Controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with delay and Poisson jumps

Size: px

Start display at page:

Download "Controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with delay and Poisson jumps"

Chrystal Anthony
5 years ago
Views:

Huan & Gao, Cogen Engineering (5, : 65585 hp://dxdoiorg/8/3396565585 SYSTEMS & CONTROL RESEARCH ARTICLE Conrollabiliy of nonlocal econd-order impulive neural ochaic funcional inegro-differenial

1 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ SYSTEMS & CONTROL RESEARCH ARTICLE Conrollabiliy of nonlocal econd-order impulive neural ochaic funcional inegro-differenial equaion wih delay and Poion jump Received: 8 March 5 Acceped: June 5 Publihed: 3 July 5 *Correponding auhor: Diem Dang Huan, Faculy of Baic Science, Bacgiang Agriculure and Forery Univeriy, Bacgiang, Vienam; Vienam Naional Univeriy, Hanoi, 44 Xuan Thuy Sree, Cau Giay, Hanoi, Vienam huandd@bafueduvn Reviewing edior: Jame Lam, Univeriy of Hong Kong, Hong Kong Addiional informaion i available a he end of he aricle Diem Dang Huan, * and Hongjun Gao 3 Abrac: The curren paper i concerned wih he conrollabiliy of nonlocal econdorder impulive neural ochaic funcional inegro-differenial equaion wih infinie delay and Poion jump in Hilber pace Uing he heory of a rongly coninuou coine family of bounded linear operaor, ochaic analyi heory and wih he help of he Banach fixed poin heorem, we derive a new e of ufficien condiion for he conrollabiliy of nonlocal econd-order impulive neural ochaic funcional inegro-differenial equaion wih infinie delay and Poion jump Finally, an applicaion o he ochaic nonlinear wave equaion wih infinie delay and Poion jump i given Subjec: Non-Linear Syem; Probabiliy Theory & Applicaion; Sochaic Model & Procee Keyword: conrollabiliy; impulive neural ochaic inegro-differenial equaion; Poion jump; coine funcion of operaor; infinie delay; Banach fixed poin heorem Mahemaic Subjec claificaion: 34A37; 93B5; 93E3; 6H; 34K5 Inroducion A one of he fundamenal concep in mahemaical conrol heory, conrollabiliy play an imporan role boh in deerminiic and ochaic conrol problem uch a abilizaion of unable yem by feedbac conrol I i well nown ha conrollabiliy of deerminiic equaion i widely Diem Dang Huan ABOUT THE AUTHORS Diem Dang Huan wa born in Bacgiang, Vienam, on 3 July 98 He received hi BS and MS degree in Mahemaic and Theory of Probabiliy and Saiic from Univeriy of Science Vienam Naional Univeriy, Hanoi, in 4 and 8, repecively From 4 o Augu, he ha been employed a Bacgiang Agriculure and Forery Univeriy Afer he go a cholarhip from he Vienamee Governmen in Augu, he ared hi PhD udy in Applied Mahemaic group in he Iniue of Mahemaic, School of Mahemaical Science, in Nanjing Normal Univeriy, China Hi reearch inere include ochaic funcional differenial equaion, ochaic parial differenial equaion, and heory conrol of dynamical yem Hongjun Gao, profeor, pecialiy in ochaic parial differenial equaion and i dynamic PUBLIC INTEREST STATEMENT Conrollabiliy play an imporan role in he analyi and deign of conrol yem Roughly peaing, conrollabiliy generally mean ha i i poible o eer dynamical conrol yem from an arbirary iniial ae o an arbirary final ae uing he e of admiible conrol I i well nown ha ochaic conrol heory i ochaic generalizaion of he claic conrol heory In hi paper, we udy he conrollabiliy of nonlocal econd-order impulive neural ochaic funcional inegro-differenial equaion wih infinie delay and Poion jump and our reul can complemen he earlier publicaion in he exiing lieraure 5 The Auhor( Thi open acce aricle i diribued under a Creaive Common Aribuion (CC-BY 4 licene Page of 6

