Approximate Controllability of Fractional Stochastic Perturbed Control Systems Driven by Mixed Fractional Brownian Motion

Transcription

1 American Journal of Applied Mahemaic and Saiic, 15, Vol. 3, o. 4, Available online a hp://pub.ciepub.com/ajam/3/4/7 Science and Educaion Publihing DOI:1.1691/ajam Approximae Conrollabiliy of racional Sochaic Perurbed Conrol Syem Driven by Mixed racional Brownian Moion Salah. Abid *, Sameer Q. aan, Uday J. Quaez Mahemaic deparmen, Educaion College, Al-Muaniriya Univeriy, Baghdad, Iraq *Correponding auhor: abidalah@gmail.com Received July 7, 15; Revied July 31, 15; Acceped Augu 13, 15 Abrac In hi paper, he approximae conrollabiliy of nonlinear racional order < < 1 Riemann- Liouville ype ochaic perurbed conrol yem driven by mixed fracional Brownian moion in a real eparable ilber pace ha been udied by uing Kranoelkii fixed poin heorem, ochaic analyi heory, fracional calculu and ome ufficien condiion. Keyword: approximae conrollabiliy, mixed fracional brownian moion, fixed poin heorem, perurbed conrol yem, mild oluion, conrol funcion Cie hi Aricle: Salah. Abid, Sameer Q. aan, and Uday J. Quaez, Approximae Conrollabiliy of racional Sochaic Perurbed Conrol Syem Driven by Mixed racional Brownian Moion. American Journal of Applied Mahemaic and Saiic, vol. 3, no. 4 (15: doi: /ajam Inroducion he aim main of hi paper i o udy he principle concep of he approximae conrollabiliy for complicaed clae of fracional order < < 1 Riemann-Liouville ype ochaic perurbed conrol yem driven by mixed fracional Brownian moion he following form i he yem under our conideraion, L D x( = (A Ax( Bu(, x(, h,, x(d G, x(, g(, x( ( d W ( σ( (, d d [, ], < < 1 L D x( = = x where, L D he Riemann-Liouville fracional derivaive of order < < 1.A i he infinieimal generaor of a compac analyic emigroup of uniformly bounded linear operaor S(,, A i a bounded linear operaor in a real eparable ilber pace X. Aume ha ρ(a A. he pace X i a ilber uch ha X = D(A A, < 1, equipped wih norm x = (A A x X. x( C 1 [, ]; L Ω, X = x(; 1 x( C [, ]; L Ω, X i a coninuou a [, ] } wih he norm x C1 = ( up ( 1 E x( 1. x i meaurable X- valued ϵ[,] random variable independen of W and W which defined on a complee probabiliy pace (Ω,, { }, P. he conrol funcion u(. L ([,]; U, U i a ilber pace and he operaor B from U ino X i a bounded linear operaor uch ha here exi conan L B >, Bu L B u. W = { W (, [, ]} i a andard cylindrical Brownian moion defined on (Ω,,{ },P wih value in a ilber pace K. Le Q be a poiive, elf adjoin and race cla operaor on K and le L (K,X be he pace of all Q -ilber-schmid operaor acing beween K and X equipped wih he ilber-schmid norm. L. W = { W (, [,] } i a Q-fracional Brownian moion wih ur index ( 1, 1 defined in a complee probabiliy pace (Ω,,{ },P wih value in a ilber pace Y, uch ha Q i a poiive,elf adjoin and race cla operaor on Y and le L (Y;X be he pace of all Q -ilber-schmid operaor acing beween Y and X equipped wih he ilber-schmid norm. L. he funcion : [,] X X X, h: [,] [,] X X, G: [,] X X L (K; X and σ: [, ] L (Y, X are coninuou funcion. Approximae conrollabiliy of ochaic conrol yem driven by fracional Brownian moion ha been inereed by many auhor; Sakhivel [19] udy for he approximae conrollabiliy of impulive ochaic yem wih fracional Brownian moion. Guendouzi and Idrii, [7] eablihed and dicued he approximae conrollabiliy reul of a cla of dynamic conrol yem decribed by nonlinear fracional ochaic funcional

