Approximate controllability of stochastic functional differential inclusions of Sobolev- type with unbounded delay in Hilbert space

Transcription

1 Global Journal of Pure and Applied Mahemaics. ISSN Volume 13, Number 9 (17), pp Research India Publicaions hp:// Approximae conrollabiliy of sochasic funcional differenial inclusions of Sobolev- ype wih unbounded delay in Hilber space R. Nirmalkumar 1 and R. Murugesu Deparmen of Mahemaics, SRMV College of Ars and Science, Coimbaore 641, Tamilnadu, India, Absrac In his paper, we consider a class of approximae conrollabiliy of sochasic funcional differenial inclusions of Sobolev ype wih unbounded delay in Hilber spaces. Using he semigroup heory and fixed poin heorem, a se of sufficien condiions is obained for he required resul of approximae conrollabiliy of sochasic funcional differenial inclusions of Sobolev ype wih unbounded delay. Finally, an example is provided o illusrae he obained resul. Keywords: Approximae conrollabiliy, Sochasic Sobolev-ype differenial inclusion, Fixed poin heorem, Unbounded delay. 1 Mahemaics Subjec Classificaion: 6A33, 34G, 93G5. 1. INTRODUCTION Differenial inclusions have wide applicaions in science, engineering, economics and in opimal conrol heory. Many auhors sudied he exisence, conrollabiliy and sabiliy of differenial inclusions [1-5, 14,, 3, 35-39]. Conrollabiliy is one of he elemenary conceps in mahemaical conrol heory, which plays a vial role in boh engineering and sciences. Conrollabiliy generally means ha i is possible o seer dynamical conrol sysems from an arbirary iniial sae o an arbirary final sae using

2 5914 R. Nirmalkumar and R. Murugesu he se of admissible conrols. There are wo basic heories of conrollabiliy can be idenified as approximae conrollabiliy and exac conrollabiliy. Mos of he crieria, which can be me in he lieraure, are formulaed for finie dimensional sysem. Bu in he infinie dimensional sysem, many unsolved problems are sill exis as for as conrollabiliy is concerned. In he case of infinie dimensional sysem, conrollabiliy can be disinguished as approximae and exac conrollabiliy. Approximae conrollabiliy means ha he sysem can be governed o arbirary small neighborhood of final sae whereas exac conrollabiliy allows o govern he sysem o arbirary final sae. In oher words he approximae conrollabiliy gives he possibiliy of governing he sysem of saes which forms a dense subspace in he sae space. Recenly, Mahmodev [18] e.al, sudied he approximae conrollabiliy of second order neural sochasic evoluion equaions using semi group mehods ogeher wih Banach fixed poin heorem. In [8], Henri quez sudied he exisence of soluions of nonauonomous second order funcional differenial equaions wih infinie delay by using Leray Schauder alernaive fixed poin heorem. In [37], Yan sudied he approximae conrollabiliy of fracional neural inegrodifferenial inclusions wih sae dependen delay in Hilber spaces and in [33] Vijayakumar e.al, discuss he approximae conrollabiliy for a class of fracional neural inegrodifferenial inclusions wih sae dependen delay using Dhage fixed poin heorem. In [7], Gudenowzi invesigaed he approximae conrollabliy for a class of fracional neural sochasic funcional inegrodifferenial inclusions using Bohnenblus-Karlin fixed poin heorem. The sochasic differenial equaions have araced grea ineres due o is applicaions in science, engineering and medical sciences. In recen years, he conrollabiliy problems of sochasic differenial equaions become a field of increasing ineres [9, 15, 17, 19]. The exisence of deerminisic conrollabiliy conceps o sochasic conrol sysems have been discussed only in limied number of publicaions. More precisely, here are less number of papers in he approximae conrollabiliy of non-linear sochasic sysems [7, 16, 3, 4, 8-31, 4]. Klamka [11,1], sudied sochasic relaive exac and approximae conrollabiliy problem for finie dimensional linear saionary dynamical sysem wih single ime-variable poin delay using open-mapping heorem. A se of necessary and sufficien condiions are esablished for he exac and sochasic conrollabiliy of linear sysem wih sae delays in [1]. In [1], Revahi e.al, sudied he exisence of sochasic funcional differenial equaions of Sobolev ype wih infinie delay. In recen years, conrollabiliy problems for various ypes of nonlinear dynamical sysems in he infinie dimensional spaces by using differen kinds of approaches have been considered in many publicaion [6, 15, 17, 18,, 3, 5-7, 34]. The approximae conrollabiliy problem for a nonlinear sochasic sysems of Sobolev ype in Hilber space has no been invesigaed largely. Moivaed by his consideraion, in his paper, we will sudy he approximae conrollabiliy problem for nonlinear

