THE theory of fractional differential equations has received

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1 Exisence of Fracional Sochasic Schrödinger Evoluion Equaions wih Poenial and Oimal Conrols Zuomao Yan and Xiumei Jia Absrac In his aer, we sudy he fracional sochasic nonlinear Schrödinger evoluion equaions wih oenial and oimal conrols in ilber saces. The exisence of mild soluions is roved by means of fracional calculus, sochasic analysis, and fixed oin heorems wih he semigrou heory. Using hese resuls, he exisence of oimal airs of sysem governed by fracional sochasic nonlinear Schrödinger equaions is also resened. An examle is given for demonsraion. Index Terms fracional sochasic schrödinger evoluion equaions, exisence, oimal conrols, oenial, fixed oin heorem. I. INTRODUCTION TE heory of fracional differenial equaions has received increasing aenion during recen years since hey can be used o describe many henomena arising in viscoelasiciy, elecrochemisry, conrol, orous media, elecromagneic, ec. Acually, for various real world roblems in science and engineering, fracional derivaives describe cerain hysical henomena more accuraely han ineger order derivaives. Some works have done on he qualiaive roeries of soluions for hese equaions; see [], [2], [3] and he references herein. Recenly, he exisence of soluions for fracional semilinear differenial equaions including delay sysems is one of he heoreical fields ha invesigaed by many auhors [4], [5], [6], [7], [8], [9]. As a resul of is widesread use, he exisence of fracional oimal conrol sysems have been discussed in ublicaions( see [], [], [2]). On he oher hand, he nonlinear Schrödinger equaion is a model of he evoluion of a one-dimensional acke of surface waves on sufficienly dee waer. I arises from he sudy of nonlinear wave roagaion in disersive and inhomogeneous media, such as lasma henomena and nonuniform dielecric media. Therefore, i is a generic equaion describing he evoluion of he slowly varying amliude of a nonlinear wave rain in weakly nonlinear, srongly disersive, and hyerbolic sysems [3]. In recen years, he fracional Schrödinger equaions have become a field of increasing ineres(see [4] and references herein). Paricularly, Guo and Xu [5] considered he fracional Schrödinger equaion wih a free aricle and an infinie square oenial. de Oliveira [6] derived he exisence of soluions o fracional Manuscri received Ocober 3, 25; revised January 2, 26. This work was suored he Naional Naural Science Foundaion of China (469), he Presiden Fund of Scienific Research Innovaion and Alicaion of exi Universiy (xz23-, XZ24-22), he Scienific Research Projec of Universiies of Gansu Province (24A-). Z. Yan and X. Jia are wih he Dearmen of Mahemaics, exi Universiy, Zhangye, Gansu 734, P.R. China, yanzuomao@63.com. Schrödinger equaion for dela oenials. Wang e al. [7] discussed he exisence, uniqueness, local sabiliy and araciviy, and daa coninuous deendence of mild soluion for fracional Schrödinger equaions wih oenial. Also, he exisence and uniqueness of oimal airs for he fracional conrolled sysems are obained. Sochasic Schrödinger equaions are frequenly used o describe many quanum measuremen rocesses and in general, quanum sysems ha are sensiive o he environmen influence (see [8]). Moreover, nonlinear sochasic Schrödinger equaions are becoming an esablished ool for numerical simulaion of he evoluion of oen quanum sysems (see [9]). Many auhors invesigaed he exisence of a mild soluion of sochasic equaions of Schrödinger in ilber saces. Grecksch and Lisei [2] considered a class of sochasic evoluion equaion of Schrödinger ye over a rile of rigged ilber saces, which includes as secial cases sochasic Schrödinger equaions. Furher, he auhors [2] also sudied he aroximaion of sochasic nonlinear equaions of Schrödinger ye by he sliing mehod. Keller [22] discussed he exisence of oimal conrols for a linear sochasic Schrödinger equaion. Pinaud [23] sudied he exisence of sochasic nonlinear Schröinger equaions driven by a fracional noise. Very recenly, much aenion has been aid o he qualiaive roeries of mild soluions o various semilinear fracional sochasic differenial and inegro-differenial equaions by using he fracional calculus and he fixed-oin echnique. For examle, Cui and Yan [24], El-Borai e al. [25], Sakhivel e al. [26], Yan and Zhang [27], Ahmed [28], Zhang e al. [29]. owever, o he bes of our knowledge, here is no work reored on he nonlinear fracional sochasic Schrödinger evoluion equaions wih oenial. To close he ga in his aer, we sudy his ineresing roblem. Firsly, we shall inroduce a suiable conce on a mild soluion for our roblems, and discuss he exisence of soluions for a class of fracional sochasic nonlinear Schrödinger equaions wih oenial by means of fracional calculus, sochasic analysis, and fixed oin heorems wih he semigrou heory. Secondly, here are several aers devoed o he exisence of an oimal conrols of sysems governed by sochasic arial differenial equaions. Among hem, Ahmed [3] considered exisence of oimal for some semilinear sochasic evoluion equaions on infinie dimensional ilber saces. Mahmudov and McKibben [3], Zhou and Liu [32] invesigaed he exisence of oimal conrols for backward sochasic arial evoluion differenial sysems in he absrac sace. Yan and Lu [33] roved he exisence of oimal conrols for a fracional sochasic arial inegro-differenial equaion. In his aer,