2 169 American Journal of Applied Mahemaic and Saiic differenial equaion in ilber pace driven by fracional Brownian moion wih ur parameer > 1. Ahmed [] inveigae he approximae conrollabiliy problem for he cla of impulive neural ochaic funcional differenial equaion wih finie delay and fracional Brownian moion wih ur parameer > 1 in a ilber pace. Abid, aan and Quaez [1] udied he Approximae conrollabiliy of fracional ochaic inegro-differenial equaion which i derived by mixed ype of fracional Brownian moion wih ur parameer > 1 and wiener proce in real eparable ilber pace. In hi paper we will udy he approximae conrollabiliy of nonlinear ochaic yem. More preciely, we hall formulae and prove ufficien condiion for he Approximae conrollabiliy of racional order < < 1 Riemann-Liouville ype ochaic perurbed conrol yem driven by mixed fracional Brownian moion in a real eparable ilber pace. he re of hi paper i organized a follow, in ecion, we will inroduced ome concep, definiion and ome lemma of emigroup heory and fracional ochaic calculu which are ueful for u here. In ecion 3, we will prove our main reul.. Preliminarie In hi ecion, we inroduce ome noaion and preliminary reul, which we needed o eablih our reul. Definiion (.1, [5]: Le be a conan belonging o (, 1. A one dimenional fracional Brownian moion B = B (, of ur index i a coninuou and cenered Gauian proce wih covariance funcion EB ( B ( = 1 (, for,.(1 If = 11, hen he incremen of B are noncorrelaed, and conequenly independen. So B i a Wiener Proce which we denoe furher by B. If ( 11, 1 hen he incremen are poiively correlaed. If (, 11 hen he incremen are negaive correlaed. B ha he inegral repreenaion B ( = K (, db ( ( where, B i a wiener proce and he kernel K (, defined a K (, = c 1 (u 3 u 1 du K (3 (, = c 1 ( 3 (4 1 c = (1, > and i a bea funcion., 1 In he cae =, we hall ue Io Iomery heorem Lemma (.1, Io iomery heorem, [11]: Le V [,] be he cla of funcion uch ha f: [, ] Ω R, f i meaurable, - adaped and E f(, ω d. hen for every f V [,], we have E f(, ωdb( = E f(, ω d (5 where B i a wiener proce. ow, we denoe by ᶓ he e of ep funcion on [, ].If Φ ᶓ hen, we can wrie i he form a: Φ( = n k=1 a k 1 [k, k 1 ](, where [,]. he inegral of a ep funcion Φ ᶓ wih repec o one dimenional fracional Brownian moion i defined where a k R, Φ(dB n = k=1 a k (B k 1 B k, = 1 < < < n1 =. Le be he ilber pace defined a he cloure of ᶓ wih repec o he calar produc <1 [,],1 [,] > = R (, = E(B B. he mapping 1 [,] {B (, [,]} can be exended o an iomery beween and pan L (Ω, [, ].i.e. he mapping L (Ω,,P, Φ B ( Φ(dB i iomery. Remark (.1: If = 1 and = L ([, ] hen by ue Io iomery, we have E Φ(dB = (Φ( d (6 If > 1, we have R (, = 1 (,, (7 R d R = ( 1 1 (8 = ( 1 dd Lemma (., [6]: or any funcion Φ, φ L [, ] L 1 [, ], we have i E Φ(dB ( φ(db ( = ( 1. Φ( φ( dd ii EdB ( db ( = R = ( 1 dd dd rom hi Lemma above, we obain E Ф db ( = ( 1 Ф Ф dd (9 Remark (., [6]: he pace conain he e of funcion Ф L [, ], uch ha,. Ф(Ф( dd <, which include L 1 ([, ]. ow, Le Ĥ be he Banach pace of meaurable funcion on [, ], uch ha Φ Ĥ=( 1. Ф(Ф( dd < (1 Lemma (.3, [1]:

3 American Journal of Applied Mahemaic and Saiic 17 L([, ] L 1 ([, ] Ĥ. Suppoe ha here exi a complee orhonormal yem {e n } n=1 in Y. Le Q L (Y, Y be he operaor defined by Qe n =λ n e n, where λ n (n=1,,. are nonnegaive real number wih finie race r Q = n=1 λ n <.he infinie dimenional fracional Brownian moion on Y can be defined by uing covariance operaor Q a = W Q ( = n=1 λ n e n B n ( W (, where B n ( are one dimenional fracional Brownian moion muually independen on (Ω,, P. In order o defined ochaic inegral wih repec o he Q-fracional Brownian moion. We inroduce he pace L (Y,X of all Q-ilber- Schmid operaor ha i wih he inner produc Φ, φ L = n=1 Φe n, φe n i a eparable ilber pace. Lemma (.4, [1]: Le {Φ(} ϵ[,] be a deerminiic funcion wih value in L (Y,X he ochaic inegral of Φ wih repec o W i defined by Φ ( = n 1 n λ Φ e ndb = n( * = λ n 1 n (K (Φe n db = n( (11 Lemma (.5, [1]: If φ: [, b] L (Y, X aifie φ( L d < hen he above um in (11 i well defined a an X-valued random variable and we have E φ( (S 1 φ( L d (1 Definiion (., [18]: he Riemann - Liouvill derivaive of order > wih lower limi zero for a funcion f can be wrien a: L D f( = 1 d n f( Γ(n d n ( 1n (13 where, >, n 1 < < nn. Definiion (.4, [18]: he Laplace ranform of he Riemann-Liouville fracional derivaion of order > give a: L{LD f(} = λ n1 L(f((λ λ k k1 k= [ D f(] = (λ(14 where, n-1< <n. Lemma (.6, [17]: Le A be he infinieimal generaor of an analyic emigroup S(, on a ilber pace X. If A i a bounded linear operaor on X hen ( A A i he infinieimal generaor of an analyic emigroup Ť(, on X. Remark (.3: Aume ha Ť(, i a compac analyic emigroup of uniformly bounded operaor in X, ha i, here exi M > 1 uch ha Ť(x M. Definiion (.5: An X -valued proce x( i called a mild oluion of he yem (1 if x( C 1 [, ]; L Ω, X and, for [, ] aifie he inegral equaion = Ť ( ( Bu d x Ť x Ť, x, h,r, x r dr d (15 (, x (, ( ( g(, x(r (r Ť G 1 Ť σ where, Ť ( = rm (r( r M (r i a Mainardi' funcion. Lemma (.7: If Ť(, i a compac analyic emigroup hen he family of operaor Ť (, have he following properie: i. or any fixed, he operaor Ť ( i a linear and bounded, i.e. for any x X, here exi M 1 uch ha Ť (x X M x Г( X. ii. or any x X, here exi M 1 uch ha Ť (x M x Г(. iii. or any x X, Ť (x, x X where, Г(, = ќ Г((11 iv. {Ť (, } i a rongly coninuou, which mean ha for every x X and 1 < hen Ť ( x Ť ( 1 x X if 1. v. he operaor Ť ( i a compac operaor in X for >. 