3 Approximae conrollabiliy of sochasic funcional differenial inclusions 5915 sochasic funcional differenial inclusions of Sobolev ype wih unbounded delay in Hilber space which are naural generalizaion of conrollabiliy conceps in he heory of infinie dimensional deerminisic conrol sysems. The paper is organized as follows: in secion, we recall some basic definiions, noaions, lemmas and some preliminary facs. In secion 3, we sudy he approximae conrollabiliy of sochasic funcional differenial inclusions of Sobolev-ype wih unbounded delay in Hilber spaces. An applicaion of our heoreical resuls is given in Secion 4.. PRELIMINARIES In his secion, he basic preliminaries, definiions, lemmas, noaions, mulivalued maps and some resuls which are needed o esablish our main resuls are discussed. Le(H, H )and (K, k )be wo real separable Hilber spaces and for convenience, we use he same noaion o denoe he norms in H and K and, o denoe he inner produc space wihou any confusion. Le L(K, H) be space of bounded linear operaors from K inoh. Le (Ω, F, {F }, P) be a complee filered probabiliy space saisfying ha F conains all P-null ses of F. Le {w(), } represens a Q Wiener process defined on (Ω, F, {F }, P)wih he co-variance operaor Q such ha Tr(Q) <. Furher, we assume ha here exiss a complee orhonormal sysem {e k } k 1 in K, a bounded sequence of nonnegaive real numbers λ k such ha Qe k = λ k e k, k=1,,... and sequence of independen Wiener processes such ha {β k } k 1 such ha w(), e K = λ k e k, e K β k (),. k=1 Le L = L (Q 1, H) be he space of all Hilber- Schmid operaors from Q 1 KoHwih he inner produc φ, ψ L = Tr[φQψ ]. In his paper, we invesigae he approximae conrollabiliy of sochasic funcional differenial inclusions of Sobolev -ype wih unbounded delay in he following form d[lx()] d ε Ax() + Bu() + F(, x ) + (s, x s )dw(s), J [, b] (.1) x() = φ() B h, (-, ] (.) where he sae x( ) akes he values in he separable real Hilber spaces H, A and Lare linear operaors on H. The hisories x (, ] B h, x (θ) = x( + θ) for belongs o he phase space B h, which will be defined laer. The iniial daa φ =

4 5916 R. Nirmalkumar and R. Murugesu {φ(), (-, ]} is an F -measurable,b h -valued sochasic process independen of W wih finie second momens. Furher F: J x B h H and : J x B h L (K, H)are appropriae mappings specified laer and he conrol funcion u(. ) is given in L(J, U), a Hilber space of admissible conrol funcions wih U as Hilber space. B is a bounded linear operaor from U ino H. The operaors A: D(A) H H and condiions: L: D(L) H H saisfy he following (A1)A and L are closed linear operaors. (A)D(L) D(A) and L is bijecive. (A3)L 1 : H D(L) is coninuous. Furher, from (A1) and (A), L 1 is closed and wih (A3) by using he closed graph heorem, we obain he boundedness of he linear operaor AL 1 : H H. Furher AL 1 generaes a srongly coninuous semigroup {T()} in H. Le us denoe max J T() = M, L 1 = M L. Definiion.1.(Phase space). Assume ha h: (, ] (, ) is a coninuous funcion wih l = h()d < + and φ is a F - measurable funcions mappings from(, ] ino H.Define he phase space B h by B h = {φ: (, ] H, for any a >, (E φ(θ) ) 1 is a bounded and measurable funcion on [ a, ] wih φ() = and h(s) sup s θ ((E φ(θ) ) 1 ) ds} <. If B h is endowed wih he norm φ Bh = h(s) sup s θ ((E φ(θ) ) 1 ) ds, φ B h, hen (B h,. Bh ) is a Banach space. Now we consider he space of B h = {x: x (, b] H)such ha x J C(J, H), x = φ B h }. Se. b be a seminorm defined by x b = φ Bh + sup s [,b] (E x(s) ) 1, x ε B h. Lemma.. Assume ha x B h, hen for all J, x B h. Moreover

5 Approximae conrollabiliy of sochasic funcional differenial inclusions 5917 where l = l(e φ(θ) ) 1 l sup sε[,] (E x(s) ) 1 + φ Bh, h(s)ds <. Definiion.3. A mulivalued map G: H H \{ } is convex(closed) valued if G(x) is convex(closed) for all x H. Gis bounded on bounded ses if G(B) = U x B G(x) is bounded in H for any bounded se B of H i.e., sup x B {sup{ y : y G(x)}} <. Definiion.4. Gis called upper semiconinuous (u.s.c for shor) on H, if for each x H, he seg(x ) is a nonempy closed subse of Hand if for each open se Nof H conaining G(x ), here exis an open neighborhood Vofx such ha G(V) N. Definiion.5. The muli-valued operaor G is called compac if G(H) is a compac subse of H. Gis called compleely coninuous if G(B)is relaively compac for every bounded subse Bof H. For more deails on Mulivalued maps, see he books of Deimling (199), Hu and Papageorgiou (1997). If he mulivalued map G is compleely coninuous wih nonempy values, hen G is u.s.c., if and only if G has a closed graph, i.e., x n x,y n y,y n G(x n ) imply y G(x ). Ghas a fixed poin if here is a x H such ha x G(x). In he following, BCC(H) denoes he se of all nonempy, bounded, closed and convex subse of H. Definiion.6. A mulivalued map G: J BCC(H)is said o be measurable if, for each x H, he funcion v: J R, defined byv() = d(x, Gx()) = inf { x z : z G()} belongs o L 1 (J, R). Definiion.7. The mulivalued map Σ J H BCC(H) is said o be L - Caraheodory if (i) (ii) (iii) Σ(, x) is measurable for each x H, x Σ(, x) is upper coninuous for almos all J, for each r >, here exiss l r L 1 (J, R) such ha Σ(, x) = sup {E σ : σ Σ(, x)} l r () for almos all J and x r. Lemma.8.([13],Lasoa and Opial). Le J be a compac real inerval, BCC(H) be he se of all nonempy, bounded, closed and convex subse of H and Σ be a mulivalued map S Σ,x and le Γ be a linear combinaion mapping from L (J, H) o C(J, H) hen, he operaor Γ os Σ : C BCC(C(J, H)), x (Γ os Σ )(x) = Γ(S Σ,x ),