2 we will consider he Lagrange roblem of sysems governed by fracional sochasic Schrödinger evoluion equaions wih oenial and he exisence resul of oimal conrols will be resened. Since many conrol sysems arising from realisic models can be described as fracional sochasic Schrödinger sysems. So i is naural o exend he conce of exisence of oimal conrols o dynamical sysems reresened by hese sysems. The known resuls aeared in [5], [6], [7] are generalized o he sochasic sysems seings, and he resuls aear are also new for deerminisic Schrödinger sysems wih oenial. The res of his aer is organized as follows. In Secion 2, we inroduce some noaions and necessary reliminaries. Secion 3 verifies he exisence of soluions for fracional sochasic conrol sysem. Secion 4 we esablish he exisence resuls for oimal airs of sysem governed by fracional sochasic conrol sysem. In Secion 5, an examle is given o illusrae our resuls. II. PRELIMINARIES The urose of his aer is o sudy he exisence of soluions for he following nonlinear fracional sochasic Schrödinger evoluion equaions wih oenial of he form i(j α x)(, y) (, y) kv (y)x(, y) = g(, x(, y)) f(, x(, y)) w() d, () y, J = [, b], x(, y) = x (y), y, (2) where 2 < α <, J α is he ( α)-order Riemann- Liouville fracional inegral oeraor, R 2 is a bounded domain wih a smooh boundary, denoes he Lalace oeraor in R 2, x is a comlex valued funcion in J R 2, k := max [,b] χ() wih χ C(J, R), is a osiive consan, and he funcion V is called oenial. x is he iniial condiion, and g, f : J R R are coninuous nonlinear funcions. The sae x(, ) akes values in a searable real ilber sace = L 2 () wih inner roduc, and norm. Define an oeraor A on L 2 () wih domain D(A) given by D(A) = 2 () (), such ha Ax = i x. By virue of he well known ille- Yosida heorem (see Pazy [34], i is obvious ha A is he infiniesimal generaor of a srongly coninuous grou {T (), < < } in. Moreover {T (), < < } can be given by (T ()x)(y) = 4πi e i y z 4 x(z)dz. (3) Furhermore, we have he following roeries. Lemma ([34]). Le {T (), } be he srongly coninuous semigrou given by (3). Then T ( ) can be exended in a unique way o a bounded oeraor from L 2 () ino L 2 () and T ()x L2 () x L2 (). Lemma 2 ([7]). Le be a measurable subse of R 2, k = max [,b] χ() and V 2 (). Then, we have kv x L 2 () k V L () x L 2 (). Throughou his aer, we use he following noaions. Le (Ω, F, P ) be a comlee robabiliy sace wih robabiliy measure P on Ω and a filraion {F } saisfying he usual condiions, ha is he filraion is righ coninuous and F conains all P -null ses. Le, K be wo real searable ilber saces and we denoe by,,, K heir inner roducs and by, K heir vecor norms, resecively. L(K, ) be he sace of linear oeraors maing K ino, and L b (K, ) be he sace of bounded linear oeraors maing K ino equied wih he usual norm and L b () denoes he ilber sace of bounded linear oeraors from o. Le {w() : } denoe an K-valued Wiener rocess defined on he robabiliy sace (Ω, F, P ) wih covariance oeraor Q, ha is E w(), x K w(s), y K = ( s) Qx, y K, for all x, y K, where Q is a osiive, self-adjoin, race class oeraor on K. In aricular, we denoe w() an K-valued Q-Wiener rocess wih resec o {F }, and x is an F -adaed, -valued random variables indeenden of w. In order o define sochasic inegrals wih resec o he Q -Wiener rocess w(), we inroduce he subsace K = Q /2 (K) of K which is endowed wih he inner roduc ũ, ṽ K = Q /2 ũ, Q /2 ṽ K is a ilber sace. We assume ha here exiss a comlee orhonormal sysem {e n } n= in K, a bounded sequence of nonnegaive real numbers {λ n } n= such ha Qe n = λ n e n. and a sequence β n of indeenden Brownian moions such ha w(), e = λn e n, e β n (), e K, J, n= and F = F w, where F w is he σ-algebra generaed by {w(s) : s }. Le L 2 = L 2 (K, ) be he sace of all ilber-schmid oeraors from K o wih he norm ψ 2 Tr((ψQ L2= /2 )(ψq /2 ) ) for any ψ L 2. Clearly for any bounded oeraors ψ L b (K, ) his norm reduces o ψ 2 Tr(ψQψ L2= ). Le L (F b, ) be he Banach sace of all F b -measurable h ower inegrable random variables wih values in he ilber sace. Le C([, b]; L (F, )) be he Banach sace of coninuous mas from [, b] ino L (F, ) saisfying he condiion su J E x() <. In aricular, we inroduce he sace C(J, ) denoe he closed subsace of C([, b]; L (F, )) consising of measurable and F -adaed -valued sochasic rocesses x C([, b]; L (F, )) endowed wih he norm x C = (su b E x() ). Then (C, C ) is a Banach sace. Le X be a Banach sace, we recall some basic definiions in fracional calculus. For more deails, see [], [2]. Definiion. The Riemann-Liouville fracional inegral of he order α > of h : J X is defined by J α h() = Γ(µ) ( s) α h(s)ds. Definiion 2. The Riemann-Liouville fracional derivaive of he order α (, ) of h : J X is defined by D α h() = d d J α h(). Definiion 3. The Cauo fracional derivaive of he order α (, ) of h : J X is defined by C D α h() = D α (h() h()).