3. Main Reul of he Approximaely Conrollable In hi ecion, we formulae and prove he reul on approximae conrollabiliy of nonlinear fracional ochaic perurbed conrol yem driven by mixed fracional Brownian moion in (1. o eablih our reul, we inroduce he following aumpion: a he operaor Ť ( i a compac for any >. b he linear fracional order yem of correponding he yem (3.41 which ha following form: L D x( = Ax( Bu(, [, ] L D x( = = x, 1 < < 1 (16 i an approximaely conrollable on [,]. c he funcion :[,] X X X, h:[,] [,] X X are aifying linear growh and Lipchiz condiion. hi mean ha, for any x, y X, here exi poiive conan K 1, K > aaaaaa K 3, K 4 > uch ha, x(, h(,, x(d,y,h,,y(d X K1xy, x(, h(,, x(d K (1 x X h(,, x h(,, y K 3 x y, h(,, x K 4 (1 x

4 171 American Journal of Applied Mahemaic and Saiic Alo, i a uniformly bounded. In oher word, here exi DD 1 > uch ha, x(, h(,, x(d < D1, for [, ] X d he funcion σ: [,] L (Y; X aifie, for every [, ] σ( L d < and here exi C 1 > uch ha up σ(( L C 1. [,] e G i adaped wih repec o, uch ha, for every [,], aify he following: i. E (Gx( L Exi. ii. E (Gx( L d <. iii. here exi D >, uch ha up [,] E (Gx( L < D. where, Gx( = G, x(, g(, x( ( Definiion (3.1: he yem (1 i aid o be approximaely conrollable on [,] if he reachable e R( i dene in he pace L (Ω, X. hi mean ha R( = L (Ω, X. where, R( = { x(,u :u L ([,]; U } ow, he conrollabiliy operaor Г aociaed wih conrol yem (16 i defined by Г = ( Ť ( BB Ť ( d (17 Alo, for any > and <, he operaor R(, Г i defined by R(, Г = (I Г 1 (18 where, B and Ť are he adjoin operaor for B and Ť repecively. Lemma (3.1, [1]: he linear fracional order deerminiic yem in (1 i an approximaely conrollable on [, ] if and only if he operaor R(, Г a, R(, Г 1. Lemma (3., [13]: or any x L (Ω,, X, here exi Ĥ L Ω; L [, ]; L (Y, X and ɸ L Ω; L [, ]; L (K; X, uch ha x = Ex ɸ( ( Ĥ(dw ( where, up [,] E ɸ( L C 1, up [,] E Ĥ( L C (19 ow, or any > and any x L (Ω,, X, we defined he conrol funcion of he yem (1 in he following form: u (,x R * * = B Ť (,Г Ex Ť x * * ( φ ( * * B Ť R,Г ˆ B Ť R,Г R(,Г Ť * * B Ť ( d, x h (,r, x ( r, R(,Г Ť * * (, x, G g(v, x(v (v ( ( B Ť ( R(,Г Ť * * B Ť ( ( σ ( Lemma (3.3: here exi poiive real conan C, uch ha for all x C [, ]; L Ω, X, E u (, x C (1 where, 4 1LBM C = E 4 (E x x (Г LB 1 LBM 1 6 M C C (Г Г ( 6LBM (Г (1( 1 6LBM (Г (1( 1 1 B 1L M (Г ( 1(1, D1, D, C1 Proof Le x C [, ]; L Ω, X and > be a fixed. rom he equaion (1, we have: E u (,x * * Ex 6E B Ť R(,Г Ť x * * ( * * ( φ ( * * ˆ 6E B Ť R, Г 6E B Ť R, Г R(,Г Ť * * 6E B Ť d,x(, h(,r,x( r 6E B Ť R(, Г Ť G, x (, g(v, x(v (v

5 American Journal of Applied Mahemaic and Saiic 17 ( R(,Г Ť * * 6E B Ť ( ( σ Applying older inequaliy and by uing Io iomery, Lemma (.5, (.7 and he aumpion (a-(e, we obain where, E u (, x C 4 1LBM C = E 4 (E x x (Г LB 1 LBM 1 6 M C C (Г Г ( 6LBM (Г (1( 1 6LBM (Г (1( 1 1 B 1L M (Г ( 1(1, D1, D, C1 ow, or any >, conider he operaor Ψ on C 1 [, ]; L Ω, X defined a follow: x = (Ψ Ť x Ť Bu (,xd ( Ť ( (, x (, h (, r, x ( r d Ť G,x(, g(v,x(v Ť σ ( (v Alo, for any δ >, he ube B δ of C 1 [, ]; L Ω, X i define a B δ = x ( C 1 [, ]; L Ω, X x( C1 δ. Lemma (3.4: for any >, here exi δ > h haaaa ΨΨ (B δ B δ. o prove ha here exi δ > h haaaa ΨΨ (B δ B δ, in oher word, Ψ x( C1 δ, for each x( B δ. Suppoe ha hi i no rue, hen for