Stochastic Maximum Principle for a PDEs with noise and control on the boundary

Sto c hastic Maxim um Principle for SPDEs with noise and con trol on the b oundary Giuseppina Guatteri, Dipartimen to di Matematica, P olitecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italia. e-mail: giuseppina. guatteri@polimi.it Abstract In t his pap er we pro ve necessary conditions for optimality of a sto c hastic control problem for a c la ss of sto c hastic par t ial differential equations tha t is controlled through the b oundary . This kind of problems can be interpreted as a sto chastic control problem for an evolution system in a Hilb ert space. The regularity of the s o lution of the adjoint equation, tha t is a backw ar d sto chastic equation in infinite dimensio n, plays a crucial ro le in the for mulation of the maximum principle. Key w ords. Sto chastic con trol, maximum pr inciple, sto chastic ev olution equation, bac kwa r d sto c hastic d ifferential equation. 1 In tro d uction The maxim u m p r inciple for sto chastic con tr ol problems in infi nite dimensions h as b een treated by Bensoussan in [1 ] using v ariational metho d . Then Hu and P eng in [12] stud ied general ev olution con trolled equations where the op erator is a generator of a C 0 - semigroup, see also the b ibliograph y therein for the results in fin ite dimension and [13] for a sto c h astic equation of fun ctional t yp e. There ha ve b een then sev eral exstensions, mainly dealing with finite dimensional systems and the few r esults regarding S PDEs alwa ys consider diffused n oise and con trol, for a comprehensive bibliograph y see [18]. Th e aim of our p ap er is to deal with control pr oblems where the con trol and noise act on the b oundary , a situation that seem interesting from the p oin t of view of app lications. As an example, let u s consider the follo wing Cauc hy p roblem for the sto c h astic heat equation            ∂ ∂ t v ( t, x ) = ∂ 2 ∂ x 2 v ( t, x ) + f ( v ( t, x )) + g ( v ( t, x )) ∂ 2 ∂ t∂ x W ( t, x ) , t ∈ [0 , T ] , x ∈ [0 , 1] ∂ ∂ x v ( t, 0) = u 1 ( t ) + ˙ W 1 ( t ) , ∂ ∂ x v ( t, 1) = u 2 ( t ) + ˙ W 2 ( t ) , v (0 , x ) = u 0 ( x ) , (1.1) where ∂ 2 ∂ t∂ x W ( t, x ) is a sp ace-time white n oise, { W i t , t ≥ 0 } , i = 1 , 2 are indep endent sta n dard real Wiener pro cesses, the un kno wn v ( t, x, ω ), repr esenting the state of the system, is a real-v al ued 1 pro cess, the con trols are t wo p redictable real-v alued pro cesses u i ( t, x, ω ), i = 1 , 2 acting at 0 and 1, and u 0 is a function defined on [0 , 1]. Equation (1.1) can b e rewritten as an ev olution equ ation in an Hilb ert sp ace but to deal with the b oundary terms one has to in tro duce unboun ded terms in the equation. Indeed if one sets H = L 2 (0 , 1) a n d A the realization of the Laplace op er ator in L 2 with Neumann conditions one can w rite equation as  dX t = ( AX t + F ( X t )) dt + ( λ − A ) Du t dt + ( λ − A ) D 1 d ˜ W t + G ( X t ) dW t t ∈ [0 , T ] X 0 = x (1.2) where λ > 0 b elongs to the resolv en t set of the op erator A , th at is generator of an analytic semigroup and D an d D 1 are maps th at transform the b oun d ary terms u and ˜ W in to elemen ts that b elong to the domain of fractional p o wer of λ − A . Hence the op erators ( λ − A ) D and ( λ − A ) D 1 are the unboun ded terms men tioned b efore and are regular enough to guarantee the existence of a mild solution of (1.2) in the sp ace H . A key p oint consists in pro vin g that the solution to the adjoint equation is more r egular and tak es v alues in th e domain of [( λ − A ) D ] ∗ so th at we can formulate the maxim um pr inciple. Since we do not assume in general th e con ve x ity of the control space w e ha ve to assume more regularit y on the co efficien ts indeed the r ate of conv ergence of the firs t ord er approxima tions is ε α