2 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ ued in many field of cience and echnology, ay, phyic and engineering (eg ee Ahmed, 4a; Balachandran & Dauer, ; Coron, 7; Curain & Zwar, 995; Zabczy, 99, and he reference herein Sochaic conrol heory i ochaic generalizaion of he claic conrol heory The heory of conrollabiliy of differenial equaion in infinie dimenional pace ha been exenively udied in he lieraure, and he deail can be found in variou paper and monograph (Ahmed, 4b; Arom, 97; Balachandran & Dauer, ; Karhieyan & Balachandran, 3; Yang, ; Zabczy, 99, and he reference herein Any conrol yem i aid o be conrollable if every ae correponding o hi proce can be affeced or conrolled in a repecive ime by ome conrol ignal If he yem canno be conrolled compleely, hen differen ype of conrollabiliy can be defined uch a approximae, null, local null, and local approximae null conrollabiliie On hi maer, we refer he reader o Ahmed (4c, Chang (7, Karhieyan and Balachandran (9, Nouya and ÓRegan (9, Sahivel, Mahmudov, and Lee (9, and he reference herein The heory of impulive differenial equaion a much a neural differenial equaion ha been emerging a an imporan area of inveigaion in recen year, imulaed by heir numerou applicaion o problem in phyic, mechanic, elecrical engineering, medicine biology, ecology, and o on The impulive differenial yem can be ued o model procee which are ubjec o abrup change, and which canno be decribed by he claical differenial yem (Lahmianham, Baǐnov, & Simeonov, 989 Parial neural inegro-differenial equaion wih infinie delay ha been ued for modeling he evoluion of phyical yem, in which he repone of he yem depend no only on he curren ae, bu alo on he pa hiory of he yem, for inance, for he decripion of hea conducion in maerial wih fading memory, we refer he reader o he paper of Gurin and Pipin (968, Nunziao (97, and he reference herein relaed o hi maer Beide, noie or ochaic perurbaion i unavoidable and omnipreen in naure a well a in man-made yem Therefore, i i of grea ignificance o impor he ochaic effec ino he inveigaion of impulive neural differenial equaion A he generalizaion of he claic impulive neural differenial equaion, impulive neural ochaic inegro-differenial differenial equaion wih infinie delay have araced he reearcher grea inere On he exience and he conrollabiliy for hee equaion, we refer he reader o (eg ee Chang, 7; Chang, Anguraj, & Arjunan, 8; Karhieyan & Balachandran, 9, 3; Par, Balachandran, & Annapoorani, 9; Par, Balaubramaniam, & Kumarean, 7; Shen & Sun, ; Yan & Yan, 3, and he reference herein Recenly, Par, Balachandran, and Arhi (9 inveigaed he conrollabiliy of impulive neural inegro-differenial yem wih infinie delay in Banach pace uing Schauder-ype fixed poin heorem Arhi and Balachandran ( eablihed he conrollabiliy of damped econd-order impulive neural funcional differenial yem wih infinie delay by mean of he Sadovii fixed poin heorem combined wih a noncompac condiion on he coine family of operaor Very recenly, alo uing Sadovii fixed poin heorem, Muhuumar and Rajivganhi (3 proved ufficien condiion for he approximae conrollabiliy of fracional order neural ochaic inegro-differenial yem wih nonlocal condiion and infinie delay By conra, here ha no been very much reearch on he conrollabiliy of econd-order impulive neural ochaic funcional differenial equaion wih infinie delay, or in oher word, he lieraure abou conrollabiliy of econd-order impulive neural ochaic funcional differenial equaion wih infinie delay i very carce To be more precie, Balaubramaniam and Muhuumar (9 dicued on approximae conrollabiliy of econd-order ochaic diribued implici funcional differenial yem wih infinie delay Mahmudova and McKibben (6 eablihed he reul concerning he global exience, uniquene, approximae, and exac conrollabiliy of mild oluion for a cla of abrac econd-order damped McKean Vlaov ochaic evoluion equaion in a real eparable Hilber pace More recenly, uing Holder inequaliy, ochaic analyi, and fixed poin raegy, Sahivel, Ren, and Mahmudov ( conidered ufficien condiion for he approximae conrollabiliy of nonlinear econd-order ochaic infinie dimenional dynamical yem wih impulive effec And Muhuumar and Balaubramaniam ( inveigaed ufficien condiion for Page of 6

