this ensures a further regularity on ( p n , ˜ p n , q n ) , namely p n ∈ L P (Ω; C ([0 , T ]; V )) ∩ L P (Ω; L (0 , T ; H ( D ))) , ˜ p n ∈ L P (Ω; C ([0 , T ]; V ∗ )) ∩ L P (Ω; L (0 , T ; V )) ,q n ∈ L P (Ω; L (0 , T ; L ( U, V ))) . In order to prove this, one should perform a further approximation on the problem depend-ing on a further parameter (let us say k , for example), write Itô’s formula for (cid:13)(cid:13) ∇ p kn (cid:13)(cid:13) H and then pass to the limit as k → ∞ . Since the procedure is quite standard, to avoidheavy notations we shall proceed formally writing Itô’s formula for k∇ p n k H : this reads k∇ p n ( t ) k H + Z Tt Z D |∇ ∆ p n ( s ) | ds + Z Tt Z D Ψ ′′ n ( y ( s )) | ∆ p n ( s ) | ds + 12 Z Tt k∇ q n ( s ) k L ( U,H ) ds − Z Tt (∆ p n ( s ) , q n ( s )) H dW ( s )= α k∇ ( y ( T ) − x T ) k H − α Z Tt Z D ( y − x Q )( s )∆ p n ( s ) ds + Z Tt ( DB ( s, y ( s )) ∗ q n ( s ) , ˜ p n ( s )) H ds . (6.1) uca Scarpa Ψ ′′ n ∈ L ∞ (Ω × Q ) (recall that here n is fixed) and we already know that p n ∈ L (Ω; L (0 , T ; Z )) , the desired regularity follows by a classical procedure based on theBurkholder-Davis-Gundy inequality. The regularity for ˜ p n follows then by comparison. First estimate.
We now prove uniform estimates independently of n and pass to thelimit as n → ∞ . First of all, taking expectations in (6.1), and performing the samecomputations as in the proof of uniqueness at the beginning of Section 6.1 yields E k ˜ p n ( t ) k V ∗ + E Z Tt Z D |∇ ˜ p n ( s ) | ds + E Z Tt k∇ q n ( s ) k L ( U,H ) ds . c ,L B k y ( T ) k L (Ω; V ) + k α x T k L (Ω; V ) + α E Z Q | y − x Q | + E Z Tt Z D | ˜ p n ( s ) | ds + E Z Tt k∇ q n ( s ) k L ( U,H ) ds . Recalling that k∇ p n k H = k∇N ˜ p n k H . k ˜ p n k V ∗ , using the compactness inequality (2.1)on the right-hand side yields, by the Gronwall lemma, k∇ q n k L (Ω; L (0 ,T ; L ( U,H ))) ≤ c . Hence, going back again in Itô’s formula (6.1), we now take supremum in time and thenexpectations: we estimate the stochastic integral using the Burkholder-Davis Gundy in-equality and integration by parts as (see. e.g. [47, Lem. 4.3]) E sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z Tt (∆ p n ( s ) , q n ( s )) H dW ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . ε E k∇ p n k C ([0 ,T ]; H ) + C ε E k∇ q n k L (0 ,T ; L ( U,H )) for every ε > , so that choosing ε sufficiently small, rearranging the terms and recallingthe estimate just proved on ( ∇ q n ) n yields, for a positive constant c independent of n , k ˜ p n k L (Ω; C ([0 ,T ]; V ∗ ))) ∩ L (Ω; L (0 ,T ; V )) + k∇ q n k L (Ω; L (0 ,T ; L ( U,H ))) ≤ c . (6.2) Second estimate.