with 1 2 < α < 1. The low er b ound 1 2 is imp osed by the presence of a noisy b oundary term, in this case we p ro ve a maximum pr inciple condition without introd ucing the second order app ro ximation of th e state unkn o wn and the additional adj oin t equation. When the control sp ace is a con v ex set, clearly this problem do es not o ccur since the first order appro ximation is of order ε . The r est of the pap er is organized as f ollo w s: in section 2 we pro vid e notation and we state th e problem in his abstract formulation sp ecifying the hyp otheses, in section 3 w e study the adjoint equation that turn s out to b e a bac kw ard sto chastic equation in the infinite d imensional sp ace H , in section 4 w e prov e the maximum pr inciple, while in the last section we pro vid e t wo examples of application of our result. Notice that, exploiting the recen t results of [8], we can deal w ith a heat equation with noisy b oun d ary conditions of Diric hlet t yp e. T h e dra w bac k is that we h a ve to work in an L 2 space w ith w eight and so w e ha ve to restrict the class of the cost functionals we can treat. 2 Preliminaries and statemen t of the problem 2.1 Notation Giv en a Banac h space X , the norm of its element s x will b e den oted by | x | X , or even by | x | wh en no confusion is p ossible. If V is another Banac h space, L ( X , V ) denotes the space of b oun ded lin ear op erators from X to V , end o wed with the us ual op erator norm. Finally w e sa y th at a mappin g F : X → V b elongs to the class G 1 ( X ; V ) if it is con tinuous, Gˆ ateaux differen tiable on X , and ∇ F : X → L ( X, V ) is strongly c on tinuous. The lett ers Ξ, H , K and U will alwa ys b e used to denote Hilb ert sp aces. The scalar p ro duct is denoted h· , ·i , equipp ed with a su bscript to sp ecify the space, if necessary . All th e Hilb ert spaces are assum ed to b e real and sep arable; L 2 (Ξ , H ) is the s pace of Hilb ert-Sc h midt op erators from Ξ to H , resp ectiv ely . Giv en an arbitrary but fixed time horizon T , w e consider all stochastic processes as defined on su bsets of the time interv al [0 , T ]. Let Q ∈ L ( K ) b e a symmetric non-n egativ e op erator, not 2 necessarily trace class and ˜ W = ( ˜ W t ) t ∈ [0 ,T ] b e a Q -Wiener pro cess w ith v alues in K , defined on a complete probability space (Ω , F , P ) and tW = ( W t ) t ∈ [0 ,T ] b e a cylindrical Wiener pro cess with v alues in Ξ, defined on the s ame probabilit y sp ace and indep endent of ˜ W . By {F t , t ∈ [0 , T ] } w e will denote the natural filtration of ( ˜ W , W ), augmen ted with the family N of P - null sets of F , see for instance [4] for its d efi nition. Obviously , the fi ltration ( F t ) satisfies the usual conditions of righ t-con tinuit y and completeness. All the concepts of measurabilit y for sto chastic pro cesses will refer to this filtration. By P w e denote th e p redictable σ -algebra on Ω × [0 , T ] and b y B (Λ) the Borel σ -algebra of any top ological space Λ. Next we define t wo classes of sto chastic p ro cesses with v alues in a Hilb ert space V . • L 2 P (Ω × [0 , T ]; V ) denotes the space of equiv alence classes of pro cesses Y ∈ L 2 (Ω × [0 , T ]; V ) admitting a pr edictable ve r sion. It is endo wed with the n orm | Y | =  E Z T 0 | Y s | 2 ds  1 / 2 . • C P ([ t, T ]; L p (Ω; S )), p ∈ [1 , + ∞ ], t ∈ [0 , T ], d enotes the space of S -v alued pro cesses Y su ch that Y : [ t, T ] → L p (Ω , S ) is con tinuous and Y has a predictable mo difi cation, endo w ed with the n orm: | Y | p C P ([ t,T ]; L p (Ω; S )) = sup s ∈ [ t,T ] E | Y s | p S Elemen ts of C P ([ t, T ]; L p (Ω; S )) are identified up to m o dification. • F or a giv en p ≥ 2, L p P (Ω; C ([0 , T ]; V )) d enotes the space of pr edictable pro cesses Y with con tinuous paths in V , suc h that the n orm k Y k p = ( E sup s ∈ [0 ,T ] | Y s | p ) 1 /p is fin ite. The elements of L p P (Ω; C ([0 , T ]; V )) are iden tified up to indistinguish abilit y . Giv en an elemen t Φ of L 2 P (Ω × [0 , T ]; L 2 (Ξ , V )) or of L 2 P (Ω × [0 , T ]; L 2 ( K, V )), the Itˆ o sto c has- tic integ r als R t 0 Φ( s ) dW ( s ) and R t 0 Φ( s ) d ˜ W ( s ), t ∈ [0 , T ], are V -v alued martingales belonging to L 2 P (Ω; C ([0 , T ]; V )). The pr evious defin itions ha ve obvio u s extensions to pro cesses defined on sub in- terv als of [0 , T ] or defined on the enti re p ositiv e real lin e R + . 