3 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ he approximae conrollabiliy of a cla of econd-order damped McKean Vlaov ochaic evoluion equaion in a real eparable Hilber pace On he oher hand, in recen year, ochaic parial differenial equaion wih Poion jump have gained much aenion ince Poion jump no only exi widely, bu alo can be ued o udy many phenomena in real live Therefore, i i neceary o conider he Poion jump ino he ochaic yem For inance, Luo and Liu (8 udied he exience and uniquene of mild oluion o ochaic parial funcional differenial equaion wih Marovian wiching and Poion jump uing he Lyapunov Razumihin echnique Ren, Zhou, and Chen ( inveigaed he exience, uniquene, and abiliy of mild oluion for a cla of ime-dependen ochaic evoluion equaion wih Poion jump More pecifically, ju recenly, here i an aricle on he complee conrollabiliy of ochaic evoluion equaion wih jump in a eparable Hilber pace dicued by Sahivel and Ren ( and in reference Ren, Dai, and Sahivel (3, Ren e al udied he approximae conrollabiliy of ochaic differenial yem driven by Teugel maringale aociaed wih a Lévy proce For more deail abou he ochaic parial differenial equaion wih Poion jump, one can ee a recen monograph of Peza and Zabczy (7 a well a paper of Cao (5, Marinelli & Rocner (, Rocner and Zhang (7, and he reference herein To he be of our nowledge, here i no wor repored on nonlocal econd-order impulive neural ochaic funcional inegro-differenial equaion wih infinie delay and Poion jump To cloe he gap, moivaed by he above wor, he purpoe of hi paper i o udy he conrollabiliy of nonlocal econd-order impulive neural ochaic funcional inegro-differenial equaion wih infinie delay and Poion jump in Hilber pace More preciely, we conider he following form: d [ x ( g (, x, σ (,, x d = [ Ax(+f (, x, σ (,, x d + Bu( d + σ(,, x dw(+ γ(, x(, v Ñ(d, dv, J: =[, T, Δx( =I (x, ={,, m} =:, m, Δx ( =I (x, =, m, x ( =x H, x( q(x, x,, x n =x = φ, for ae J : = (,, ( where < < < < n < T, n N; x( i a ochaic proce aing value in a real eparable Hilber pace H; A: D(A H H i he infinieimal generaor of a rongly coninuou coine family on H The hiory x : J H, x (θ =x( + θ for, belong o he phae pace, which will be decribed in Secion Aume ha he mapping f, g: J H H, σ: J J, σ i : J J H, i =,, I, I : H, =, m, q: n, and γ: J H H are appropriae funcion o be pecified laer The conrol funcion u( ae value in L (J, U of admiible conrol funcion for a eparable Hilber pace U and B i a bounded linear operaor from U ino H Furhermore, le = < < < m < m+ = T be prefixed poin, and Δx( =x( + x( repreen he jump of he funcion x a ime wih I, deermining he ize of he jump, where x( + and x( repreen he righ and lef limi of x( a =, repecively Similarly x ( + and x ( denoe, repecively, he righ and lef limi of x ( a Le φ( (Ω, and x ( be H-valued -meaurable random variable independen of he Wiener proce {w(} and he Poion poin proce p( wih a finie econd momen The main echnique ued in hi paper include he Banach conracion principle and he heorie of a rongly coninuou coine family of bounded linear operaor The rucure of hi paper i a follow: in Secion, we briefly preen ome baic noaion, preliminarie, and aumpion The main reul in Secion 3 are devoed o udy he conrollabiliy for he yem ( wih heir proof An example i given in Secion 4 o illurae he heory In Secion 5, concluding remar are given Page 3 of 6