6 5918 R. Nirmalkumar and R. Murugesu is a closed graph operaor in C C, where S Σ,x is known as he seleced operaor se from Σ, is given by σ S Σ,x = {σ L (L(K, H)): σ() Σ(, x) for a.e J}. Lemma.9(Bohnenblus-Karlin). Le D be a nonempy subse of H, which is bounded, closed and convex. Suppose G: D H \{ } is u.s.c wih closed, convex values and such ha G(D) is compac. Then G has a fixed poin. Definiion.1. A coninuous H- valued process x is said o be a mild soluion of (.1)-(.) if (i) (ii) x() is F - adaped and {x : [, b]} is B h -valued. for each J, x() saisfies he following inegral equaion: x() = L 1 T()Lφ() + L 1 T( s)f(s, x s )ds + L 1 T( s)bu(s)ds (iii) + L 1 T( s)σ(s, x s )dw(s), J x() = φ() on (, ] saisfying φ Bh <. 3. APPROXIMATE CONTROLLABILITY RESULTS In his secion, we shall formulae and prove sufficien condiions for he approximae conrollabiliy for a class of sochasic differenial inclusion of Sobolev ype wih unbounded delay of he form (.1)-(.) by using Bohnenblus-Karlin fixed poin heorem. Firs we prove he exisence of soluions for he conrol sysem and hen show ha under cerain assumpions, he approximae conrollabiliy of he sochasic conrol sysem (.1)-(.) is implied by he approximae conrollabiliy of he associaed linear par. Definiion 3.1. Le x b (φ, u)be he sae value of (.1)-(.) a he erminal ime b corresponding o he conrol u and he iniial value φ. Inroduce he se R(b, φ) = {x b (φ; u)(): u(. ) L(J, U)}, which is called he reachable se of (.1)-(.) a he ime b and is closure in H is denoed by R(b, ). φ The sysem (.1)-(.) is said o be approximaely conrollable on J if ) R(b, φ = H.

7 Approximae conrollabiliy of sochasic funcional differenial inclusions 5919 Inorder o sudy he approximae conrollabiliy of he sysem (.1)-(.), we consider he linear sysem d[lx()] d εax() + Bu(), [, b] (3.1) x() = φ() B h (3.) I is convienen a his poin o inroduce he conrollabiliy and relevan operaors associaed wih (3.1)-(3.), b γ b = L 1 T(b s)bb L 1 T (b s)ds: H H, R(α, γ b ) = (αi + γ b ) 1 : H H where B denoes he adjoin of B and T () is he adjoin of T(). I is sraigh forward ha he operaor γ b is a linear bounded operaor. In order o esablish he resul, we need he following hypoheses: (H1)T(), > is compac. (H) The funcion F: J B h H saisfies he following:f(, ψ): J H measurable for is each ψ B h and F(, ): B h H is coninuous for a.e J and for ψ B h, F(, ): J H is srongly measurable and here exiss a consan M f > such ha E F(, ψ) M f ( ψ Bh ). (H3) The mulivalued map Σ: J B h BCC(x) is an L caraheodory funcion which saisfies he following condiions: (i) For each J, he funcion Σ(,. ) is u.s.c and for x B h, he funcion Σ(, ψ) is measurable. And for each fixed ψ B h, he se is nonempy. S Σ,x = {σ L (L(K, H)): σ() Σ(, ψ) for a.e J} (ii) For each posiive number r here exiss a posiive funcion l r : J R + such ha

8 59 R. Nirmalkumar and R. Murugesu sup{e σ : σ() Σ(, ψ)} l r () a.e J. and lim inf l r(s)ds = Λ < r r Lemma 3.. For any x b L (F b, H), here exiss φ L F (Ω, L (J, L(K, H))) b such ha x b = Ex b + φ (s)dw(s). Now for any α >, x b L (F b, H)and for σ S Σ,ψ, we define he conrol funcion u α (, x) = B L 1 T (b s)r(α, γ b ) {Ex b + φ (s)dw(s) L 1 T()Lφ() L 1 T( s)f(s, y s + φ s)ds + L 1 T( s) σ(s)dw(s)ds} (s)ds Theorem 3.3. Suppose ha he hypoheses (H1)-(H3) are saisfied, hen he sysem (.1)-(.) has a mild soluion on J provided ha 1M L M l b M f [1 + 5 ( M L M M B ) ] + 3M M L Tr(Q)Λ [5 ( M M L M B ) + 1] < 1 (3.3) and where B = M B. α α Proof. For any ε >, we consider he operaor Φ ε : B h B h defined by Φ ε x he se of z B h such ha z() = { φ() (, ] L 1 T()Lφ() + L 1 T( s)f(s, y s + φ s)ds + L 1 T( s)bu α (s, y s + φ s)ds + L 1 T( s)σ(s)dw(s) ds, J