2.2 The optimal con trol problem and t he state equation Let H b e a sep arab le real Hilb ert sp ace, and U a separable Hilb ert, called the space of con trols. Let U ad a non empty set of U . W e set the space L 2 P (Ω × [ 0 , T ]; U ad ) the sp ace of admissible controls, and we denote it by U . W e make the follo wing, assum p tions that we denote by ( A ): (A.1) A : D ( A ) ⊂ H → H is a linear, unboun ded op er ator that generate a C 0 -semigroup that is also analytic, { e tA } t ≥ 0 suc h that | e tA | L ( H,H ) ≤ M e ω t , t ≥ 0 for some M > 0 and ω ∈ R . T his means in particular that ev ery λ > ω b elongs to the resolve nt set of A . 3 (A.2) F : R + × H → H , G : R + × H → L (Ξ , H ), are measurable functions suc h that for h = F , G , t → h ( t, x, y ) is con tinuous for ev ery fixed x ∈ H , y ∈ K . F urthermore, there are constant s L , ∆ and γ ∈ [0 , 1 / 2[ suc h that: | F ( t, x ) − F ( t, u ) | K ≤ L | x − u | H | e sA [ G ( t, x ) − G ( t, u )] | L 2 (Ξ ,H ) ≤ L (1 ∧ s ) γ | x − u | H , | F ( t, 0) | K ≤ ∆ , | e sA G ( t, x ) | L 2 (Ξ ,H ) ≤ ∆ (1 ∧ s ) γ (1 + | x | H ) , for eve r y x, u ∈ H and s, t ∈ R + . (A.3) F ( t, · ) ∈ G 1 ( H ; H ); for ev ery s > 0, e sA G ( t, · ) ∈ G 1 ( H ; L 2 (Ξ , H )) and | F x ( t, x ) − F x ( t, u ) | K ≤ L | x − u | H | e sA [ G x ( t, x ) − G x ( t, u )] | L 2 (Ξ ,H ) ≤ L (1 ∧ s ) γ | x − u | H , (2.1) for eve r y x, y ∈ H ; (A.4) There exists a con tin u ous linear op erator D : U → D (( λ − A ) α ) f or some 1 2 < α < 1 and λ > ω , see for ins tance [16] or [17] for the d efinition of the fractional p o wer of the op erator A . (A.5) There exists a linear op erator D 1 : U → H an d there is a constan t 0 < β < 1 2 suc h that the follo wing holds : | e tA ( λ − A ) D 1 p Q | L 2 ( K,H ) ≤ C t β for some λ > 0. W e consider in the Hilb ert space H the sto chasti c differen tial equatio n for the un kno wn pro cess X t , t ∈ [0 , T ]:  dX t = ( AX t + F ( t, X t )) dt + ( λ − A ) Du t dt + ( λ − A ) D 1 d ˜ W t + G ( t, X t ) dW t t ∈ [0 , T ] X 0 = x (2.2) As u s ual, see also [4], w e m ean by mild solution to this equ ation a ( F t )- p redictable pro cess X t , t ∈ [0 , T ] with contin uous p ath in H su c h that P - a.s. X t = e tA x + Z t 0 e ( t − s ) A F ( s, X s ) ds + Z t 0 e ( t − s ) A ( λ − A ) D u s ds + Z t 0 e ( t − s ) A ( λ − A ) D 1 d ˜ W s + Z t 0 e ( t − s ) A G ( s, X s ) dW s , t ∈ [0 , T ] Prop osition 2.1 U nder the assumptions ( A ) , for every u ∈ U ther e exists a uniq ue pr o c ess X ∈ C P ([0 , T ]; L 2 (Ω; H )) mild solution of e qu ation (2.2) . 4 Pro of. Th e only p oint to c hec k in order to p erform the fixed p oint argumen t, s ee[4], theorem 7.6, or [9], prop osition 3.2 , is that pro cesses ( R t 0 e ( t − s ) A ( λ − A ) D u s ds ) t ∈ [0 ,T ] and ( R t 0 e ( t − s ) A ( λ − A ) D 1 d ˜ W s ) t ∈ [0 ,T ] b elong to the space C P ([0 , T ]; L 2 (Ω; H )). W e h a ve indeed: sup t ∈ [0 ,T ] E    Z t 0 e ( t − s ) A ( λI − A ) D u s ds    2 ≤ C 2 T 1 − 2 α E Z T 0 | u s | 2 ds < + ∞ Moreo v er: sup t ∈ [0 ,T ] E    Z t 0 e ( t − s ) A ( λI − A ) D 1 d ˜ W s    2 ≤ C 2 E h Z T 0 ρ − 2 β ds i 2 < + ∞ where b oth C is defined in ( A ). W e asso ciate to this state equation the follo wing cost fun ctional: J ( x, u ) = E Z T 0 l ( t, X t , u t ) dt + E h ( X T ) (2.3) where l and h verify ( B) : (B.1) ( i ) l : [0 , T ] × H × U → R is m easur able and ther e exist a constant L > 0 and a mo dulus of con tinuit y ¯ ω : [0 , + ∞ ) → [0 , + ∞ ), su ch that: | l ( t, x, u ) − l ( t, x ′ , u ′ ) | ≤ ( L | x − x ′ | + ¯ ω ( k u − u ′ k U )) (2.4) for all t ∈ [0 , T ], x, x ′ ∈ H and u, u ′ ∈ U . ( ii ) Moreov er for all t ∈ [0 , T ] and all u ∈ U l ( t, · , u ) ∈ G 1 ( H ; R ) suc h that | l x ( t, x, u ) − l x ( t, x ′ , u ) | ≤ L | x − x ′ | (2.5) for all t ∈ [0 , T ], x, x ′ ∈ H and u ∈ U . (B.2) ( i ) h : H → R , is measurable and there exist a constan t L > 0 such that | h ( x ) − h ( x ′ ) | ≤ L | x − x ′ | (2.6) for all x, x ′ ∈ H . ( ii ) Moreov er h ∈ G 1 ( H ; R ) and | h x ( x ) − h x ( x ′ ) | ≤ L | x − x ′ | (2.7) for all x, x ′ ∈ H . The optimal con tr ol problem consists in minimizing J o v er all u ∈ U . W e will seek f or n ecessary conditions fulfilled by an optimal couple, wh en ev er it exists, ( ¯ X , ¯ u ) ∈ C P ([0 , T ]; L 2 (Ω; H )) × U suc h that inf u ∈U J ( x, u ) = E Z T 0 l ( t, ¯ X t , ¯ u t ) dt + E h ( ¯ X T ) (2.8) where ¯ X is the mild solution to:  d ¯ X t = ( A ¯ X t + F ( t, ¯ X t )) dt + ( λ − A ) D ¯ u t dt + ( λ − A ) D 1 d ˜ W t + G ( t, ¯ X t ) dW t t ∈ [0 , T ] ¯ X 0 = x (2.9) 5 3 Regularit y results for the Adjoin t equation In this sect ion we co n sider the follo wing bac kwa r d sto c hastic d ifferen tial equation, the so-ca lled adjoint e quation :  − d Y t = ( A T Y t + F x ( t, ¯ X t ) T Y t ) dt + G x ( t, ¯ X t ) T Z t dt − l x ( t, ¯ X t , ¯ u t ) dt − Z t dW t − ˜ Z t d ˜ W t t ∈ [0 , T ] Y T = − h x ( ¯ X T ) (3.1) Thanks to hyp otheses (A) on the deriv ativ es F x and G x and hypotheses (B) on the deriv ativ es l x and h x this equation is affine with unif orm ly b ounded co efficients (in the linear part) and in tegrable forcing term and integrable final data. The generator A is an unboun ded op erator b ut generates a C 0 -semigroup, so existence and uniqueness for the solution in L 2 P (Ω; C ([0 , T ]; H )) × L 2 P ([0 , T ] × Ω; L 2 ( K × Ξ , H )) to this equ ation is a w ell kno w n result, see [11]. It remains to pr o ve an ext ra regularit y p rop erty for the Y comp onen t. Prop osition 3.1 U nder assumptions (A) a nd (B) th e r e exists a unique mild solution ( Y , Z ) in L 2 P (Ω; C ([0 , T ]; H )) × L 2 P ([0 , T ] × Ω; L 2 ( K × Ξ , H )) . M or e over for every t ∈ [0 , T [ and P − a.s. , Y t ( ω ) b elongs to the domain of D T A T . Pro of. Th e mild solution exists by [11], theorem 3.1 or [14], theorem 4.4 th at is a couple ( Y , Z ) = ( Y , ( ˆ Z , ˜ Z )) in L 2 P (Ω; C ([0 , T ]; H )) × L 2 P ([0 , T ] × Ω; L 2 ( K × Ξ , H )) such that: Y t = − e ( T − t ) A T h x ( ¯ X T ) + Z T t e ( s − t ) A T ( F x ( s, ¯ X s ) T Y s + G x ( s, ¯ X s ) T Z s ) ds − Z T t e ( s − t ) A T l x ( s, ¯ X s , ¯ u s ) ds − Z T t e ( s − t ) A T Z s dW s − Z T t e ( s − t ) A T ˜ Z s d ˜ W s , t ∈ [0 , T ] Let u s now p ro ve the regularit y result. W e h a ve : Y t = E F t Y t = − e ( T − t ) A T E F t h x ( ¯ X T ) + Z T t e ( s − t ) A T E F t ( F x ( s, ¯ X s ) T Y s + G x ( s, ¯ X s ) T Z s ) ds − Z T t e ( s − t ) A T E F t l x ( s, ¯ X s , ¯ u s ) ds (3.2) W e ha v e to ev aluate || D T A T Y t || U = sup u ∈ U, || u || =1 h D T A T Y t , u i U = sup u ∈ U, || u || =1 h Y t , AD u i U . W e ha ve: |h E F t h x ( ¯ X T ) , Ae ( T − t ) A D u i| ≤ E F t | h x ( ¯ X T ) || Ae ( T − t ) A D u | ≤ C ( T − t ) 1 − α (1 + E F t | ¯ X T | ); | Z T t h E F t ( F x ( s, ¯ X s ) T Y s + G x ( s, ¯ X s ) T Z s ) , Ae ( s − t ) A D u i ds | ≤ C Z T t E F t ( | Y s | + | Z s | ) ( s − t ) 1 − α ds ≤ C  Z T 0 ( E F t | Y s | 2 + E F t | Z s | 2 ) ds  1 / 2 T 2 α − 1 ; | Z T t h E F t l x ( s, ¯ X s , ¯ u s ) , Ae ( s − t ) A D u i ds | ≤ Z T t C ( s − t ) 1 − α (1 + E F t | ¯ X s | + E F t | ¯ u s | ) ds ≤ C  Z T 0 (1 + E F t | ¯ X s | 2 + E F t | ¯ u s | 2 ) ds  1 / 2 T 2 α − 1 . 