3 Definiion 4 ([35]). The Miag-Leffler funcion is defined by E α,β (z) = n= z n Γ(αn β) ), α, β >, z C, (4) where C denoes he comlex lane. When β =, se E α (z) = E α, (z). Definiion 5 ([]). The Mainardi s funcion is defined by ( z) n M α (z) = n!γ( αn α ), < α <, z C. (5) n= The Lalace ransform of he Mainardi s funcion M α (ξ) is (see [36]): By (4) and (6), i is clear ha e ξλ M α (ξ)dξ = E α ( λ). (6) M α (ξ)dξ =, < α <. (7) On he oher hand, M α (z) saisfies he following equaliy (see [32]) and he equaliy (see [36]) and ξ θ M α (ξ)dξ = For < α <, se T α ()x = S α ()x = α ξ α M α(/ξ α )e λξ dξ = e λ α. (8) Γ(θ ), θ >, < α <. (9) Γ(αθ ) M α (ξ)t ( α ξ)xdξ,, x, () αξm α (ξ)t ( α ξ)xdξ,, x. () I is easy o see ha T α () and S α () are srongly coninuous on R. Moivaed by he Lemma 3. in [4], we resen he following definiion of mild soluions o ()-(2). Definiion 6. An F -adaed sochasic rocess x(, y) : [, b] is called a mild soluion of he sysem ()-(2) if x(, y) = x (y), x(, y) C(J, ) and (i) x(, y) is measurable and adaed o F,. (ii) x(, y) has càdlàg ahs on J a.s and for each J, x(, y) saisfies x(, y) = T α ()x (y) kv (y)x(s, y)ds g(s, x(s, y))ds f(s, x(s, y))dw(s). Lemma 3 (Bochner s Theorem [37]). A measurable funcion Λ : J is Bochner inegrable, if Λ is Lebesgue inegrable. Lemma 4 ([38]). For any and for arbirary L 2-valued redicable rocess φ( ) such ha s 2 su E φ(v)dw(v) ((2 )) s [,] ( (E φ(s) ds) 2 L2) /, [, ). In he res of his aer, we denoe by C = (( )/2) /2. Lemma 5 (Schaefer s fixed oin heorem [39]). Le X be a normed linear sace. Le G : X X be a comleely coninuous oeraor, ha is, i is coninuous and he image of any bounded se is conained in a comac se and le ζ(g) = {x X : x = λgx for some < λ < }. Then eiher ζ(g) is unbounded or G has a fixed oin. III. EXISTENCE OF SOLUTIONS FOR FRACTIONAL STOCASTIC CONTROL SYSTEMS In his secion, we rove he exisence of soluions for fracional sochasic conrol sysem ()-(2). Consider he sace Z = {x(, y) : [, b] ; x(, y) = x (y), x(, y) C(J, )} endowed wih he uniform convergence oology. In order o obain he resul, we inroduce he following assumions: () The funcion g : J is coninuous and here exiss a consan L g > such ha E g(, x ) g(, x 2 ) L ge x x 2 for all J, x, x 2. (2) The funcion f : J L 2 is coninuous and here exiss a consans L f > such ha E f(, x ) f(, x 2 ) L 2 L f E x x 2 for all J, x, x 2. (3) Le L = 3 b α (Γ(α)) (α ) (k V L () L g C L f b /2 ) be such ha < L <. Theorem. Le V 2 (). Assume ha ()-(3) are saisfied, hen sysem ()-(2) has a unique mild soluion x(, y) Z. Proof. Define he oeraor Φ : Z Z by (Φx)(, y) = T α ()x (y) kv (y)x(s, y)ds g(s, x(s, y))ds f(s, x(s, y))dw(s), J. (2) I is clear ha Φ is a well-defined oeraor from Z ino Z. We show ha Φ has a fixed oin, which in urn is a mild soluion of he roblem ()-(2). By (9)-(), and Lemma, we have T α () Lb (), S α () Lb () Γ(α),.