4 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ Preliminarie In hi ecion, we briefly recall ome baic definiion and reul for ochaic equaion in infinie dimenion and coine familie of operaor For more deail on hi ecion, we refer he reader o Da Prao and Zabczy (99, Faorini (985, Proer (4, and Travi and Webb (978 Le (H, H,, H and (K, K,, K denoe wo real eparable Hilber pace, wih heir vecor, norm, and heir inner produc, repecively We denoe by (K;H he e of all linear bounded operaor from K ino H, which i equipped wih he uual operaor norm In hi paper, we ue he ymbol o denoe norm of operaor regardle of he pace poenially involved when no confuion poibly arie Le (Ω,, F ={ }, P be a complee filered probabiliy pace aifying he uual condiion (ie i i righ coninuou and conain all P-null e Le w =(w( be a Q-Wiener proce defined on he probabiliy pace (Ω,, F, P wih he covariance operaor Q uch ha Tr(Q < We aume ha here exi a complee orhonormal yem {e } in K, a bounded equence of nonnegaive real number λ uch ha Qe = λ e, =,,, and a equence of independen Brownian moion {β } uch ha w(, e K = λ e, e K β (, e K, Le = (Q K; H be he pace of all Hilber Schmid operaor from Q K ino H wih he inner produc Ψ, φ = Tr[ΨQφ, where φ i he adjoin of he operaor φ Le p = p(, D p (he domain of p( be a aionary -Poion poin proce aing i value in a meaurable pace (, B( wih a σ-finie ineniy meaure λ(dv by N(d, dv he Poion couning meaure aociaed wih p, ha i, N(, = = D p, I (p( for any meaurable e B(K {}, which denoe he Borel σ-field of (K {} Le Ñ(d, dv: = N(d, dv λ(dvd be he compenaed Poion meaure ha i independen of w( Denoe by (J ; H he pace of all predicable mapping γ: J H for which E γ(, v Hλ(dvd < We may hen define he H-valued ochaic inegral γ(, vñ(d, dv, which i a cenered quareinegrable maringale For he conrucion of hi ind of inegral, we can refer o Proer (4 The collecion of all rongly meaurable, quare-inegrable H-valued random variable, denoed by (Ω, H, i a Banach pace equipped wih norm x = ( E x Le C(J, (Ω, H be he Banach pace of all coninuou map from J o (Ω, H, aifying he condiion up J E x( < An imporan ubpace i given by (Ω, H ={f (Ω, H: f i -meaurable} Furher, le ( } F (, T; H = {g: J Ω H: g( i F -progreively meaurable and E J g( H d < Nex, o be able o acce conrollabiliy for he yem (, we need o inroduce he heory of coine funcion of operaor and he econd-order abrac Cauchy problem Page 4 of 6

5 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ Definiion ( The one-parameer family {C(} R (H i aid o be a rongly coninuou coine family if he following hold: (i C( =I, I i he ideniy operaor in H; (ii C(x i coninuou in on R for any x H; and (iii C( + +C( =C(C( for all, R ( The correponding rongly coninuou ine family {S(} R (H, aociaed o he given rongly coninuou coine family {C(} R (H i defined by S(x = C(xd, R, x H (3 The infinieimal generaor A: H H of {C(} R (H i given by Ax = d d C(x =, for all x D(A ={x H: C( C (R, H} I i well nown ha he infinieimal generaor A i a cloed, denely defined operaor on H, and he following properie hold (ee Travi & Webb, 978 Propoiion Suppoe ha A i he infinieimal generaor of a coine family of operaor {C(} R Then, he following hold: (i There exi a pair of conan M A and α uch ha C( M A e α, and hence S( M A e α ; (ii A r S(uxdu =[C(r C(x, for all r < ; and (iii There exi N uch ha S( S(r N r eα d, r < Than o he Propoiion and he uniform boundedne principle ha we ee a direc conequence ha boh {C(} J and {S(} J are uniformly bounded by M = M A e α T The exience of oluion for he econd-order linear abrac Cauchy problem { x ( =Ax(+h(, J, x( =z, x ( =w, ( where h: J H i an inegrable funcion ha ha been dicued in Travi and Webb (977 Similarly, he exience of oluion of he emilinear econd-order abrac Cauchy problem ha been reaed in Travi and Webb (978 Definiion The funcion x( given by x( =C(z + S(w + S( h(d, J, i called a mild oluion of (, and ha when z H, x( i coninuouly differeniable and x ( =AS(z + C(w + C( h(d, J Page 5 of 6