9 Approximae conrollabiliy of sochasic funcional differenial inclusions 591 where σ S Σ,x. We shall show ha he operaor Φ ε has a fixed poin, which is hen a soluion of (.1)-(.). Clearly x 1 = x(b) (Φ ε x)(b), which means ha u α (, x) seers sysem (.1)-(.) from x o x b in finie ime b. For φ B h, we define φ by φ(), (, ] φ () = { L 1 T()Lφ() J hen φ B h. Le x() = y() + φ (), < b. I is easy o see ha y saisfies y = and y() = L 1 T( s)f(s, y s + φ s)ds + L 1 T( s)σ(s)dw(s) ds, J if and only if x saisfies + L 1 T( s)bu α (s, y s + φ s)ds x() = L 1 T()Lφ() + L 1 T( s)f(s, y s + φ s)ds + L 1 T(b )BB L 1 T (b s) R(α, γ b ) {Ex b + φ (s)dw(s) L 1 T()Lφ() L 1 T( s)f(s, y s + φ s)ds + L 1 T( s)σ(s)dw(s) ds and x() = φ(), (, ]. Le B h = {y B h : y = B h }. For any y B h, we have } (s)ds + L 1 T( s)σ(s)dw(s)ds, J y b = y Bh + sup s [,b] {(E y(s) ): s b} = sup s [,b] {(E y(s) ): s b}. hus (B h, b ) is a Banach space. Se B r = {y B h : y b r} for some r >, hen B r B h is a uniformly bounded and for y B r,from Lemma. we have y + φ Bh ( y Bh + φ Bh ) 4 (l sup s [,] (E y(s) ) + y Bh + l sup s [,] (E φ (s) ) + φ Bh ) 4l (r + M E φ() ) + 4 φ Bh r Define he mulivalued map ψ: B h B h defined by ψy he se of z B h and here exis σ L (L(K, H)) such ha σ S Σ,x and

10 59 R. Nirmalkumar and R. Murugesu z () = (, ] L 1 T( s)f(s, y s + φ s)ds + L 1 T( s)bu α (s, y s + φ s)ds + L 1 T( s)σ(s)dw(s)ds, J { Obviously, he operaor Φ ε has a fixed poin if and only if ψ has a fixed poin. So our aim is o show ha ψ has a fixed poin. For he sake of convience, we subdivide he proof ino several seps. Sep 1:ψis convex for each x B r. In fac, if φ 1, φ belongs o ψ(x), hen here exiss σ 1, σ S Σ,x such ha for each J, we have φ() = L 1 T( s)f(s, y s + φ s)ds + L 1 T(b )BB L 1 T (b s)r(α, γ b ) {Ex b + φ (s)dw(s) L 1 T()Lφ() L 1 T( s)f(s, y s + φ s)ds + L 1 T( s) σ(s)dw(s) Le λ [,1]. Then for each J, we ge (λφ 1 + (1 λ)φ )() = L 1 T( s)f(s, y s + φ s)ds + L 1 T(b )BB L 1 T (b s)r(α, γ b ) ds} (s)ds + L 1 T( s) σ(s)dw(s) ds {Ex b + φ (s)dw(s) L 1 T()Lφ() L 1 T( s)f(s, y s + φ s)ds + L 1 T( s)[λσ 1 (s) + (1 λ)σ (s)]dw(s) ds} (s)ds + L 1 T( s)[λσ 1 (s) + (1 λ)σ (s)]dw(s) ds I is easy o see ha S Σ,x is convex since Σ has convex values. Soλσ 1 (s) + (1 λ)σ (s) S Σ,x. Thus λφ 1 + (1 λ)φ ψ(x). Sep :We show ha here exis some r > such ha ψ(b r ) B r. if i is no rue, hen here exiss ε > such ha for every posiive number r and J, here exiss a

11 Approximae conrollabiliy of sochasic funcional differenial inclusions 593 funcion y r B r, bu ψ B r, ha is (ψ(y r ))() rfor some J. For such ε >,elemenary inequaliy can show ha r < E (ψy r )() 3 {E L 1 T( s)f(s, y s + φ s)ds 3 {E L 1 T( s)f(s, y s + φ s)ds + E L 1 T( s)bu α (s, y s + φ s)ds + E L 1 T( s)σ(s)dw(s)ds } + E L 1 T( s)bb T (b s)r(α, γ b ) {Ex b + φ (s)dw(s) L 1 T()Lφ() L 1 T( s)f(s, y s + φ s)ds + L 1 T( s) σ(s)dw(s) 3M L M b M f (r ) + 3 {5 ( M L M M B ds }+ E L 1 T( s)σ(s)dw(s)ds M L M E φ() + M L M b M f (r ) + M L M TrQ l r (s)ds}} + 3 M L M TrQ l r (s)ds α ) {E x b + E φ (s)dw(s) Dividing boh sides of he above inequaliy by r and aking r we have 1M L M l b M f [1 + 5 ( M L M M B ) ] + 3M M α L Tr(Q)Λ [5 ( M M L M B ) + 1] 1 α which is a conradicion o our assumpion. Hence, for some posiive number r > and some σ S Σ,x, ψ(b r ) B r. Sep 3:ψ(B r ) is equiconinuous. Indeed ε > be small, τ 1 τ b. for each y B r and z belongs o ψ 1 y, here exiss σ S Σ,x such ha for each J, we have E z (τ ) z (τ 1 ) = 9 {E τ L 1 T(τ s)f(s, y s + φ s)ds τ 1 τ 1 τ 1 ε + E L 1 [T(τ s) T(τ 1 s)]f(s, y s + φ s)ds τ 1 ε + E L 1 [T(τ s) T(τ 1 s)f(s, y s + φ s)ds