4 Le [, b] and x (, y), x (, y) Z. From (),(2) and Lemmas 2, 4, we have E (Φx )(, y) (Φx )(, y) 3 E kv (y) [x (s, y) x (s, y)]ds 3 E [g(s, x (s, y)) g(s, x (s, y))]ds 3 E [f(s, x (s, y)) f(s, x (s, y))]dw(s) 3 (Γ(α)) k b V L () ( s) (α ) E x (s, y) x (s, y) ds 3 (Γ(α)) b ( s) (α ) E g(s, x (s, y)) g(s, x (s, y)) [ ds 3 C (Γ(α)) [( s) (α ) E f(s, x (s, y)) f(s, x (s, y)) L 2] 2/ ds ] /2 3 (Γ(α)) k b V L () ( s) (α ) ds su E x (s, y) x (s, y) s [,b] 3 (Γ(α)) b L g ( s) (α ) E x (s, y) x (s, y) ds 3 C (Γ(α)) L f b/2 E x (s, y) x (s, y) ds ( s) (α ) 3 b (α ) (Γ(α)) (α ) (k b V L () L gb C L f b/2 ) su E x (s, y) x (s, y) s [,b] = 3 b α (Γ(α)) (α ) (k V L () L g C L f b /2 ) x x C. Taking suremum over, Φ x Φ x C L x x C, where L = 3 b α (Γ(α)) (α ) (k V L () L g C L f b /2 ) <, which imlies ha Φ is a conracion on Z. ence by he Banach fixed oin heorem, Φ has a unique fixed oin x(, y) in Z, and x(, y) is he unique mild soluion of sysem ()-(2). The roof is comlee. We use he below condiion insead of () and (2) o avoid he Lischiz coninuiy of g, f used in Theorem. (B) The oeraor families T α () and S α () are comac for all >. (B2) The funcion g(, ) : is coninuous for each J, and for every x, he funcion g(, x) is srongly measurable. (B3) There exiss a osiive funcion m g L (J, R ) such ha E g(, x) m g() for all J, x. (B4) The funcion f(, ) : L 2 is coninuous for each J, and for every x, he funcion f(, x) is srongly measurable. (B5) There exiss a osiive funcion m f L (J, R ) such ha E f(, x) L 2 m f () for all J, x. Theorem 2. Le V 2 (). If he assumions (B)-(B5) are saisfied. Then he sysem ()-(2) has a leas one mild soluion on J rovided ha 2 (α ) >. Proof. We define he ma Φ on he sace Z as in Eq. (2). We shall show ha Φ saisfies all condiions of Lemma 5. The roof will be given in several ses. Se. The se ζ = {x(, y) Z : λ (, ), x(, y) = λφ(x(, y))} is bounded. Indeed, le λ (, ) and le x(, y) Z be a ossible soluion of x(, y) = λφ(x(, y)) for some < λ <. This imlies by (2) ha for each [, b] we have x(, y) = λt α ()x (y) λ λ λ kv (y)x(s, y)ds g(s, x(s, y))ds f(s, x(s, y))dw(s), J. By (B2), (B5), and Lemmas 2 and 4, we have for J E x(, y) 4 E T α ()x (y) 4 E kv (y)x(s, y)ds 4 E g(s, x(s, y))ds 4 E

5 where f(s, x(s, y))dw(s) 4 E x 4 (Γ(α)) k b ( s) (α ) V (y) E x(s, y) ds 4 (Γ(α)) b ( s) (α ) E g(s, x(s, y)) [ ds 4 C (Γ(α)) [( s) (α ) E f(s, x(s, y)) L 2] 2/ ds ] /2 4 E x 4 (Γ(α)) k b V L () ( s) (α ) E x(s, y) ds 4 (Γ(α)) b 4 C (Γ(α)) b/2 ( s) (α ) m f (s, y)ds 4 E x 4 (Γ(α)) k b V L () ( s) (α ) m g (s, y)ds ( s) (α ) E x(s, y) ds ( 4 (Γ(α)) b ( s) ( ) (m g (s, y)) ds 4 C ( ( s) (Γ(α)) b/2 2 (α ) ( (m f (s, y)) ds ) ds ) 2 (α ) M(y) 4 (Γ(α)) k b V L () ( s) (α ) E x(s, y) ds, M(y) = 4 E x 4 (Γ(α)) b ( ) 2 (α ) ( b 2 (α ) b (m g (s, y)) ds 4 C (Γ(α)) b/2 ) ds ) ( 2 (α ) ( (m f (s, y)) ds Consider he funcion defined by ) 2 (α ) b ). v(, y) = su{e x(s, y) : s }, b. For each [, b], we have v(, y) M(y) 4 (Γ(α)) k b V L () ( s) (α ) v(s, y)ds. Alying Gronwall s inequaliy in he above exression, we obain v(, y) { M(y)ex 4 (Γ(α)) k b } V L () ( s) (α ) ds { M(y)ex 4 b α (Γ(α)) (α ) k } V L (), and herefore { x(, y) C M(y)ex 4 (Γ(α)) b α } (α ) k V L () <. Thus he roof of boundedness of he se ζ is comlee. Se 2. The se {(Φx)(, y) : x B r } is relaively comac in. We noe ha (Φ(B r ))(, y) is relaively comac in Z for =. Le < s b be fixed and ε a real number saisfying < ε < for x B r. We define (Φ ε x)(, y) = T α ()x (y) ε kv (y)x(s, y)ds ε ε g(s, x(s, y))ds f(s, x(s, y))dw(s). Using he comacness of T α (), S α () for >, we deduce ha he se U ε () = {(Φ ε x)(, y) : x B r } is relaively comac in for every ε, < ε <. Moreover, for every