each δ >, here exi x( B δ uch ha Ψ x( C1 > δδ, for [, ], may depending upon δ. owever, on he oher hand, we have E Ψx( 5 E Ť x 5 E Ť Bu (,xd Ť 5 E d, x (, h (,r, x ( r Ť 5 E G, x (, g(v, x(v (v 5 E Ť σ Applying older inequaliy and by uing Io iomery, Lemma (.5, (.7, (3.3 and he aumpion (a-(e, we obain ence, [,] up E Ψ x( M 5L 5 Ex (Г( ( 1(1 5 D1, 5,D ( 1( ( 1(1 Ψ x( B C,,C1 1 C 5LB C, M 5 Ex (Г( ( 1(1 5 D1, 5,D ( 1( ( 1(1 1,C1 herefore, M 5L δ < 5 Ex (Г( ( 1(1 5 D1, 5,D ( 1( ( 1(1 B C,,C1 By dividing boh ide of above inequaliy by δ and aking he limi a δ, which i a conradicion. hu, for each >, here exi poiive number δ uch ha Ψ (B δ B δ. ow, Le Ψ = Ψ 1 Ψ where,

6 173 American Journal of Applied Mahemaic and Saiic Ť (Ψ1x = d, x (, h (,r, x ( r = (Ψ x Ť x Ť Bu (,xd Ť ( G, x (, g(v, x(v (v Ť σ (3 (4 Lemma (3.5: Aume ha he aumpion (a (e hold, hen for any >, and for any x, y B δ, (Ψ 1y( ( Ψ x( B δ, for [,]. Proof Le x, y B δ and, we have E(Ψ1y( (Ψx( 5 E Ť x 5 E Ť ( ( Bu (, xd Ť 5 E d, y(, h (,r, y( r Ť 5 E G, x (, g(v, x(v (v 5 E Ť σ Applying older inequaliy and by uing Io iomery, Lemma (.5, (.7, (3.3 and he aumpion (a-(e, we obain 1 5LB C, E(Ψ y( (Ψ x( M 5 Ex (Г( ( 1(1 5 D1, 5,D ( 1( ( 1(1,C1 which mean ( Ψ 1 y( ( Ψ x( C1 δ,hen ( Ψ 1 y ( ( Ψ x ( B δ Lemma (3.6: or any [, ], he operaor Ψ 1 i a conracion on B δ, provided ha γ =, K 1 ((1 Le [, ], and y 1, y B δ, we have E Ψ y Ψ y < 1 Ť, y1(, h (,r, y1( r = E d, y(, h (,r, y( r By uing lemma (.7 and aumpion (c, we ge E Ψ1y1 Ψ1y, ( K1E y1 y d 1 By aking he upremum over [,] for boh ide, we have up [,] { E Ψ1y1 Ψ1y ( }, K1 up E y y ( 1(1 [,] herefore, Where, γ =, { 1 } Ψ1y1 Ψ1y C 1, K1 y 1( y ( ( 1(1 C 1 γ y1 y C 1 K 1 ((1 < 1, hence Ψ 1 i a conracion. Lemma (3.7: Aume ha he aumpion (a (e hold, hen he operaor Ψ map bounded e ino bounded e in B δ. Le x ( B δ, for [, ], we have E Ψx( 4 E Ť x 4 E Ť Bu (,xd Ť 4 E ( G, x (, g(v, x(v (v 4 E Ť σ

7 American Journal of Applied Mahemaic and Saiic 174 Applying older inequaliy and by uing Io iomery, Lemma (.5, (.7, (3.3 and he aumpion (a-(e, we obain where, Ð = 4 4, E Ψ x( Ð D M (Г( E x 4 L B C, ((1 8 1 ((1 ((1, C 1 By aking he upremum over [,] for boh ide, we ge up E Ψ x( Ð [,] herefore, for each x B δ, we ge Ψ x( C1 Ð. hen Ψ map bounded e ino bounded e in B δ. Lemma (3.8: Aume ha he aumpion (a (e hold, hen Ψ i a coninuou on B δ. Le {x n } n=1 be a equence in B δ uch ha x n x a n in C 1 [, ]; L Ω, X. or each [,], we have E(Ψx n(-(ψx( Ť ( ( E d B[u (, xn u (, x ] 1 Ť E [(Gx n ( (Gx(] rom Io iomery and lemma (.7, we obain E(Ψx n(-(ψx( ( LB, d E u (,xn u (,x (, d E (Gx n ( (Gx( herefore, i follow from he coninuiy of G and u ha for each [, ], G, x n (, g(v, x n (v (v G, x(, g(v, x(v (v and u (, x n u (, x, uing he Lebegue dominaed convergence heorem ha for all [, ], we conclude (Ψ x n (Ψ x( C1, a n, Implying ha Ψ x n Ψ x C1, a n. ence, Ψ i coninuou on B δ. Lemma (3.9: If he aumpion (a (e are hold, hen for x B δ, he e {(Ψ x(, [, ]} i equiconinuou. Le 1, [, ] uch ha < 1 <. hen, from he equaion (4, we have E Ψx Ψx( 1 7E [(Ť ( Ť ( 1 ]x 1 Ť ( 14E ( Bu (, xd Ť ( E Ť 1 Bu (, xd ( 1 7E Ť ( ( Bu (,xd ( ( 1 Ť 14E ( (Gx( Ť E Ť 1 ( ( 1 (Gx( ( 7E Ť ( ( (Gx( 1 1 Ť ( 14E ( σ ( Ť E Ť 1 σ ( 1 7E Ť ( ( σ 1 ow, from Lemma (.7, noing he fac ha for every ϵ >, here exi τ > uch ha, whenever 1 < τ, for every 1, [,], Ť ( Ť ( 1 < ϵ. herefore, when 1 < τ, we have 1 E Ψ x Ψ x LB C 14 7 Ex LBC, 1 ( 14 d 1 ( 1 D C, B ( 1 7 L d ,D ( ( 1 d 1 7 D d 8, ( 1 C ( 8 C1, ( 1 d ( 1 8 C d 1 1, 1