6 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ For addiional deail abou coine funcion heory, we refer he reader o Travi and Webb (977, 978 Since he yem ( ha impulive effec, he phae pace ued in Balaubramaniam and Nouya (6 and Par e al (7 canno be applied o hee yem So, we need o inroduce an abrac phae pace, a follow: Aume ha l:j (, + i a coninuou funcion wih l = J l(d < For any a >, we define { : = ψ:j H:(E ψ(θ i a bounded and meaurable funcion on [ a, and } J l( up θ [, (E ψ(θ d < + If i endowed wih he norm ψ = l( up (E ψ(θ d, ψ, J θ [, hen, i i clear ha (, i a Banach pace (Hino, Muraami, & Naio, 99 Le J T = (, T We conider he pace { T : = x:j T H uch ha x C(J, H and here exi x( and x(+ wih } x( =x(+, x( q(x, x,, x n =φ, =, m, where x i he rericion of x o J =(, +, =, m Se T be a eminorm in T defined by x T = φ + up(e x(, x T J Now, we recall he following ueful lemma ha appeared in Chang (7 Lemma (Chang, 7 Aume ha x T, hen for J, x Moreover, ( l E x( x x + l up (E x( [, Nex, we give he definiion of mild oluion for ( Definiion 3 An -adaped càdlàg ochaic proce x:j T H i called a mild oluion of ( on J T if x( q(x, x,, x n =x = φ and x ( =x H, aifying φ, x, q (Ω, H; he funcion C( g(, x, σ (, τ, x dτ and S( f (, x τ, σ (, τ, x τ dτ are inegrable on [, T uch ha he following condiion hold: (i {x : J} i a -valued ochaic proce; (ii For arbirary J, x( aifie he following inegral equaion: x( =C([φ(+q(x, x,, x n ( + S([x g(, x, + C( g(, x, σ (, τ, x τ dτd + S( f (, x, σ (, τ, x τ dτd + S( Bu(d + S( σ(, τ, x τ dw(τd + S( γ (, x(, v Ñ(d, dv + C( I (x + S( I (x ; and < < < < ( Page 6 of 6

7 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ (iii Δx( =I (x, Δx ( =I (x, =, m Definiion 4 The yem ( i aid o be conrollable on he inerval J T, if for every iniial ochaic proce φ defined on J, x ( =x H and y H; here exi a ochaic conrol u L (J, U which i adaped o he filraion { } J uch ha he oluion x( of he yem ( aifie x(t =y, where y and T are he preaigned erminal ae and ime, repecively To prove our main reul, we li he following baic aumpion of hi paper (H There exi poiive conan M C,, and M σ uch ha for all, J,x, y C( M C, S( ; E [σ (,, x σ (,, yd Mσ x y (H The funcion g:j H H i coninuou and here exi a poiive conan M g uch ha for all J,x, x, y, y (Ω, H E g(, x, y g(, x, y M g ( x x + E y y (H3 For each (, J J, he funcion σ :J J H i coninuou and here exi a poiive conan M σ uch ha for all, J,x, y E [σ (,, x σ (,, yd Mσ x y (H4 The funcion f :J H H i coninuou and here exi a poiive conan M f uch ha for all J,x, x, y, y (Ω, H E f (, x, y f (, x, y M f ( x x + E y y (H5 The funcion I, I C(, H, =, m and here exi poiive conan, and uch ha for all x, y E I (x, E I (x ; E I (x I (y x y, E I (x I (y x y,, (H6 For each φ, h( =lim c σ(,, φdw( exi and coninuou Furher, here exi c a poiive conan M h uch ha E h( M h (H7 The funcion σ:j J (K, H i coninuou and here exi poiive conan M σ, M σ uch ha for all, J and x, y E σ(,, x M σ ; E σ(,, x σ(,, y M σ x y (H8 The funcion q: n i coninuou and here exi poiive conan M q, M q uch ha for all x, y, J E q(x, x,, x n ( M q ; Page 7 of 6