We write Itô’s formula for ( k∇ p n k H ) , getting k∇ p n ( t ) k H + 34 Z Tt k∇ p n ( s ) k H k∇ ˜ p n ( s ) k H ds + 34 Z Tt k∇ p n ( s ) k H Z D Ψ ′′ ( y ) | ˜ p n ( s ) | ds + 38 Z Tt k∇ p n ( s ) k H k∇ q n ( s ) k L ( U,H ) ds + 32 Z Tt k∇ p n ( s ) k H kh− ∆ p n ( s ) , q n ( s ) i V k L ( U, R ) ds − Z Tt k∇ p n ( s ) k H (∆ p n ( s ) , q n ( s )) H dW ( s )= 18 k α ∇ ( y ( T ) − x T ) k H + 34 α Z Tt Z D k∇ p n ( s ) k H ( y − x Q )( s )˜ p n ( s ) ds + 34 Z Tt k∇ p n ( s ) k H ( q n ( s ) , DB ( s, y ( s ))˜ p n ( s )) L ( U,H ) ds , (6.3)8 Optimal control of a stochastic CH equation
By the Hölder and Young inequalities and the definition of ˜ p n , for all δ > we have α Z Tt Z D k∇ p n ( s ) k H ( y − x Q )( s )˜ p n ( s ) ds ≤ α Z Tt k∇ p n ( s ) k H k ˜ p n ( s ) k H k ( y − x Q )( s ) k H ds ≤ δ Z Tt k∇ p n ( s ) k H k∇ ˜ p n ( s ) k H ds + k α ( y − x Q ) k L (0 ,T ; H ) + C δ Z Tt k∇ p n ( s ) k H ds . Moreover, recalling that Ψ ′′ ≥ − c ad k ˜ p n k V ∗ . k∇ p n k H , for every δ > we have − E Z Tt k∇ p n ( s ) k H Z D Ψ ′′ ( y ( s )) | ˜ p n ( s ) | ds ≤ c E Z Tt k∇ p n ( s ) k H k ˜ p n ( s ) k H ds ≤ δ E Z Tt k∇ p n ( s ) k H k∇ ˜ p n ( s ) k H ds + C δ Z Tt E k∇ p n ( s ) k H ds . Similarly, by assumptions (A4)–(A5), recalling that ( DB ( · , y )˜ p n ) D = 0 and writing q = q − q D + q D , and arguing as in the the proof of (6.2) we have E Z Tt k∇ p n ( s ) k H ( q n ( s ) , DB ( s, y ( s ))˜ p n ( s )) L ( U,H ) ds . L B δ E Z Tt k∇ p n ( s ) k H k∇ q n ( s ) k L ( U,H ) ds + δ E Z Tt k∇ p n ( s ) k H k∇ ˜ p n ( s ) k H ds + C δ Z Tt k∇ p n ( s ) k H ds . Hence, taking expectations in (6.3), recalling the assumptions on x T and x Q and that y ∈ L (Ω; L ∞ (0 , T ; V )) , choosing δ > sufficiently small, the Gronwall lemma yields k∇ p n k C ([0 ,T ]; L (Ω; H ))) + E Z T k∇ p n ( s ) k H k∇ q n ( s ) k L ( U,H ) ds ≤ c . At this point, we go back to (6.3), take supremum in time and then expectations: esti-mating the stochastic integral through the Burkolder-Davis-Gundy inequality as E sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z Tt k∇ p n ( s ) k H (∆ p n ( s ) , q n ( s )) H dW ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . E (cid:18)Z T k∇ p n ( s ) k H k∇ q n ( s ) k L ( U,H ) ds (cid:19) / ≤ E k∇ p n k C ([0 ,T ]; H ) (cid:18)Z T k∇ p n ( s ) k H k∇ q n ( s ) k L ( U,H ) ds (cid:19) / ≤ δ E k∇ p n k C ([0 ,T ]; H ) + C δ E Z T k∇ p n ( s ) k H k∇ q n ( s ) k L ( U,H ) ds . Choosing δ > , rearranging the terms and taking into account the estimate alreadyproved, we get k∇ p n k L (Ω; C ([0 ,T ]; H )) + E Z T k∇ p n ( s ) k H k∇ q n ( s ) k L ( U,H ) ds ≤ c . (6.4) uca Scarpa rd-power and then expectations: using again (A2)and the Young inequality we get E sup r ∈ [ t,T ] k∇ p n ( r ) k H + E k∇ ˜ p n k L ( t,T ; H ) + E k∇ q n k L ( t ; T ; L ( U,H )) . c ,L B E k α ∇ ( y ( T ) − x T ) k H + E k ˜ p n k L ( t,T ; H ) + E k α ( y − x Q ) k L (0 ,T ; H ) + E sup r ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z Tr (∆ p n ( s ) , q n ( s )) H dW ( s ) (cid:12)(cid:12)(cid:12)(cid:12) , where the last term is estimated thanks to the Burkholder-Davis-Gundy inequality by E (cid:18)Z Tt k∇ p n ( s ) k H k∇ q n ( s ) k L ( U,H ) ds (cid:19) / ≤ σ E k∇ q n k L ( t,T ; L ( U,H )) + C σ E k∇ p n k C ([0 ,T ]; H ) for every σ > and for a certain C σ > . Hence, noting also that E k ˜ p n k L ( t,T ; H ) ≤ σ E k∇ ˜ p n k L ( t,T ; H ) + ˜ C σ E k∇ p n k L ( t,T ; H ) for a certain ˜ C σ > , choosing σ > sufficiently small, rearranging the terms and taking(6.4) into account, by the Gronwall lemma we deduce that k ˜ p n k L (Ω; L (0 ,T ; V )) + k∇ q n k L (Ω; L (0 ,T ; L ( U,H ))) ≤ c . (6.5) Third estimate.