8 175 American Journal of Applied Mahemaic and Saiic he righ hand of he inequaliy above end o a 1 and εε. ence for x B δ, he e { Ψ x, [, ]} i equiconinuou. Lemma (3.1: If he aumpion (a i hold. hen for each [, ], he e Ü ( = { Ψ x, x B δ } i relaively compac in B δ. Le (,] be a fixed and < ȃ <, for every γγ >, x B δ, we define a,γ ˆ x = γ Ψ rm r Ť( r x dr aˆ γ (5 rm r Ť(( r Bu, x drd aˆ γ rm r Ť(( r (Gx(dr aˆ γ W rm r Ť(( r σ drd hen, from he definiion of emigroupť( r, r >, we can eaily be wrien in he form, Ť( r= Ť(ȃ γ Ť( r ȃ γ, from he equaliy (5, we have â,γ x = ( ˆ ( ˆ γ Ψ Ť a γ rm r Ť r a γ x dr ( ˆ Ť a γ ( ˆ Ť a γ ( ˆ Ť a γ aˆ rm r Ť(( r aˆ γ drd ( Bu (, x γ ˆ a γ ( aˆ γ rm r Ť(( r aˆ γ dr (Gx( rm ( r Ť(( r dr â γ ( σ hen, from he compacne of Ť(ȃ γ, we obain ha he e Ψ ȃ,γ x(, x B δ i relaively compac in X for every ȃ uch ha < ȃ <. Moreover, for x B δ, we can eaily prove ha Ψ ȃ,γ x( i convergen o Ψ x( in B δ, a ȃ and γ, hence he e Ü ( = { Ψ x, x B δ } i relaively compac in B δ. Lemma (3.11: If he aumpion (a (e are hold. hen Ψ i a compleely coninuou. rom lemma (3.7 and (3.9, for each x B δ he operaor Ψ i uniformly bounded and he e { Ψ x, [, ]} i equiconinuou by Appling he Arzela Acoli heorem, i reul ha for x B δ, he e { Ψ x, [, ]} i relaively compac. we obain ha Ψ i a compleely coninuou. heorem (3.1: If he aumpion (a-(e are aified. hen for each >, he conrol yem (1 ha a mild oluion on [, ], provided ha γ =, K 1 ((1 < 1. or any > and for any x, y B δ, ( Ψ 1 y( ( Ψ x( B δ, for [,]. By uing lemma (3.6 and (3.11 wih applying Kranoelkii fixed poin heorem, we conclude ha he operaor Ψ ha a fixed poin, which give rie o mild oluion of yem (1 wih ochaic conrol funcion given in (. hi complee he proof. heorem (3.: If he aumpion (a (e are aified. hen he ochaic conrol yem (1 i approximaely conrollable on [,]. or every >, le x be a fixed poin of he operaor Ψ in he pace C 1 [, ]; L Ω, X, which i a mild oluion under he ochaic conrol funcion in ( of he ochaic conrol yem (1. hen, we have = ( ( R (,Г aˆ R (,Г Ť x x R,Г Ex Ť x d, x (, h (,r, x ( r ( ( ( R,Г Ť G, x (, g(v, x (v (v R (,Г Ť ( ( σ (5 By uing he aumpion (c he funcion i uniformly bounded, uch ha E (, x, h(,, xd D 1, for any. hen, for all (,ω [, ] Ω, here i a ubequence of equence, x (, h, r, x (rdr denoed by, x (, h, r, x (rdr which i weakly converging o ay ( in X. Alo, here i a ubequence of equence G, x (, g(v, x (v (v denoed by G, x (, g(v, x (v (v which i weakly converging o ay G( in L (K, X. On he oher hand, from he aumpion (b, for all < he operaor R(, Г Srongly a and R(, Г 1. E x x ( ( 8E R,Г Ex Ť x ( φ 8E R,Г ( ˆ 8E R,Г ( ( ( R,Г Ť 8E d [, x (, h (,r, x ( r ( ] ( 8E R,Г Ť d