8 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ E q(x, x,, x n ( q(y, y,, y n ( M q x y (H9 The linear operaor W:L (J, U L (Ω, H defined by Wu = J S(T Bu(d ha an induced invere W which ae value in L (J, U KerW (ee Carimichel & Quinn, 984 and here exi wo poiive conan M B and uch ha B M B and W (H The funcion γ:j H H i a Borel meaurable funcion and aifie he Lipchiz coninuiy condiion, he linear growh condiion, and here exi poiive conan M γ, M γ uch ha for any x, y F (, T;H, J ( ( E γ(, x(, v λ(dvd H E γ(, x(, v 4 λ(dvd H M γ E ( E ( E ( + x( H d; γ(, x(, v γ(, y(, v H λ(dvd γ(, x(, v γ(, y(, v 4 λ(dvd H M γ E x( y( d H 3 Main reul In hi ecion, we hall inveigae he conrollabiliy of nonlocal econd-order impulive neural ochaic funcional inegro-differenial equaion wih infinie delay and Poion jump in Hilber pace The main reul of hi ecion i he following heorem Theorem 3 Aume ha he aumpion (H (H hold If Ξ < and Θ <, hen he yem ( i conrollable on J T, where Ξ: = 3 ( { } + 9T M B l [M T C M g ( + M σ + M f ( + M σ + TM γ C, { ( [ ( Θ: = 98l T M B M C M q + 4l + 7T M B T M C M g + Mσ ( + T M f + Mσ + T 3 M σ Tr(Q+ TM C γ m + mm C l = + m m = } Proof Uing he aumpion (H9, for an arbirary funcion x(, we define he conrol proce Page 8 of 6

9 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ u T x ( =W {y C(T[φ(+q(x, x,, x n ( S(T[x g(, x, T C(T g(, x, σ (, τ, x τ dτd C(T I (x < <T T S(T f (, x, σ (, τ, x τ dτd S(T I (x T S(T [h(+ σ(, τ, x τ dw(τ d T S(T γ (, x(, v } Ñ(d, dv ( < <T (3 We ranform ( ino a fixed poin problem Conider he operaor Π: T T defined by Πx( =φ(+q(x, x,, x n (, J ; Πx( =C([φ(+q(x, x,, x n ( + S([x g(, x, + C( g(, x, σ (, τ, x τ dτd + S( f (, x, σ (, τ, x τ dτd + S( Bu T (d x + S( [h(+ σ(, τ, x τ dw(τ d + S( γ (, x(, v Ñ(d, dv + C( I (x + S( I (x, for ae J < < < < In wha follow, we hall how ha uing he conrol u T x (, he operaor Π ha a fixed poin, which i hen a mild oluion for yem ( Clearly, Πx(T =y For φ, we defined φ by φ(+q(x, x φ ( ={,, x n ( if J, C( [ φ(+q(x, x,, x n ( if J, hen φ T Se x( =z(+ φ (, J T I i eay o ee ha x aifie ( if and only if z aifie z =, x ( =x = z ( =z and z( =S([z g(, φ, + C( g(, z + φ, σ (, τ, z τ + φ τ dτd + S( f (, z + φ, σ (, τ, z τ + φ τ dτd + S( Bu T (d z+ φ + S( [h(+ σ(, τ, z τ + φ τ dw(τ d + S( γ (, z( + φ (, v Ñ(d, dv + < < < < C( I (z + φ + S( I (z + φ, J, Page 9 of 6

10 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ where u T z+ φ ( i obained from (3 by replacing x = z + φ Le ={y :y = } For any y T T T, we have y T = y + up(e y( J = up (E y(, J and hu (, T T i a Banach pace Se B r ={y T : y T r} for ome r, hen B r i uniformly bounded, and for u B T r, by Lemma, we have z + φ ( z + φ 4 ( l up (E z( + z + l up (E φ ( + φ [, [, ( [ 4l r + M C E φ( + M q + 4 φ (3 : = r Define he map Π: T T defined by Πz( =, for J and Πz( =S([z g(, φ, + C( g(, z + φ, σ (, τ, z τ + φ τ dτd + S( f (, z + φ, σ (, τ, z τ + φ τ dτd + S( Bu T (d z+ φ + S( [h(+ σ(, τ, z τ + φ τ dw(τ d + S( γ (, z( + φ (, v Ñ(d, dv + < < < < C( I (z + φ + S( I (z + φ, J Obviouly, he operaor Π ha a fixed poin which i equivalen o prove ha Π ha a fixed poin Noe ha, by our aumpion, we infer ha all he funcion involved in he operaor are coninuou, herefore Π i coninuou Le z, z T From (3, by our aumpion, Hölder inequaliy, he Doob maringale inequaliy, and he Burholder Davi Gundy inequaliy for pure jump ochaic inegral in Hilber pace (ee Luo & Liu, 8, Lemma, and in view of (3, for J, we obain he following eimae E u T z+ φ ( { [ 9 E y + M C (E φ( + M q + E x + (M g φ + C [ ( [ ( + T M C M g [ + Mσ r + C + C + T M f [ + Mσ r + C 3 + C4 + T r (M h + TTr(QM σ +TM γ C( + +mm C l m = + m m = } : = l, and Page of 6