6 x B r we have E (Φx)(, y) (Φ ε x)(, y) 3 E ε kv (y)x(s, y)ds 3 E ε g(s, x(s, y))ds 3 E ε f(s, x(s, y))dw(s) 3 (Γ(α)) k ε ε V (y) E x(s, y) ds ( s) (α ) 3 (Γ(α)) ε ( s) (α ) ε E g(s, x(s, y)) [ ds 3 C (Γ(α)) [( s) (α ) ε E f(s, x(s, y)) L 2] 2/ ds ] /2 3 (Γ(α)) k ε V L () ε ( s) (α ) E x(s, y) ds 3 (Γ(α)) ε 3 C ε (Γ(α)) ε/2 ( s) (α ) m f (s, y)ds ε 3 (Γ(α)) k ε V L () ε ( s) (α ) rds ( 3 (Γ(α)) ε ( (m g (s, y)) ds ε 3 C (Γ(α)) ε/2 ( ( s) ε ( 2 (α ) (m f (s, y)) ds ε ( s) (α ) m g (s, y)ds ( s) ε ) ) ds ), 2 (α ) ) ds and here are relaively comac ses arbirarily close o he se {(Φx)(, y) : x B r }, and (Φ(B r )() is a relaively comac in. By he Arzelá-Ascoli heorem, we can conclude ha he se {(Φx)(, y) : x B r } is relaively comac in for every J. Se 3. Φ mas bounded ses ino equiconinuous ses of Z. Le < ε < < b. From se 2, (ΦB r )(, y) is relaively comac for each and by he srong coninuiy of T α (), S α (), we can choose < δ < b wih T α ( h)x T α ()x ε, S α ( h)x S α ()x ε for x (ΦB r )(, y) when < h < δ. For any x B r, we have E (Φ 2 x)( h, y) (Φ 2 x)(, y) 4 E [T α ( h s) T α ( s)]x (y) h 4 E ( h s) α S α ( h s) = kv (y)x(s, y)ds kv (y)x(s, y)ds h 4 E ( h s) α S α ( h s) g(s, x(s, y))ds g(s, x(s, y))ds h 4 ( h s) α S α ( h s) f(s, x(s, y))dw(s) f(s, x(s, y))dw(s) 4 I i. i= In view of (B3), (B5) and ölder s inequaliy, i follows ha I 2 I = 4 T α ( h s)x (y) T α ( s)x (y) 4 ε, 3 E [( h s) α ( s) α ] S α ( h s)kv (y)ds 3 E [S α ( h s) S α ( s)] ( s) α kv (y)ds h 3 E ( h s) α S α ( h s) kv (y)ds 3 b ( h s) α ( s) α S α ( s) E kv (y) ds 3 b S α ( h s)k

7 I 4 I 3 S α ( s)k ( s)(α ) E V (y) ds h 3 b ( h s) (α ) S α ( h s) E kv (y) ds 3 b (Γ(α)) k V L () ( h s) α ( s) α ds 3 b ε V L () 3 b (Γ(α)) k V L () h ( h s) (α ) ds, ( s) (α ) ds 3 E [( h s) α ( s) α ] S α ( h s)g(s, x(s, y))ds 3 E [S α ( h s) S α ( s)] ( s) α g(s, x(s, y))ds h 3 E ( h s) α S α ( h s) g(s, x(s, y))ds 3 b ( h s) α ( s) α S α ( s) E g(s, x(s, y)) ds 3 E S α ( h s)g(s, x(s, y)) S α ( s)g(s, x(s, y)) ( s)(α ) ds h 3 ( h s) (α ) S α ( h s) E g(s, x(s, y)) ds 3 b (Γ(α)) ( h s) α ( s) α m g (s)ds 3 b ε ( s) (α ) ds 3 b (Γ(α)) h ( h s) (α ) m g (s, y)ds, 3 E [( h s) α ( s) α ] S α ( h s)f(s, x(s, y))dw(s) Since 3 E [S α ( h s) S α ( s)] ( s) α f(s, x(s, y))dw(s) h 3 E ( h s) α S α ( h s) f(s, x(s, y))dw(s) [ 3 C [ ( h s) α ( s) α S α ( s) E f(s, x(s, y)) L 2] 2/ ds [ 3 C [ S α ( h s)f(s, x(s, y)) S α ( s)f(s, x(s, y)) ( s)(α ) ] 2/ ds [ h 3 C [( h s) (α ) S α ( h s) E f(s, x(s, y)) L 2] 2/ ds 3 C (Γ(α)) b/2 ] /2 ] /2 ( h s) α ( s) α m f (s, y)ds 3 C ε b /2 ( s) (α ) ds 3 C h (Γ(α)) b/2 ( h s) (α ) m f (s, y)ds. ( h s) α ( s) α m g (s, y)ds 2 ( h s) (α ) m g (s, y)ds 2 ( s) (α ) m g (s, y)ds, and by ölder s inequaliy, we have ( s) (α ) m g (s)ds ( ( s) 2 (α ) ds ( 2 (α ) ( (m g (s, y)) ds Similarly, we have ) ( ) 2 (α ) b ) <. (m g (s, y)) ds ( h s) (α ) m g (s, y)ds <. ] /2 )

8 Then by he dominaed convergence heorem, In he same way, we can ge ( h s) α ( s) α m g (s, y)ds as h. ( h s) α ( s) α ds as h, ( h s) α ( s) α m f (s, y)ds as h. We see ha E Φx( h, y) Φx(, y) ends o zero indeendenly of x B r as h and sufficienly small osiive number ε. Thus, he se {Φx(, y) : x B r } is equiconinuous. Se 4. Φ : Z Z is coninuous. Le {x (n) } B r wih x (n) x(n ) in Z. By he assumions (B) and (B4), we have g(s, x (n) (s, y)) g(s, x(s, y)) as n, f(s, x (n) (s, y)) f(s, x(s, y)) as n for each s [, ], and since E g(s, x (n) (s, y)) g(s, x(s, y)) 2 m g (s, y), s [, b], E f(s, x (n) (s, y)) f(s, x(s, y)) L 2 2 m f (s, y), s [, b], and ( s)(α ) ds <, ( s)(α ) m g (s, y)ds <, ( s)(α ) m f (s, y)ds <. Then he dominaed convergence heorem ensures ha E (Φx (n) )(, y) (Φx)(, y) 3 E kv (y) [x (n) (s, y) x(s, y)]ds 3 E [g(s, x (n) (s, y)) g(s, x(s, y))]ds 3 E [f(s, x (n) (s, y)) f(s, x(s, y))]dw(s) 3 b (Γ(α)) ( s) (α ) k V (y) E x(n) (s, y) x(s, y) ds 3 b (Γ(α)) ( s) (α ) E g(s, x (n) (s, y)) g(s, x(s, y)) [ ds 3 C (Γ(α)) [( s) (α ) Then E f(s, x (n) (s, y)) f(s, x(s, y)) L 2] 2/ ds ] /2 3 b (Γ(α)) k V (y) E x (n) (s, y) x(s, y) ds 3 b (Γ(α)) ( s) (α ) ( s) (α ) E g(s, x (n) (s, y)) g(s, x(s, y)) ds 3 C (Γ(α)) b/2 ( s) (α ) E f(s, x (n) (s, y)) f(s, x(s, y)) ds. L 2 Φ 2 x (n) (, y) Φ 2 x(, y) C = su E (Φ 2 x (n) )(, y) (Φ 2 x)(, y) J as n. Therefore, Φ is coninuous. These argumens enable us o conclude ha Φ is comleely coninuous. We can now aly Lemma 5 o conclude ha Φ has a leas fixed oin x(, y) Z, which is a mild soluion of roblem ()-(2). The roof is comlee. IV. EXISTENCE OF FRACTIONAL STOCASTIC OPTIMAL CONTROLS In his secion we consider a conrol roblem and resen a resul on he exisence of fracional sochasic oimal conrols. le Y is a searable reflexive ilber sace from which he conrols u ake he values. L (J, L(Y, )) denoe he sace of oeraor valued funcions which are measurable in he srong oeraor oology and uniformly bounded on he inerval J. Le L F (J, Y ) is he closed subsace of L F (J Ω, Y ), consising of all measurable and F - adaed, Y -valued sochasic rocesses saisfying he condiion E u() Y d <, and endowed wih he norm u L F (J,Y ) = (E u() Y d). Le U be a nonemy closed bounded convex subse of Y. We define he admissible conrol se U ad = {v(, y) L F (J, Y ); v(, y) U a.e. J}. Consider he following conrolled nonlinear fracional sochasic Schrödinger evoluion equaions wih oenial of he form i(j α x)(, y) (, y) kv (y)x(, y) = B(, y)u(, y) g(, x(, y)) f(, x(, y)) w() d, (3) y, (, b], u U ad, x(, y) = x (y), y. (4) We will assume ha (S) B L (J, L(Y, )). Then, Bu L (J, ) for all u U ad. By Theorem 2, we have he following resul.

9 Theorem 3. Assume ha assumions of Theorem 2 hold and, in addiion, he assumion (S) is saisfied. For every u U ad, sysem (3)-(4) has a mild soluion corresonding o u given by he soluion of he following inegral equaion x u (, y) = T α ()x (y) kv (y)x(s, y)ds B(s, y)u(s, y)ds g(s, x(s, y))ds f(s, x(s, y))dw(s), J. Proof. Consider he sace Z endowed wih he uniform convergence oology and define he oeraor Θ : Z Z by (Θx)(, y) = T α ()x (y) kv (y)x(s, y)ds B(s, y)u(s, y)ds g(s, x(s, y))ds f(s, x(s, y))dw(s), J. Using (S) and he ölder inequaliy, we have E B(s, y)u(s, y)ds [ E ( s) α S α ( s) B(s, y) u(s, y) ds ] (Γ(α)) B [ E ( s) α u(s, y) Y ds ( ) (Γ(α)) B ( s) (α ) ds E u(s, y) Y ds (Γ(α)) B b α u L (J,Y ), F ( ) α where B is he norm of oeraor B in Banach sace L (J, L(Y, )) and α >. From Lemma 3, i follows ha ( s) α S α ( s)b(s, y)u(s, y) is Lebesgue inegrable wih resec o s [, ] for all J. ence we conclude ha Θ is a well-defined oeraor from Z ino Z. The roofs of he ] oher ses are similar o hose in Theorem 2. Therefore, we omi he deails. The roof is comlee. Le x u (, y) denoe he mild soluion of sysem (3)-(4) corresonding o he conrol u(, y) U ad. We consider he Lagrange roblem (P): Find an oimal air (x (, y), u (, y)) BC U ad such ha J (x (, y), u (, y)) J (x u (, y), u(, y)) for all u(, y) U ad, where he cos funcion J (x u (, y), u(, y)) = E l(, x u (, y), u(, y))d, and x u (, y) denoes he mild soluion of sysem (3)- (4) corresonding o he conrol u(, y) U ad. We inroduce he following