We write Itô’s formula for k p n k H , getting for every t ∈ [0 , T ] , P -a.s. k p n ( t ) k H + Z Tt Z D |∇ p n ( s ) | ds + Z Tt Z D Ψ ′′ n ( y ( s ))˜ p n ( s ) p n ( s ) ds + 12 Z Tt k q n ( s ) k L ( U,H ) ds + Z Tt ( p n ( s ) , q n ( s )) H dW ( s )= α k y ( T ) − x T k H + α Z Tt Z D ( y − x Q )( s ) p n ( s ) ds + Z Tt ( q n ( s ) , DB ( s, y ( s )) p n ( s )) L ( U,H ) ds . Taking expectations, using the Young inequality, the boundedness of DB and the esti-mates (6.2)–(6.5), we infer that, for every t ∈ [0 , T ] , E k p n ( t ) k H + E Z Tt Z D |∇ p n ( s ) | ds + E Z Tt k q n ( s ) k L ( U,H ) ds . E Z Q | Ψ ′′ n ( y )˜ p n | , where the implicit constant is independent of n . Now, by (A2) and the Hölder and Younginequalities, we have E Z Q | Ψ ′′ ( y )˜ p n | . E Z Q | ˜ p n | + E Z Q | y | | ˜ p n | . E Z Q | ˜ p n | + E k y k L ∞ (0 ,T ; V ) k ˜ p n k L (0 ,T ; V ) ≤ E k ˜ p n k L (0 ,T ; H ) + E k y k L ∞ (0 ,T ; V ) + E k ˜ p n k L (0 ,T ; V ) , Optimal control of a stochastic CH equation so that by (6.2) and (6.5) we get k q n k L (Ω; L (0 ,T ; L ( U,H ))) + k Ψ ′′ n ( y )˜ p n k L (Ω; L (0 ,T ; H )) ≤ c . (6.6)Now we go back to Itô’s formula for k p n k H : instead of taking expectations straight away,we take at first supremum in time and then expectations, getting Performing the usualcomputation as before using the Young inequality we get, for every t ∈ [0 , T ] , E sup r ∈ [ t,T ] k p n ( t ) k H . E sup r ∈ [ t,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z Tr ( p n ( s ) , q n ( s )) H dW ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . The Burkholder-Davis-Gundy and Young inequalities ensure that, for every δ > , E sup r ∈ [ t,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z Tr ( p n ( s ) , q n ( s )) H dW ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ E sup r ∈ [ t,T ] k p n ( t ) k H + C δ E k q n k L ( t,T ; L ( U,H )) , so that choosing δ sufficiently small and using (6.6) we infer that k p n k L (Ω; C ([0 ,T ]; H )) ≤ c . (6.7) Passage to the limit.