12 594 R. Nirmalkumar and R. Murugesu τ + E L 1 T(τ s)bu r α (η, s)dηds τ 1 τ 1 τ 1 ε + E L 1 [T(τ s) T(τ 1 s)]bu α r (η, s)dηds τ 1 ε + E L 1 [T(τ s) T(τ 1 s)]bu α r (η, s)dηds τ + E L 1 T(τ s)σ(s)dw(s)ds τ 1 τ 1 τ 1 ε + E L 1 [T(τ s) T(τ 1 s)]σ(s)dw(s)ds τ 1 ε + E L 1 [T(τ s) T(τ 1 s)]σ(s)dw(s)ds } τ 9 {M M L E f(s, y s + φ s) ds τ 1 + M τ 1 L T(τ s) T(τ 1 s) E f(s, y s + φ s) ds τ 1 ε + M τ 1 ε L T(τ s) T(τ 1 s) E f(s, y s + φ s) ds + M M τ L Bu r α (η, s) dηds τ 1 + M τ L T(τ s) T(τ 1 s) 1 E Bu r α (η, s) dη ds τ 1 ε + M τ 1 ε L T(τ s) T(τ 1 s) E Bu r α (η, s) dη ds τ + M M L Tr(Q)E σ(s)dw(s) ds τ 1 + M τ 1 L T(τ s) T(τ 1 s) Tr(Q)E σ(s)dw(s) ds τ 1 ε τ 1 ε + M L T(τ s) T(τ 1 s) Tr(Q)E σ(s)dw(s) ds} Therefore for ε sufficienly small, we can verify ha he righ-hand side of he above inequaliy ends o zero as τ τ 1. On he oherhand, he compacness of T() for > implies he coninuiy in he uniform operaor opology. Thus ψ maps B r ino equiconinuous family of funcions. Sep 4: The se () = {φ(): φ ψ(b r )} is relaively compac in H. Le [, b] be fixed and ε a real number saisfying < ε <. For x B r, we

13 Approximae conrollabiliy of sochasic funcional differenial inclusions 595 define ε φ ε () = L 1 T( s)f(s, y s + φ s)ds + ε L 1 T(b )BB L 1 T (b s)r(α, γ b ) {Ex b + φ (s)dw(s) L 1 T()Lφ() L 1 T( s)f(s, y s + φ s)ds + L 1 T( s)σ(s)dw(s) ds} (s)ds + ε φ ε () = T(ε) L 1 T( s ε)f(s, y s + φ s) ε ε L 1 T( s)σ(s)dw(s) + T(ε) L 1 T( s ε)bb L 1 T (b s)r(α, γ b ) {Ex b + φ (s)dw(s) L 1 T()Lφ() L 1 T( s)f(s, y s + φ s)ds b + L 1 T( s)σ(s)dw(s) ds} (s)ds + T(ε) b ds ε L 1 T( s ε)σ(s)dw(s) for σ S Σ,x. SinceT() is a compac operaor, he se ε () = {φ ε (): φ ε ψ(b r )} is relaively compac in H for each ε, < ε <. Moreover, for each < ε <, we have E φ() φ ε () Therefore ε L 1 T( s)f(s, y s + φ s)ds + E L 1 T( s)bb T (b s)r(α, γ b ) ϵ {Ex b + φ (s)dw(s) L 1 T()Lφ() L 1 T( s)f(s, y s + φ s)ds + L 1 T( s) σ(s)dw(s) ds } (s)ds+ ε φ() φ ε () as α + L 1 T( s)σ(s)dw(s)ds Hence here exiss relaively compac ses arbirarily close o he se () = {φ(): φ ψ(b r )} and he se () is relaively compac in H for all [, b]. Since i is compac a =, hence () is relaively compac in H for all [, b]. Sep 5. ψ has a closed graph. Le y n y as n, z n ψy n for each y n B r and z n z as n. We shall show ha z ψy. Since z n ψy n, here exiss a σ n ds