assumion on l. (P) The funcional l : J Y R { } is Borel measurable. (P2) l(,, ) is sequenially lower semiconinuous on Y for almos all J. (P3) l(, x, ) is convex on Y for each x and almos all J. (P4) There exis consans d, d 2 >, µ is nonnegaive and µ L (J, R) such ha l(, x, u) µ() d x d 2 u Y. To rove he exisence of soluion for roblem (P), we need he following imoran lemma. Lemma 6. Oeraor Ψ : L (J, Y ) Z for some α( ϑ) > given by (Ψu)(, y) = S α ( s)b(s)u(s)ds is comleely coninuous. Proof. Suose ha u n L F (J, Y ) is bounded, we define Λ n (, y) = (Ψu n )(, y), J. Similar o he roof of Theorem 2, one can know ha for any fixed J and, E Λ n (, y) is bounded. By using (B)-(B5), i is ease o verify ha Λ n (, y) is relaively comac in and is also equiconinuous. Due o Ascoli-Arzela Theorem again, {Λ n (, y)} is comac in. Obviously, Ψ is linear and coninuous. ence, Ψ is a comleely coninuous oeraor. The roof is comlee. Nex we can give he following resul on exisence of oimal conrols for roblem (P). Theorem 4. If he assumions (P)-(P4) and he assumions of Theorem 3 hold. Then he Lagrange roblem (P) admis a leas one oimal air on Z U ad. Proof. Wihou loss of generaliy, we assume ha inf{j (x u (, y), u(, y)) u(, y) U ad } = ε <. Oherwise, here is nohing o rove. By assumions (P)-(P4), we have J (x u (, y), u(, y)) ϕ()d d E x u (, y) d d 2 E x(, y) Y d η >,

10 where η > is a consan. ence, ε η >. On he oher hand, by using definiion of infimum, here exiss a minimizing sequence of feasible air {(x m (, y), u m (, y))} A ad, such ha J (x m (, y), u m (, y)) ε as m, where A ad = {(x(, y), u(, y)) x(, y) is a mild soluion of sysem (3)-(4) corresonding o u(, y) U ad }. For {u m (, y)} U ad. {u m (, y)} is bounded in L F, so here exiss a subsequence, relabeled as {u m (, y)}, and u L F(J,Y ) such ha u m (, y)(w) u (, y) in L F as m. Since U ad is closed and convex, by Marzur Lemma, we conclude ha u (, y) U ad. Now we suose ha x m (, y) are he mild soluions of sysem (3)-(4) corresonding o u m (, y)(m =,, 2,...), and x m (, y) saisfied he following inegral equaion x m (, y) = T α ()x (y) kv (y)x m ((s, y)ds B(s, y)u m (s, y)ds g(s, x m (s, y))ds f(s, x m (s, y))dw(s), J. Le g m (s, y) g(s, x m (s, y)), f m (s, y) f(s, x m (s, y)). Then by (B4) and (B5), we obain ha g m (, y) L (J,) ( ) = E g m (s, y) ds = E g m (s, y) ds ( m h (s, y)ds b f m (, y) L (J,L b (K,) ( ) = E f m (s, y) L b (K,) ds = E f m (s, y) L b (K,) ds ( m f (s, y)ds b ) (m g (s, y)) ds, ) (m f (s, y)) ds. Tha is o say, h m (, y) : J and f m (, y) : J L b (K, ) are bounded coninuous oeraors. ence, h m (, y) L (J, ), f m (, y) L (J, L b (K, )). Furhermore, {h m (, y)}, {f m (, y)} is bounded in L (J, ) and in L (J, L b (K, )), and here are subsequences, relabeled as {h m (, y)}, {f m (, y)}, and ĥ(, y) L (J, ), f(, y) L (J, L b (K, )) such ha h m (, y)(w) ĥ(, y) in L (J, ) as m, f m ( )(w) f(, y) in L (J, L b (K, )) as m. Nex we urn o consider he following conrolled sysem i(j α x)(, y) (, y) kv (y)x(, y) = B(y, )u (y, ) ĝ(, y) f(, y) w() d y, [, b], u U ad, (5) x(, y) = x (y), y. (6) By Theorem 3, i is easy o see ha sysem (5)-(6) has a mild soluion x(, y) = T α ()x (y) kv (y) x(s, y)ds B(s, y)u (s, y)ds ĝ(s, y)ds f(s, y)dw(s), J. For each J, x m (, y), x(, y) Z, we have E x m (, y) x(, y) µ () m (, y) µ (2) m (, y) µ (3) m (, y) µ (4) m (, y), where µ () m (, y) = 4 E [kv (y)x m (s, y) kv (y) x(s, y)]ds µ (2) m (, y) = 4 E B(s, y)[u m (s, y) u (s, y)]ds µ (3) m (, y) = 4 E [g m (s, y) ĝ(s, y)]ds, µ (4) m (, y) = 4 E [f m (s, y) f(s, y)]dw(s).,,