6 This implies that for some constan t C > 0 that d ep ends on T and the qu an tities defined in (A) and ( B) : E || D T A T Y t || U ≤ C ( T − t ) 1 − α (1 + sup t ∈ [0 ,T ] E | X t | 2 + E sup t ∈ [0 ,T ] | Y t | 2 + E Z T 0 | Z t | 2 dt ) < + ∞ . 4 The Maxim um Principle 4.1 V aria tion of the tra ject ory Let ( ¯ X , ¯ u ) b e an optimal couple of p roblem (2.2) and (2. 8). Fix v ∈ U ad and ¯ t ∈ [0 , T ] and for ev ery 0 < ε < T − ¯ t defin e u ε ( t ) =  v t ∈ E ε := [ ¯ t, ¯ t + ε ]; ¯ u ( t ) t / ∈ E ε (4.1) Let u s consider the follo w ing equations:  dX ε t = ( AX ε t + F ( t, X ε t )) dt + ( λ − A ) Du ε t dt + ( λ − A ) D 1 d ˜ W t + G ( t, X ε t ) dW t t ∈ [0 , T ] X ε 0 = x (4.2) and  d ˜ X ε t = ( A ˜ X ε t + F x ( t, ¯ X t ) ˜ X ε t ) dt + ( λ − A ) D ( u ε t − ¯ u t ) dt + G x ( t, ¯ X t ) ˜ X ε t dW t t ∈ [0 , T ] ˜ X ε 0 = 0 (4.3) W e hav e: Prop osition 4.1 U nder hyp othesis (A) for every ε > 0 ther e exist a unique mild solution X ε ∈ C P ([0 , T ]; L 2 (Ω; H )) of e quation (4 .2 ) and a unique solution ˜ X ε ∈ C P ([0 , T ]; L 2 (Ω; H )) of e quation (4.3) . Mor e over for al l p ≥ 1 : E sup t ∈ [0 ,T ] | ˜ X ε t | p < + ∞ . (4.4) Pro of. Th e existence and uniqueness of th e solutions are guaranteed by theorem 7.6 of [4]. Let us no w prov e (4.4). W e ha ve ˜ X ε t = Z t 0 e ( t − s ) A F x ( s, ¯ X s ) ˜ X ε s ds + Z t 0 e ( t − s ) A G x ( s, ¯ X s ) ˜ X ε s dW s + Z t 0 e ( t − s ) A ( λI − A ) D ( u ε s − ¯ u s ) ds so E sup 0 ≤ t ≤ r | ˜ X ε t | p ≤ c ( p ) h E sup 0 ≤ t ≤ r    Z t 0 e ( t − s ) A F x ( s, ¯ X s ) ˜ X ε s ds    p + E su p 0 ≤ t ≤ r    Z t 0 e ( t − s ) A G x ( s, ¯ X s ) ˜ X ε s dW s    p + E su p 0 ≤ t ≤ r    Z t 0 e ( t − s ) A ( λI − A ) D ( u ε s − ¯ u s ) ds    p i ≤ c ( p )[ I 1 + I 2 + I 3 ] 7 F or I 1 w e ha ve, thanks to h y p otheses (A.1) and (A.2) there exists a constan t C d ep endin g on T , p and the q u an tities in ( A ) suc h that: I 1 ≤ C E sup 0 ≤ t ≤ r Z t 0 sup 0 ≤ σ ≤ s | ˜ X ε | p ds ≤ C Z r 0 E su p 0 ≤ σ ≤ s | ˜ X ε | p ds Then, h a ving that | Ae ( t − s ) A D | L ( U,H ) ≤ C ( t − s ) 1 − α for soma costan t C > 0, thanks to ( A.4) we ha ve I 3 ≤ C p  Z T 0 1 s 1 − α ds  p | v | p U = C p T αp | v | p U Ev entually to treat term I 2 w e us e the factorization metho d , see [3]. T ak e p > 2 and ρ ∈ (0 , 1) such that 1 p < ρ < 1 2 − γ , an d let c − 1 ρ = R t s ( t − σ ) ρ − 1 ( σ − s ) − ρ dσ . Hence I 2 = E sup 0 ≤ t ≤ r    Z t 0 e ( t − s ) A G x ( s, ¯ X s ) ˜ X ε s dW s    p = E sup 0 ≤ t ≤ r    Z t 0 e ( t − σ ) A ( t − σ ) ρ − 1 dσ h Z σ 0 e ( σ − s ) A ( σ − s ) − ρ G x ( s, ¯ X s ) ˜ X ε s dW s i    p ≤ E sup 0 ≤ t ≤ r  Z t 0 ( t − σ ) ( ρ − 1) q  p/q Z t 0    Z σ 0 e ( σ − s ) A ( σ − s ) − ρ G x ( s, ¯ X s ) ˜ X ε s dW s    p dσ ≤ C Z r 0  Z σ 0 ( σ − s ) − 2( ρ + γ ) ds  p/ 2 E su p 0 ≤ s ≤ σ | ˜ X ε σ | p dσ . for some constan t C > 0, dep en ding on T , p and the parameter defin ed in ( A ) . So com b ining all these estimates together, we obtain th at E sup 0 ≤ t ≤ r | ˜ X ε t | p ≤ C h Z r 0 E su p 0 ≤ σ ≤ s | ˜ X ε σ | p ds + | v | p U T αp . i Th us by Gron w all theorem we can conclude. W e claim that: Prop osition 4.2 U nder hyp othesis (A) ther e is a c onstant δ > 0 indep endent of ε such that: ∆ ε := E sup 0 ≤ t ≤ T | X ε t − ¯ X t | ≤ δ ε α (4.5) Pro of. W e hav e th at:    d ( X ε t − ¯ X t ) = [ A ( X ε t − ¯ X t ) + F ( t, X ε t ) − F ( t, ¯ X t )] dt + ( λ − A ) D ( u ε t − ¯ u t ) dt +( G ( t, X ε t ) − G ( t, ¯ X t ) dW t , t ∈ [0 , T ] X ε 0 − ¯ X 0 = 0 (4.6) F ollo win g p r op osition 4.1, we hav e for p > 2 : E sup t ∈ [0 ,r ] | X ε t − ¯ X t | p ≤ C h Z r 0 E sup σ ∈ [0 , s ] | X ε σ − ¯ X σ | p ds + sup ¯ t ∈ [0 ,T ]  Z ¯ t + ε ¯ t | v | U ( t − s ) 1 − α ds  p i 8 where C as usu al is indep endent of ε and m and it is function of