14 596 R. Nirmalkumar and R. Murugesu S Σ,yn such ha z n() = L 1 T( s)f(s, (y n ) s + φ s)ds + L 1 T(b )BB L 1 T (b s)r(α, γ b ) {Ex b + φ (s)dw(s) L 1 T(b)Lφ() L 1 T( s)f(s, (y n ) s + φ s)ds + L 1 T( s)σ n (s)dw(s) we mus prove ha here exiss σ S Σ,y such ha z () = L 1 T( s)f(s, (y ) s + φ s)ds ds} (s)ds + L 1 T( s)σ n (s)dw(s) ds, + L 1 T(b )BB L 1 T (b s)r(α, γ b ) {Ex b + φ (s)dw(s) L 1 T(b)Lφ() L 1 T( s)f(s, (y ) s + φ s)ds + L 1 T( s)σ (s)dw(s) ds} (s)ds + L 1T( s)σ (s)dw(s) ds, Now, for every J, since g is coninuous and he from he definiion of u ε we ge (z n() L 1 T( s)f(s, (y n ) s + φ s)ds + L 1 T(b )BB L 1 T (b s)r(α, γ b ) J J {Ex b + φ (s)dw(s) L 1 T(b)Lφ() L 1 T( s)f(s, (y n ) s + φ s)ds + L 1 T( s)σ n (s)dw(s) ds} (s)ds + L 1 T( s)σ n (s)dw(s) ds) (z () L 1 T( s)f(s, (y ) s + φ s)ds + L 1 T(b )BB L 1 T (b s)r(α, γ b )

15 Approximae conrollabiliy of sochasic funcional differenial inclusions 597 {Ex b + φ (s)dw(s) L 1 T(b)Lφ() L 1 T( s)f(s, (y ) s + φ s)ds + L 1 T( s)σ (s)dw(s) ds} (s)ds + L 1T( s)σ (s)dw(s) ds) as n. Consider he linear coninuous operaor κ: L 1 (J H) C(J H), (κσ)() = L 1 T( s)σ(s)dw(s) ds L 1 T(b )BB L 1 T (b s)r(α, γ b ) L 1 T( s)σ(s)dw(s) ds From Lemma.8 i follows ha κos Σ is a closed graph operaor. Also, from he definiion of κ, we have ha (z n() L 1 T( s)f(s, (y n ) s + φ s)ds + L 1 T(b )BB L 1 T (b s)r(α, γ b ) {Ex b + φ (s)dw(s) L 1 T(b)Lφ() L 1 T( s)f(s, (y n ) s + φ s)ds + L 1 T( s)σ n (s)dw(s) κ(s Σ,yn ) ds} (s)ds + L 1 T( s)σ n (s)dw(s) ds) Since y n y for some y S Σ,y, i follows from Lemma.8 ha (z () L 1 T( s)f(s, (y ) s + φ s)ds + L 1 T(b )BB L 1 T (b s)r(α, γ b ) {Ex b + φ (s)dw(s) L 1 T(b)Lφ() L 1 T( s)f(s, (y ) s + φ s)ds + L 1 T( s)σ (s)dw(s) for some σ S Σ,y. Therefore ψ has a closed graph. ds} (s)ds + L 1 T( s)σ (s)dw(s) ds)

16 598 R. Nirmalkumar and R. Murugesu As a consequences of sep 1 o sep 5 ogeher wih he Arzela-Ascoli heorem, we conclude ha ψ is a compac mulivalued map, u.s.c wih convex closed values. As a consequences of Lemma.8, we can deduce ha ψ has a fixed poin x which is a mild soluion of (.1)-(.). Furher, in order o prove he approximae conrollabiliy resul, he following addiional assumpion is required. (H4) The linear inclusion (3.3)-(3.4) is approximaely conrollable. (H5) αr(α, γ b ) = α(αi + γ b ) 1 as α + in he srong operaor opology. Theorem 3.4.Assume ha he assumpion of Theorem 3.3 hold and in addiion, hypohesis (H1)-(H5) are saisfied and hen he nonlinear sochasic differenial inclusion (.1)-(.) is approximaely conrollable on J. Proof. Le x α( ) be a fixed poin of Φ ε in B r. By Theorem 3.3 any fixed poin Φ ε is a mild soluion of (.1)-(.) under he conrol x α(b) = x b R(α, γ b ) {Ex b + φ (s)dw(s) L 1 T()Lφ() L 1 T( s)f(s, y s + φ s)ds + L 1 T( s)σ(s)dw(s) ds} Moreover by assumpion on σand Dunford-Peis Theorem, we have ha he {σ α (s)} is weakly compac in L(J, H), so here is a subsequence, sill denoed by {σ α (s)}, ha converges weakly o σ(s) say in L 1 (J, H). Now we have E x α(b) x b = 5E R(α, γ b ) {Ex b + φ (s)dw(s) L 1 T()Lφ() L 1 T( s)f(s, y s + φ s)ds L 1 T( s)σ(s)dw(s) ds} for s b he operaor α(αi + γ b ) 1 srongly as α + and moreover α(αi + γ b ) 1 1. I follows from Lebesgue dominaed convergence heorem and he compacnesss of T() ha E x α(b) x b asα +. This proves he approximae conrollabiliy of he differenial inclusion (.1)-(.).