11 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ E u T z+ φ ( ut z+ φ ( 4l M ( ( W {M C M q + T M C M g + Mσ + T M f + Mσ + T 3 M σ Tr(Q+ TM C γ m + mm C l = where C > i a poiive conan and + m m = } up E z( z(, J C : = T up σ (,,, C (, J J C 3 : = T up σ (,,, C (, J J : = up g(,,, J : = up 4 f (,, J Lemma 3 Under he aumpion of Theorem 3, here exi r > uch ha Π(B r B r Proof If hi propery i fale, hen for each r >, here exi a funcion z r ( B r, bu Π(z r B r, ie Π(z r ( > r for ome J However, by our aumpion, Hölder inequaliy and he Burholder Davi Gundy inequaliy, we have r < E Π(z r ( [ 8 [ E x + (M g φ + C ( + T M C [M g [ + Mσ r + C + C [ ( + T M f [ + Mσ r + C 3 + C4 + T M B l + T r (M h + TTr(QM σ +TM γ C( m m + +mm C + m, ( M + 8( + 9T M B [T M C M g ( + M σ + M f ( + M σ l = = + TM γ C l r, (33 where M [ : = 7( + 9T M B E y + M C (E φ( + M q + 8( + 9T M B [ [ E x + (M g φ + C + T M C (M g C + C +T (M f C 3 + C 4 + T (M h + TTr(QM σ +TM γ C + mmc m Dividing boh ide of (33 by r and noing ha ( [ r = 4l r + M C E φ( + M q + 4 φ r and aing he limi a r, we obain Ξ = + m m = which conradic our aumpion Thu, for ome poiive number r, Π(B r B r Thi complee he proof of Lemma 3 Lemma 3 Under he aumpion of Theorem 3, Π: T T i a conracion mapping Proof Le z, z T Then, by our aumpion, Hölder inequaliy, Burholder Davi Gundy inequal- Page of 6

12 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ iy, Lemma, and ince z = and z =, for each J, we ee ha E (Πz( (Πz( 4l {T M C M g ( + Mσ + T M f ( + Mσ + T 3 M σ Tr(Q+ TM γ C + mm C m = + m m = Taing he upremum over, we obain } { 98l T M B M C M q + 4l l up E z( z( + 7T M B E u T z+ φ ( ut z+ φ ( J ( + 7T M B ( ( [T M C M g + Mσ + T M f + Mσ + T 3 M σ Tr(Q+ TM C γ + mm C m = + m m (Πz (Πz T Θ z z T = } up E z( z( J l By our aumpion, we conclude ha Π i a conracion on T Thu, we have compleed he proof of Lemma 3 On he oher hand, by Banach fixed poin heorem, here exi a unique fixed poin x( T uch ha (Πx( =x( Thi fixed poin i hen he mild oluion of he yem ( Clearly, x(t =(Πx(T =y Thu, he yem ( i conrollable on J T The proof for Theorem 3 i hu complee Now, le u conider a pecial cae for he yem ( If γ (, x(, v, he yem ( become he following nonlocal econd-order impulive neural ochaic funcional inegro-differenial equaion wih infinie delay wihou Poion jump: d [ x ( g (, x, σ (,, x d = [ Ax(+f (, x, σ (,, x d + Bu( d + σ(,, x dw(, J: =[, T, Δx( =I (x, ={,, m} =:, m, Δx ( =I (x, =, m, x ( =x H, x( q(x, x,, x n =x = φ, for ae J : = (,, (34 Corollary 3 Aume ha all aumpion of Theorem 3 hold excep ha (H and Ξ, Θ replaced by Ξ, Θ uch ha Ξ: = 56l T( [ + 8T M B M C M g ( + M σ + M f ( + M σ, and { ( Θ: [ ( = 7l T M B M C M q + l + 6T M B T M C M g + Mσ + T M f ( + Mσ + T 3 M σ Tr(Q+mM C m = + m m If Ξ < and Θ <, hen he yem (34 i conrollable on J T = } Page of 6