T , p , | v | U and the qu an tities in tro d uced in ( A ). T h erefore by Gronw all lemma w e can conclude. W e set η ε t := X ε t − ¯ X t − ˜ X ε t , and we end the section with the follo wing r esu lt Prop osition 4.3 U nder hyp othesis (A) ther e is a c onstant δ 1 > 0 indep endent of ε su ch that: ∆ 1 ε := E su p 0 ≤ t ≤ T | η ε t | ≤ δ ε 2 α (4.7) Pro of. W e hav e:                dη ε t = d ( X ε t − ¯ X t − ˜ X ε t ) = Aη ε t dt + F x ( t, ¯ X t ) η ε t dt + G x ( t, ¯ X t ) η ε t dW t + h Z 1 0 ( F x ( t, ¯ X t + θ ( X ε t − ¯ X t )) − F x ( t, ¯ X t ))( X ε t − ¯ X t ) dθ i dt + h Z 1 0 ( G x ( t, ¯ X t + θ ( X ε t − ¯ X t )) − G x ( t, ¯ X t ))( X ε t − ¯ X t ) dθ i dt, X ε 0 − ¯ X 0 = 0 (4.8) Notice that, thank s to (A.3) , we hav e that for some γ ∈ [0 , 1 2 [ | e sA [ G x ( t, x ) − G x ( t, u )] | L 2 (Ξ ,H ) ≤ L (1 ∧ s ) γ | x − u | H for eve r y x, y ∈ H and s ∈ R + . Thus f or ev er y p > 2, as in p rop osition 4.1, we obtain that E s up t ∈ [0 ,r ] | η ε t | p ≤ C h E sup σ ∈ [0 , T ] | X ε σ − ¯ X σ | 2 p + Z r 0 E sup σ ∈ [0 , s ] | η ε σ | p ds i and we conclude. 4.2 Main r esult No w we are able to state and pr o v e the M aximum Principle for our optimal control prob lem. Theorem 4.4 Assume hyp otheses (A) and ( B) . L et ( ¯ X , ¯ u ) b e a optimal p air of Pr oblem (2.8) and (2.2) . Then ther e exists a unique solution ( Y , Z ) ∈ L 2 P (Ω; C ([0 , T ]; H )) × L 2 P ([0 , T ] × Ω; L 2 (Ξ , H )) of e quation (3.1) and H ( t, ¯ X t , ¯ u t , Y t ) ≥ H ( t, ¯ X t , v , Y t ) , ∀ v ∈ U ad , a.e. t ∈ [0 , T ] , P − a.s. (4.9) wher e H ( t, x, v , p ) := h D ∗ ( λ − A ) ∗ p, v i H − l ( t, x, v ) , ( t, x, v , p ) ∈ [0 , T ] × H × U ad × D ( D ∗ ( λ − A ) ∗ ) , λ > ω Pro of. S in ce ( ¯ X , ¯ u ) is optimal for ev ery ε > 0 and x ∈ H w e ha ve: 0 ≤ J ( x, u ε ) − J ( x, ¯ u ) = E Z T 0 ( l ( t, X ε t , u ε t ) − l ( t, ¯ X t , ¯ u t )) dt + E ( h ( X ε T ) − h ( ¯ X T )) = I 1 + I 2 . 9 Let u s considet I 1 , add ing and s ubstractiong we get: E Z T 0 ( l ( t, X ε t , u ε t ) − l ( t, ¯ X t , ¯ u t )) dt = E Z T 0 ( l ( t, X ε t , u ε t ) − l ( t, ¯ X t , u ε t )) dt + E Z T 0 ( l ( t, ¯ X t , u ε t ) − l ( t, ¯ X t , ¯ u t )) dt = J 1 + J 2 Let u s concen tr ate on J 1 , we hav e, th anks to prop ositions 4.2 an d 4.3,: E Z T 0 ( l ( t, X ε t , u ε t ) − l ( t, ¯ X t , u ε t )) dt = E Z T 0 Z 1 0 [ l x ( t, ¯ X t + θ ( X ε t − ¯ X t ) , u ε t ) − l x ( t, ¯ X t , u ε t )]( X ε t − ¯ X t ) dθ dt + E Z T 0 [ l x ( t, ¯ X t , u ε t ) − l x ( t, ¯ X t , ¯ u t )]( X ε t − ¯ X t ) dt + E Z T 0 l x ( t, ¯ X t , ¯ u t ) η ε t dt + E Z T 0 l x ( t, ¯ X t , ¯ u t ) ˜ X ε t dt ≤ C ε 2 α + E Z T 0 l x ( t, ¯ X t , ¯ u t ) ˜ X ε t dt Com bin ing all these relations w e end up with: E Z T 0 ( l ( t, X ε t , u ε t ) − l ( t, ¯ X t , ¯ u t )) dt ≤ C ε 2 α + E Z T 0 l x ( t, ¯ X t , ¯ u t ) ˜ X ε t dt + E Z T 0 ( l ( t, ¯ X t , u ε t ) − l ( t, ¯ X t , ¯ u t )) dt. Similarly w e get: E h ( X ε T ) − E h ( ¯ X T ) ≤ C ε 2 α + E h x ( ¯ X T ) ˜ X ε T and thus: E Z T 0 ( l ( t, X ε t , u ε t ) − l ( t, ¯ X t , ¯ u t )) dt + E h ( X ε T ) − E h ( ¯ X T ) ≤ C ε 2 α + E Z T 0 l x ( t, ¯ X t , ¯ u t ) ˜ X ε t dt + E h x ( ¯ X T ) ˜ X ε T + E Z T 0 ( l ( t, ¯ X t , u ε t ) − l ( t, ¯ X t , ¯ u t )) dt No w, computing d h ˜ X ε t , Y t i , solutions resp ectiv ely of equations (4.3) and (3.1) we obtain: − E h ˜ X ε T , h x ( ¯ X T ) i = E Z T 0 l x ( t, ¯ X t , ¯ u t ) ˜ X ε t dt + E Z T 0 h AD ( u ε t − ¯ u t ) , Y t i dt Therefore: 0 ≤ J ( x, u ε ) − J ( x, ¯ u ) = E Z T 0 ( l ( t, X ε t , u ε t ) − l ( t, ¯ X t , ¯ u t )) dt + E h ( X ε T ) − E h ( ¯ X T ) ≤ C ε 2 α + E Z ¯ t + ε ¯ t h ( ¯ u t − v ) , D ∗ ( λ − A ) ∗ Y t i dt + E Z ¯ t + ε ¯ t ( l ( t, ¯ X t , v ) − l ( t, ¯ X t , ¯ u t )) dt Then dividing b y ε and recalling that α > 1 2 , using a lo calizatio n pro cedure, see [2], and exploiting the contin uit y of l w e can conclude. 