17 Approximae conrollabiliy of sochasic funcional differenial inclusions AN APPLICATION: Consider a conrol sysem of sochasic differenial inclusion wih unbounded delay of he form [z(, x)] ε z xx(, x) + F(, z( r), x)ds + μ(, x) +G(, y( r), x)dw(), [,1], r >, x [,1] (4.1) z(, ) = z(, 1) =, 1 (4.) z(, x) = φ(, x), x 1, (4.3) where w() denoes a sandard cylindrical wiener process in Hdefined on a sochasic process (ω, F) and H = K = L ([,1]). Define he operaors A: D(A) H H and L: D(L) H H by Ay = y and Ly = y y, where each domain D(A) and D(L) is given by {y H, y, y are absoluely coninuous y H, y() = y(1) = }. Furher A and L can be wien as Ay = n=1 n y, z n z n, y D(A), Ly = n=1 (1 + n ) y, z n z n, y D(L), where z n (x) = sin nx, n = 1,,3, is he π orhogonal se of vecors of A. Also for z H, we have and and L 1 z = n z, z n z n n=1 n AL 1 z = 1 + n z, z n z n n=1 n T()z = exp 1 + n z, z n z n n=1 Furher, we consider he phase space B h, wih norm φ Bh = g(s)sup s θ (E φ() ) 1/ ds where g(s) = e s, s < and g(s)ds = 1. Le z()(x) = z(, x). Define he funcion F: J B h H andσ: J B h L Q by F(, z)( ) = F(, z( )), Σ(, y( )) = G(, y( )) and he bounded linear operaor Bu()(x) = μ(, x) respecively.

18 593 R. Nirmalkumar and R. Murugesu Moreover, i can be easily seen ha AL 1 is compac and bounded wih L 1 1 and AL 1 generaes a srongly coninuous semigroup T(), wih T() e 1. Thus wih he above choices (4.1)-(4.3) can be wrien in he absrac from of (.1)- (.). Furher, we can impose some suiable condiions on he above defined funcions o verify he assumpions on Theorem 3.4, we can conclude ha (4.1)-(4.3) is approximaely conrollable on [, b]. REFERENCES [1] Abada, N., Benchohra, M., and Hammouche, H., 9, "Exisence and conrollabiliy resuls for nondensely defined impulsive semilinear funcional differenial inclusions," Journal of DifferenialEquaion, 46(1), [] Balasubramaniam, P., Nouyas, S.K., 6, "Conrollabiliy for neural sochasic funcional differenial inclusions wih infinie delay in absrac space," Journal of Mahemaical Analysis and Applicaions, 34, [3] Balasubramaniam, P., Nouyas, S.K., and Vinayagam, D., 5, "Exisence of soluions of semilinear sochasic delay evoluion inclusions in a Hilber space," Journal of Mahemaical Analysis and Applicaions, 35, [4] Balasubramaniam, P.,Vinayagam, D., 5, "Exisence of soluions of nonlinear neural sochasic differenial inclusions in a Hilber space," Sochasic Analysis and Applicaions, 3, [5] Chang, Y.K., Nieo, J.J., 9, "Exisence of soluions for impulsive neural inegro-differenial inclusions wih nonlocal iniial condiions via fracional operaors," Numerical Funcional Analysis Opimizaion, 3, [6] Dauer, J.P.,Mahmudov, N.I., 4, "Conrollabiliy of sochasic semilinear funcional differenial sysems in Hilber spaces," Journal of Mahemaical Analysis and Applicaions, 9, [7] Guendouzi, T., Bousmaha, L., 14, "Approximae Conrollabiliy of fracional neural sochasic funcional inegrodifferenial inclusions wih infinie delay," Qualiaive heory of Dynamical Sysems, DOI 1.17/S y. [8] Henri quez, H.R.,11, "Exisence of soluions of non-auonomous second order funcional differenial equaions wih infinie delay," Nonlinear Anal,TMA, 74, [9] Klamka, J., 7, "Sochasic conrollabiliy of linear sysems wih delay in conrol," Bullein of he Polish Academy of Sciences: Technical Sciences, 55, 3-9. [1] Klamka, J., 7, "Sochasic conrollabiliy of linear sysems wih sae delays," Inernaional Journal of Applied Mahemaics and Compuer science,