9 American Journal of Applied Mahemaic and Saiic 176 ( ( ( R, Г Ť 8E G, x (v G( (, g(v, x (v ( By uing he Lebegue dominaed convergence heorem, we obain E x ( x a. hi give he approximae conrollabiliy. Reference [1] Abid S.., aan S. Q. and Quaez U. J. Approximae conrollabiliy of racional Sochaic Inegro-Differenial Equaion Driven by Mixed racional Brownian Moion, American Journal of Mahemaic and Saiic 15, Vo., PP:7-81, 15. [] Ahmed M. amdy, Approximae Conrollabiliy Of Impulive eural Sochaic Differenial Equaion Wih racional Brownian Moion in A ilber Space, Advance in Difference Equaion, Springer Open Journal, 14:113, 14. [3] Balachandran K., Kiruhika S. and rujillo J. On racional Impulive Equaion Of Sobolev ype wih onlocal Condiion in Banach Space, Compuer and Mahemaic wih Applicaion, o. 6, PP: , 11. [4] Engel K. J, and agel R. One Parameer Semigroup or Linear Evoluion Equaion, Springer-Verlag, ew York, Berlin,. [5] Gani J., eyde C.C., Jager P. and Kurz.G., Probabiliy and I Applicaion, Springer-Verlag London Limied, 8. [6] Grippenberg, G. and orro I., On he Predicion Of racional Brownian Moion, Journal of Applied Probabiliy, Vol. 33, o.,pp: 4-41, [7] Guendouzi. and Idrii S., Approximae Conrollabiliy of racional Sochaic uncional Evoluion Equaion Driven By A racional Brownian Moion, Romai J., Vo.8, o.,pp:13-117, 1. [8] Kerboua M., Debbouche A. and Baleanu D., Approximae Conrollabiliy Of Sobolev ype racional Sochaic onlocal onlinear Differenial Equaion in ilber Space, Elecronic Journal Of Qualiaive heory of Differenial Equaion, o. 58, PP: 1-16, 14. [9] Li C., Qian D. and Chen Y., On Riemann- Liouville and Capuo Derivaive, indawi Publihing Corporaion, Vol. 11, Aricle ID 56494, 15 page.11. [1] Li K., Sochaic Delay racional Evoluion Equaion Driven By racional Brownian Moion, Mahemaical Mehod in he Applied Science, 14. [11] Maden enrik, io inegral, 6. [1] Mahmudov. and Zorlu S., Approximae Conrollabiliy Of racional Inegro-Differenial Equaion Involving onlocal Iniial Condiion, Boundary Value Problem, Springer Open Journal, 13:118, 13. [13] Mahmudov., Conrollabiliy of Linear Sochaic Syem in ilber Space, Journal of Mahemaical Analyi and Applicaion Vo. 59, PP: 64-8, 1. [14] Mihura Y. S., Sochaic Calculu for racional Brownian Moion and Relaed Procee, Lec,oe in Mah., 199, Springer, 8. [15] ourdin I.., Selec Apec of fracional Brownian Moion, Springer-Verlag Ialia, 1. [16] ualar D., racional Brownian moion: ochaic calculu and Applicaion, Proceeding of he Inernaional Congre of Mahemaician, Madrid, Spain, European Mahemaical Sociey, 6. [17] Pazy, A., Semigroup of Linear Operaor and Applicaion o Parial Differenial Equaion, Springer-Verlag, ew York, [18] Podlubny I., racional Differenial Equaion, Academic Pre, San Diego. California, USA, [19] Sakhivel R., Approximae Conrollabiliy Of Impulive Sochaic Evoluion Equaion, unkcialaj Ekvacioj, Vol. 5(9, PP: , 9. [] Zhou Y., Wang J. and eckan M. Conrollabiliy Of Sobolev ype racional Evoluion yem, Dynamic of PDE, Vol. 11, o. 1, PP: 71-87, 14.