10 4.3 The case when U ad is a con v ex set In th is paragraph we assume that the non empty s ubset U ad is a c onvex set . The space U of admissible control s and the optimal control problem are the same. Besides ( A . 1 ) − ( A . 4 ) − ( A . 5 ) w e assum e ( C ): (C.1) F : R + × H → H , G : R + × H → L (Ξ , H ), are measurable functions suc h that for h = F , G , t → h ( t, x ) is contin uous for eve ry fixed x ∈ H . Moreo v er F ( t, · ) ∈ G 1 ( H ; H ); for every s, t > 0, e sA G ( t, · ) ∈ G 1 ( H ; L 2 (Ξ , H )) and there is a constant L > 0 and γ ∈ [0 , 1 2 [ such that: | F x ( t, x ) | K ≤ L | e sA G x ( t, x ) | L 2 (Ξ ,H ) ≤ L (1 ∧ s ) γ , (4.10) for eve r y x ∈ H and eve r y s, t ∈ R + ; (C.2) F or all t ∈ [0 , T ] and all u ∈ U , l ( t, · , u ) ∈ G 1 ( H ; R ) and for all t ∈ [0 , T ] and all x ∈ H , l ( t, x, · ) ∈ G 1 ( U ; R ) and there is a constan t ∆ > 0 s uc h that: | l x ( t, x, u ) | + | l u ( t, x, u ) | ≤ ∆(1 + | x | H + | u | U ) (4.11) for all t ∈ [0 , T ], x ∈ H and u ∈ U . (C.3) h ∈ G 1 ( H ; R ) and th ere is a constan t ∆ > 0 suc h that: | h x ( x ) | ≤ ∆ (1 + | x | H ) (4.12) for all x ∈ H . W e hav e the follo w in g v ariational inequ ality . Lemma 4.5 The c ost functional J is Gate aux-differ entiable and the f ol lowing variational ine quality holds: d dθ J ( u ( · ) + θ v ( · )) | θ = 0 = E h x ( ¯ X T ) ˜ X T + E Z T 0 [ l x ( s, ¯ X s , ¯ u s ) ˜ X s + l u ( s, ¯ X s , ¯ u s ) v s ] ds ≥ 0 (4.13) wher e v ∈ L 2 P ((0 , T ); U ) satisfies ¯ u ( · ) + v ( · ) ∈ U , and ˜ X is the solution to the fol lowing line ar e quation: ˜ X t = Z t 0 e A ( t − s ) F x ( s, ¯ X s ) ˜ X s ds + Z t 0 e A ( t − s ) G x ( s, ¯ X s ) ˜ X s dW s + Z t 0 ( λ − A ) Dv s ds (4.14) Pro of. Th e pro of is similar to [12]. It is clear that the results of Prop osition 3.1 still hold und er these hyp otheses, so w e can assert the m axim um prin ciple: 11 Theorem 4.6 L et ( ¯ u, ¯ X ) b e an optimal p air the pr oblem (2.8) and (2.2) . Then ther e exists a unique p air ( Y , Z ) ∈ L 2 P (Ω; C ([0 , T ]; H )) × L 2 P ([0 , T ] × Ω; L 2 (Ξ , H )) solution of e quation (3.1) such that: h H u ( t, ¯ X t , ¯ u t , Y t ) , v − ¯ u t i ≤ 0 , ∀ v ∈ U ad , a.e. t ∈ [0 , T ] , P − a.s. (4.15) wher e H ( t, x, u, p ) := h D ∗ ( λ − A ) ∗ p, u i H − l ( t, x, u ) , ( t, x, u, p ) ∈ [0 , T ] × H × U ad × D ( D ∗ ( λ − A ) ∗ ) , λ > ω Pro of. F rom (4.13) w e h a ve th at: E h h x ( ¯ X T ) , ˜ X T i + E Z T 0 h l x ( s, ¯ X s , ¯ u s ) , ˜ X s i + h l u ( s, ¯ X s , ¯ u s ) , v s i ds ≥ 0 (4.16) for ev ery v ∈ L 2 P ((0 , T ); U ) s atisfies ¯ u ( · ) + v ( · ) ∈ U and ˜ X is the solution to (4 .14 ). Moreo ver ev aluating R T 0 d h Y t , ˜ X t i dt , we get that: − E h h x ( ¯ X T ) , ˜ X T i − E Z T 0 h l x ( s, ¯ X s , ¯ u s ) , ˜ X s i = E Z T 0 h Y s , ( λ − A ) D v s i ds (4.17) Th us com b in ing (4.16) and (4.17) we end up with : E Z T 0 h l u ( s, ¯ X s , ¯ u s ) , v s i ds ≥ E Z T 0 h D ∗ ( λ − A ) ∗ Y s , v s i ds Th us using a lo calization pro cedu re, see [2], we can conclude c hosing v t = v − ¯ u t , with v ∈ U ad . 5 Applications W e pro vide t wo examples to whic h our result apply . 5.1 Example 1 Let u s consider the follo w ing problem:            ∂ ∂ t y ( t, x ) = ∂ 2 ∂ x 2 y ( t, x ) + f ( y ( t, x )) + g ( y ( t, x )) ∂ 2 ∂ t∂ x W ( t, x ) , t ∈ [0 , T ] , x ∈ [0 , 1] ∂ ∂ x y ( t, 0) = u 1 ( t ) + ˙ W 1 ( t ) , ∂ ∂ x y ( t, 1) = u 2 ( t ) + ˙ W 2 ( t ) , y (0 , x ) = u 0 ( x ) , (5.1) where ∂ 2 ∂ t∂ x W ( t, x ) is a s pace-time white noise, { W i t , t ≥ 0 } , i = 1 , 2 are indep endent sta n dard real Wiener pro cesses, the un kno wn y ( t, x, ω ), rep r esen ting the state of the system, is a real-v alued pro cess, the con trols are tw o pr edictable real-v alued p ro cesses u i ( t, x, ω ), i = 1 , 2 acting at 0 and 1 and h a ving v alues in {− 1 , 0 , 1 } ; u 0 ∈ L 2 (0 , 1). W e assum e that f and g are C 1 , 1 ( R ). No w we write (5.1) as an ev olution equ ation in the sp ace H = L 2 (0 , 1). This is done in [5 ], see also bib liography therein and [7]. W e d efine the op erator A in H b y setting D ( A ) = { y ∈ H 2 (0 , 1) : ∂ ∂ x y (0) = ∂ ∂ x y (1) = 0 } , Ay ( x ) = ∂ 2 ∂ x y ( x ) , f or y ∈ D ( A ) . 