11 Using he ölder inequaliy, we can obain µ () m (, y) 4 b ( s) (α ) S α ( s) kv (y) E xm (s, y) x(s, y) ds 4 b (Γ(α)) k V L () ( s) (α ) E x m (s, y) x(s, y) ds. By (S) and using he ölder inequaliy again, we have µ (2) m (, y) [ 4 E ( s) α B(s, y) and S α ( s)[u m (s, y) u (s, y)] ds 4 B 4 B ( ) ( s) (α ) ds ] E S α ( s)[u m (s, y) u (s, y)] ds ( ) b α α E S α ( s)[u m (s, y) u (s, y)] ds, µ (3) m (, y) [ 4 E ( s) α S α ( s) [g m (s, y) ĝ(s, y)] ds ] ( ) 4 ( s) (α ) ds E S α ( s)[g m (s, y) ĝ(s, y)] ds ( ) 4 b α α µ (4) m (, y) E S α ( s)[g m (s, y) ĝ(s, y)] ds, [ 4 C [( s) (α ) E S α ( s) ] /2 [f m (s, y) f(s, y)] ds] 2/ L 2 ( ) /2 4 ( s) /2(α ) /2 ds E S α ( s)[f m (s, y) f(s, y)] ds ( ) /2 /2 4 b (/2)α (/2)α E S α ( s)[f m (s, y) f(s, y)] ds, By Lemma 6 and Lebesgue s dominaed convergence heorem, Therefore, we obain E S α ( s)[u m (s, y) u (s, y)] ds as m, E S α ( s)[g m (s, y) ĝ(s, y) ]ds as m, E S α ( s)[f m (s, y) f(s, y) ]ds as m. µ (2) m (, y), µ (3) m (, y), µ (4) m (, y) as m. Then we have E x m (, y) x(, y) 4 b (Γ(α)) k V L () ( s) (α ) E x m (s, y) x(s, y) ds µ (2) m (, y) µ (3) m (, y) µ (4) m (, y). Using Gronwall s inequaliy again, E x m (, y) x(, y) [µ (2) m (, y) µ (3) m (, y) µ (4) m (, y)] { ex 4 b α (Γ(α)) (α ) } k V L (), which imlies ha x m (, y) x(, y) in Z as m. Furher, by (B3) and (B5), we can obain g m (y, ) g(, x(, y) in Z as m, f m (, y) f(, x(y, ) in Z as m. Using he uniqueness of limi, we have ĝ(, y) = g(, x(, y)), Thus, x(, y) can be given by x(, y) = T α ()x (y) f(, y) = f(, x(, y)). kv (y) x(s, y)ds B(y, s)u (s, y)ds g(s, x(s, y))ds f(s, x(s, y))dw(s), J,

12 which is jus a mild soluion of sysem (3)-(4) corresonding o u (, y). Since Z L (J, ), using (P)-(P4) and Balder s heorem ([4]), we can obain ε = lim E l(, x m (, y), u m (, y))d m E l(, x(, y), u (, y))d J ( x(, y), u (, y)) ε. This shows ha J aains is minimum a ( x(, y), u (, y)) Z U ad and he roof is comlee. V. APPLICATION Consider he following conrolled fracional sochasic Schrödinger equaion of he form i(j α z)(, y) (, y) kv (y)z(, y) = k(y, τ)u(, τ)dτ e γ e sin x(, y) e e γ2 e cos x(, y)w() e d, (7) y, (, b], u U ab, x(, y) = x (y), y, (8) where 2 < α <, J α is he ( α)-order Riemann- Liouville fracional inegral oeraor, R 2 is a bounded domain wih a smooh boundary, denoes he Lalace oeraor in R 2, and k C(, R), γ and γ 2 are osiive real consan. w() denoes a one-dimensional sandard Wiener rocess in defined on a sochasic sace (Ω, F, P ). Le X = Y = (L 2 (), ). The oeraor A : D(A) defined by D(A) = 2 () () such ha Ax = i x. I is obvious ha A is he infiniesimal generaor of a comac, analyic semigrou T ( ) of uniformly bounded linear oeraor. For d >, we define he admissible conrol se U ad = {u(, y) J Y measurable, F -adaed sochasic rocesses, and u L F (J,Y ) d}. Define x()y = x(, y), and and g(, x(, y)) = e γ e sin x(, y), e f(, x(, y)) = e γ2 e cos x(, y), e (Bu)(, y) = k(y, τ)u(, τ)dτ. Thus sysem (7)-(8) can be ransformed ino (3)-(4) wih he cos funcion J (x, u) = E x(, y) 2 dy E u(, y) 2 dy. Obviously, E g(, x) = e γ, E f(, x) = e γ2. For u L 2 ([, b] ), we have Bu(, y) 2 ddy [ ] k(y, τ)u(s, τ) 2 dsdτ ddy M 2 k b 2 (mes()) 2 u(s, τ) 2 dsdτ <, where M k = max (y,τ) k(y, τ). This imlies ha he oeraor B : L 2 ([, b] ) L 2 ([, b] ), and Bu L 2 ([,b]) M kb mes() u L 2 ([,b]). Then we can conclude ha B is a bounded oeraor in L 2 ([, b] ). Furher, we can imose some suiable condiions on he above-defined funcions o verify he assumions on Theorem 4. Therefore, he roblem (7)-(8) has a leas one oimal air. 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