We deduce that there is ( p, ˜ p, q ) with p ∈ L ∞ (0 , T ; L (Ω; V )) ∩ L P (Ω; L (0 , T ; Z ∩ H ( D ))) , ˜ p ∈ L ∞ (0 , T ; L (Ω; V ∗ )) ∩ L P (Ω; L (0 , T ; V )) ,q ∈ L P (Ω; L (0 , T ; L ( U, V ))) , such that ˜ p = − ∆ p and p n ∗ ⇀ p in L ∞ (0 , T ; L (Ω; V )) ∩ L (Ω; L (0 , T ; Z ∩ H ( D ))) , ˜ p n ∗ ⇀ ˜ p in L ∞ (0 , T ; L (Ω; V ∗ )) ∩ L (Ω; L (0 , T ; V )) ,q n ⇀ q in L (Ω; L (0 , T ; L ( U, V ))) . Now, we know from [34, Lem. 2.1] that the stochastic integral operator is linear continuous(hence also weakly continuous) from the space L P (Ω; L (0 , T ; L ( U, V ))) to the space L (Ω; W s, (0 , T ; V )) : consequently, we deduce that Z · q n ( s ) dW ( s ) ⇀ Z · q ( s ) dW ( s ) in L (Ω; W s, (0 , T ; V )) . Finally, by (A2), the embedding
V ֒ → L ( D ) and the fact that y ∈ L (Ω; L ∞ (0 , T ; V )) itis immediate to check that Ψ ′′ ( y ) ∈ L (Ω; L ∞ (0 , T ; L ( D ))) , so in particular Ψ ′′ n ( y ) → Ψ ′′ ( y ) in L (Ω × Q ) . Hence, since by the convergences of (˜ p n ) n we have ˜ p n ⇀ ˜ p in L (Ω × Q ) , so that Ψ ′′ n ( y )˜ p n ⇀ Ψ ′′ ( y )˜ p in L / (Ω × Q ) . Similarly, it is a standard matter to check that the weak convergence of ( q n ) n and theboundedness of DB imply DB ( · , y ) ∗ q n ⇀ DB ( · , y ) ∗ q in L (Ω; L (0 , T ; H )) . Hence, passing to the weak limit as n → ∞ , we get that ( p, ˜ p, q ) is a solution to the(2.18)–(2.22). Finally, note the extra regularities p ∈ C w ([0 , T ]; L (Ω; V )) and ˜ p ∈ C w ([0 , T ]; L (Ω; V ∗ )) follow a posteriori by comparison in the limit equation. uca Scarpa In this final section we prove the last Theorem 2.5 containing the simpler version of first-order necessary conditions on optimality. The main idea is to remove the dependence on z in the variational inequality of Theorem 2.4 by using the adjoint problem and a suitableduality relation between z and ˜ p .Let then ¯ u ∈ U be an optimal control and ¯ y := S (¯ u ) be the corresponding solutionto the state equation. Then we know that the adjoint problem admits a solution ( p, ˜ p, q ) solving (2.18)–(2.22), where ˜ p is uniquely determined. Let v ∈ U be arbitrary and set h := v − ¯ u : the main point is to prove the duality relation α E Z Q (¯ y − x Q ) z h + α E Z D (¯ y ( T ) − x T ) z h ( T ) = E Z Q ˜ ph . If we are able to prove such duality result, then it is clear that Theorem 2.5 follows directlyfrom Theorem 2.4.Let ( z nh ) n and ( p n , ˜ p n , q n ) n be the approximated solutions introduced in Sections 5.1and 6.1: then we have z nh ∈ L (Ω; C ([0 , T ]; H ) ∩ L (0 , T ; Z )) , ˜ p n ∈ C ([0 , T ]; L (0 , T ; V ∗ )) ∩ L (Ω; L (0 , T ; V )) , with z nh (0) = 0 , p n ( T ) = α (¯ y ( T ) − x T ) , and dz nh − ∆( − ∆ z nh + Ψ ′′ n (¯ y ) z nh − h ) dt = DB (¯ y ) z nh dW , − dp n − ∆˜ p n dt + Ψ ′′ (¯ y )˜ p n dt = α (¯ y − x Q ) dt + D (¯ y ) ∗ q n dt − q n dW , where the equations are intended in the Hilbert triplet ( Z, H, Z ∗ ) . We deduce in particularthat d ( z nh , p n ) H = ( z nh , dp n ) H + h dz nh , p n i Z + ( DB (¯ y ) z nh , q n ) L ( U,H ) dt , where ( z nh , p n ) H (0) = 0 , ( z nh , p n ) H ( T ) = α Z D (¯ y ( T ) − x T ) z nh ( T ) . Writing Itô’s formula for ( z nh , p n ) H yields then α E Z D (¯ y ( T ) − x T ) z nh ( T ) = − E Z Q ∆ z nh ˜ p n + E Z Q Ψ ′′ n (¯ y )˜ p n z nh − α E Z Q (¯ y − x Q ) z nh − E Z T ( DB ( s, ¯ y ( s )) ∗ q n ( s ) , z nh ( s )) H ds − E Z Q ∆ z nh ∆ p n + E Z Q Ψ ′′ n (¯ y ) z nh ∆ p n − E Z Q h ∆ p n + E Z T ( DB ( s, ¯ y ( r )) z nh ( s ) , q n ( s )) L ( U,H ) ds , from which, recalling the definition of DB ( · , ¯ y ) ∗ and that − ∆ p n = ˜ p n , α E Z Q (¯ y − x Q ) z nh + α Z D (¯ y ( T ) − x T ) z nh ( T ) = E Z Q h ˜ p n ∀ n ∈ N . The thesis now follows letting n → ∞ .2 Optimal control of a stochastic CH equation
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