12 Moreo v er for ev ery λ > 0, D (( λ − A ) α ) = H 2 α (0 , 1) , for 0 < α < 3 4 . F or ev ery fixed λ > 0 there exists d i ∈ H 2 α (0 , 1), s ee for instance [7] that solv es the follo wing Neumann p roblems:                ∂ 2 ∂ x 2 d i ( x ) = λd i ( x ) , x ∈ [0 , 1] , i = 1 , 2 ∂ ∂ x d 1 (0) = 1 , ∂ ∂ x d 1 (1) = 0 , ∂ ∂ x d 2 (0) = 0 , ∂ ∂ x d 2 (1) = 1 . (5.2) Th us we set U = K = R 2 and U ad = {− 1 , 0 , 1 } × {− 1 , 0 , 1 } , the co v ariance matrix Q = I and D = D 1 : R 2 → D (( λ − A ) α ), su c h that D u ( t )( x ) = d 1 ( x ) u 1 ( t ) + d 2 ( x ) u 2 ( t ) and D ˜ W ( t )( x ) = d 1 ( x ) W 1 t + d 2 ( x ) W 2 t . W e set Ξ = L 2 (0 , 1) and ∂ ∂ x W ( t, · ) = W ( t ) is a cylind rical Wiener pr o cess in Ξ = L 2 ([0 , 1]), see for instance [4 ]. W e fi n ally set X t = y ( t, · ) and F ( ξ )( · ) = f ( ξ ( · )) and G ( ξ )( · ) = g ( ξ ( · )) for all ξ ∈ H , then system (5.1) can b e written as  dX t = ( AX t + F ( X t )) dt + ( λ − A ) Du t dt + ( λ − A ) D 1 d ˜ W t + G ( X t ) dW t t ∈ [0 , T ] X 0 = u 0 , (5.3) It is then easy to sho w that all h yp otheses (A) are fulfilled, n ote that we can c h ose α > 1 2 . Let us in tro d uce the follo win g finite h orizon cost J ( x, u 1 ( · ) , u 2 ( · )) = E Z T 0 Z 1 0 l( y ( s, x ) , u 1 ( s ) , u 2 ( s )) dx ds + E Z 1 0 h( y ( T , x )) dx. with l( x, u 1 , u 2 ) : R × R × R → R , contin uous and deriv able, w ith b oun ded and Lipsc hitz con tinuous deriv ativ es in x unif orm ly w ith resp ect to u 1 , u 2 and Lipschitz con tinuous w ith resp ect to ( u 1 , u 2 ) uniformly with resp ect to x . Moreo v er w e assume h ∈ C 1 , 1 ( R ). W e set: l ( ξ , u 1 , u 2 ) = Z 1 0 l( ξ ( x ) , u 1 , u 2 ) dx, for ξ ∈ H , u 1 , u 2 ∈ U h ( ξ ) = Z 1 0 h( ξ ( x )) dx, for ξ ∈ H . Hence also hyp othesis (B) is fu lfilled and theorem 4.4 h olds. 5.2 Example 2 No w w e consid er a b ound ary con trol problem for a s to chastic heat equation w ith Diric h let condition p ertur b ed by a sto c h astic pro cess. More precisely w e h a v e:            ∂ ∂ t v ( t, x ) = ∂ 2 ∂ x 2 v ( t, x ) + f ( v ( t, x )) , t ∈ [0 , T ] , x ∈ [0 , + ∞ ) v ( t, 0) = u ( t ) + ˙ W ( t ) , v (0 , x ) = v 0 ( x ) , (5.4) 13 where { W t , t ≥ 0 } , is a standard real Wiener pr o cess, the un kno w n v ( t, x, ω ), representi n g the state of the system, is a real-v alued p ro cess, the con trol is a pr edictable real-v alued pro cess u ( t, x, ω ), acting at 0; v 0 ∈ L 2 (0 , 1). W e assume that f is C 1 ( R ) with b oun ded deriv ativ e, and clearly g = 0. It is well kno wn that it is not p ossible to rewrite the Cauc hy problem (5.4) as an evol u tion equation in the space L 2 (0 , + ∞ ), see [5]. In [15 ] it is sho wn that the Diric hlet map tak es v alues in the domain of ( − A ) α , for a certain α > 1 2 , w hen it is considered in the Hilb ert space L 2 (0 , + ∞ ; ( ρ θ + 1 ∧ 1) dρ ). More p recisely if we set H = L 2 (0 , + ∞ ; ( ρ θ + 1 ∧ 1) dρ ), th e op erator A 0 that is the realizatio n of the L ap lacian with Dirichlet conditions in L 2 (0 , + ∞ ) extends to an op erator A that generates an analytic op eratorin H . F or every fixed λ > 0 there exists d ∈ D (( λ − A ) α )) for some α > 1 2 :    ∂ 2 ∂ x 2 d ( x ) = λd ( x ) , x ≥ 0 d (0) = 1 , (5.5) Th us w e set U = U ad = K = R , the co v ariance matrix Q = 1 and D = D 1 : R 2 → D (( λ − A ) α )), suc h that D u ( t )( x ) = d ( x ) u ( t ) and D ˜ W ( t )( x ) = d ( x ) W t . 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