19 Approximae conrollabiliy of sochasic funcional differenial inclusions , [11] Klamka, J., 8, "Sochasic conrollabiliy of sysems wih variable delay in conrol," Bullein of he Polish Academy of Sciences: Technical Sciences, [1] Klamka, J., 9, " Sochasic conrollabiliy of sysems wih muliple delays in conrol," Inernaional Journal of Applied Mahemaics and Compuer Science, 19, [13] Lasoa, A., Opial, Z., 1965, "An applicaion of he Kakuani-Ky Fan heorem in he heory of ordinary differenial equaions," Bull. Acad. Pol. Sci. Ser. Sci. Mah. Asron. Phys, 13, [14] Lin, A., Hu, L., 1, "Exisence resuls for impulsive neural sochasic funcional inegro-differenial inclusions wih nonlocal iniial condiions," Compuers&Mahemaics Applicaions, 59, [15] Mahmudov, N.I., 1, " Conrollabiliy of linear sochasic sysems in Hilber spaces,"journal of Mahemaical Analysis and Applicaions, 59, [16] Mahmudov, N.I., 3, " Approximae conrollabiliy of semilinear deerminisic and sochasic evoluion equaions in absrac spaces," SIAM Journal of Conrol Opimizaion, 4, [17] Mahmudov, N.I., Denker, A.,, "On conrollabiliy of linear sochasic sysems," Inernaional Journal of Conrol, 73, [18] Mahmudov, N.I., McKibben, M.A., 6, "Approximae conrollabiliy of secondorder neural sochasic evoluion equaions," Dynamicsof Coninuous, Discree Impulsive Sysems, 13(5), [19] Mahmudov, N.I., Zorlu, S., 3, "Conrollabiliy of nonlinear sochasic sysems," Inernaional Journal of Conrol, 76, [] Ren, Y., Hu, L., Sakhivel, R., 11, "Conrollabiliy of impulsive neural sochasic funcional differenial inclusions wih infinie delay," Journal of Compuaional and Applied Mahemaics, 35, [1] Revahi, P., Sakhivel, R., Yong Ren., 16, " Sochasic funcional differenial equaions of Sobolev-ype wih infinie delay," Saisics and Probabiliy Leers, 19, [] Rykaczewski, K., 1, "Approximae conrollabiliy of differenial inclusions in Hilber spaces," Nonlinear Analysis, 75, [3] Sakhivel, R., Ren, y, Mahmudov, N.I., 1, "Approximae conrollabiliy of second-order sochasic differenial equaions wih impulsive effecs," Modern Physics Leers B, 4, [4] Sakhivel, R., Nieo, J.J., Mahmudov, N.I., 1, "Approximae conrollabiliy of nonlinear deerminisic and sochasic sysems wih unbounded delay," Taiwanese Journal of Mahemaics, 14, [5] Sakhivel, R., Ren, Y., 13, "Approximae conrollabiliy of fracional

20 593 R. Nirmalkumar and R. Murugesu differenial equaions wih sae dependen delay," Resuls in Mahemaics, 63, [6] Sakhivel, R., Ganesh, R., Ren, Y., Anhoni, S.M., 13, "Approximae conrollabiliy of nonlinear fracional dynamical sysems," Communicaion in Nonlinear Science and Numerical Simulaion, 18, [7] Sakhivel, R., Ganesh, R., Anhoni, S.M., 13, "Approximae conrollabiliy of fracional nonlinear differenial inclusions," Applied Mahemaics and Compuaion, 5, [8] Sakhivel, R., Ganesh, R., Suganya, S., 1, "Approximae conrollabiliy of fracional neural sochasic sysem wih infinie delay," Repors on Mahemaical Physics, 7, [9] Sakhivel, R., Suganya, S., Anhoni, S.M., 1, "Approximae conrollabiliy of fracional sochasic evoluion equaions," Compuers&Mahemaics Applicaions, 63, [3] Sakhivel, R., Ren, Y., 11, "Complee conrollabiliy of sochasic evoluion equaions wih jumps", Repors on Mahemaical Physics, 68, [31] Sakhivel, R., 9, "Approximae conrollabiliy of impulsive sochasic evoluion equaions", Funkcial Ekvac, 5, [3] Vijayakumar, V., Ravichandran, C., Murugesu, R., 13, "Nonlocal conrollabiliy of mixed Volerra-Fredholm ype fracional semilinear inegrodifferenial inclusions in Banach spaces," Dynamics of Coninuous, Discree Impulsive Sysems B: Applicaion and Algorihms, (4), [33] Vijayakumar, V., Ravichandran, C., Murugesu, R., 13, "Approximae conrollabiliy for a class of fracional neural inegro-differenial inclusions wih sae dependen delay," Nonlinear Sudies, (4), [34] Vijayakumar, V., Selvakumar, A., Murugesu, R., 14, "Conrollabiliy for a class of fracional neural inegro-differenial equaions wih unbounded delay," Applied Mahemaics and Compuaion, 3, [35] Vijayakumar, V., Ravichandran, C., Murugesu, R., Trujillo, J.J., 14, "Conrollabiliy resuls for a class of fracional semilinear inegro-differenial inclusions via resolven operaors," Applied Mahemaics and Compuaion, 47, [36] Wang, J.R., Zhou, Y., 11, "Exisence and conrollabiliy resuls for fracional semilinear differenial inclusion," Nonlinear Analysis Real World Applicaions, 1, [37] Yan, Z., 1, "Approximae conrollabiliy of fracional neural inegrodifferenial inclusions wih sae-dependen delay in Hilber spaces," IMA Journal of Mahemaics and Conrol Informaion, DOI:1.193/imamci/dns33. [38] Yan, Z.,Yan, X., 13, "Exisence of soluions for a impulsive nonlocal

21 Approximae conrollabiliy of sochasic funcional differenial inclusions 5933 sochasic funcional inegrodifferenial inclusions in Hilber spaces," Zeischrif für angewande Mahemaik und Physik, 64, [39] Yan, Z., Zhang, H., 13, "Exisence of soluions o impulsive fracional parial neural sochasic inegrodifferenial inclusions wih sae-dependen delay," Elecronic Journal of Differenial Equaions, 81, 1-1. [4] Zang, Y., Li, J., 13, "Approximae conrollabiliy of fracional impulsive neural sochasic differenial equaions wih nonlocal condiions," Boundary Value Problems, 193.

22 5934 R. Nirmalkumar and R. Murugesu