13 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ Applicaion In hi ecion, he eablihed previou reul are applied o udy he conrollabiliy of he ochaic nonlinear wave equaion wih infinie delay and Poion jump Specifically, we conider he following conrollabiliy of nonlocal econd-order impulive neural ochaic funcional inegro-differenial equaion wih infinie delay and Poion jump of he form: [ [ y(, ξ δ (, ξ,, P (y(, ξd b ( τp (y(τ, ξdτd = y(, ξ+ δ (, ξ,, G (y(, ξd + b ( τg ξ (y(τ, ξdτd +b(ξu( d + δ( y(, ξdβ(+ y(, ξvñ(d, dv, J, ξ [, π, Δy( (ξ = η ( y(, ξd, =, m, ξ [, π, Δy ( (ξ = ρ ( y(, ξd, =, m, ξ [, π, y(,=y(, π =, J, y(, ξ =x (ξ, ξ [, π, y(, ξ n π p (ξ, ζy(, ζdζ = φ(, ξ, J i= i i, ξ [, π, where β( i a andard one-dimenional Wiener proce in H, defined on a ochaic bai (Ω,, P; ={v R: < v R a, a > }; < < < < n < T, n N; = < < < m < m+ < T are prefixed number, and φ Le p = p(, D p be a K-valued σ-finie aionary Poion poin proce (independen of β( on a complee probabiliy pace wih he uual condiion (Ω,, (, P Le Ñ(d, dv: = N(d, dv λ(dvd, wih he characeriic meaure λ(dv on B(K {} Aume ha (4 v λ(dv < and v 4 λ(dv < To rewrie (4 ino he abrac from of (, we conider he pace H = L ([, π wih he norm Le e n (ξ: = in nξ, n =,, 3, denoe he compleed orhogonal baic in H and π β( = n= λn β n (e n,,λ n >, where {β n (} n are one-dimenional andard Brownian moion muually independen on a uual complee probabiliy pace (Ω,, (, P and Defined A:H H by A =, wih domain D(A =H ([, π H ξ ([, π, where H ([, π={w L ([, π: w z L ([, π, w( =w(π =} H ([, π = {w L ([, π: w z, w z L ([, π} Then, Ax = n x, e n e n, n= x D(A, (4 (ee Travi & Webb, 987, Example 5 Uing (4, one can eaily verify ha he operaor C( defined by C(x = co(n x, e n e n, R, n= Page 3 of 6

14 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ from a coine funcion on H, wih aociaed ine funcion in(n S(x = x, e n n e n, R n= I i clear ha (ee Travi & Webb, 977, for all x H, R,C( x and S( x are periodic funcion wih C( and S( Thu, (H i rue Now, we give a pecial -pace Le l( =e,, hen l = J l(d = and define ψ = l( up (E ψ(θ d, ψ J θ [, I follow from Hino e al (99 ha (, i a Banach pace Hence, for (, ψ J, where ψ(θx = ψ(θ, x, (θ, x J [, π Le y((ξ =y(, ξ To udy he yem (4, we aume ha he following condiion hold: (i Le B (R, H be defined a Bu(ξ =b(ξu, ξ π, u R, b(ξ L ([, π (ii The linear operaor W: L (J, U H defined by Wu = J S(T b(ξu(d i a bounded linear operaor bu no necearily one-o-one Le KerW ={u L (J, U:Wu = } be null pace of W and [KerW be i orhogonal complemen in L (J, U Le W :[KerW Range(W be he rericion of W o [KerW, W i necearily one-o-one operaor The invere mapping heorem ay ha (W i bounded ince [KerW and Range(W are Banach pace Since he invere operaor W i bounded and ae value in L (J, U KerW, he aumpion (H9 i aified (iii The funcion p i :[, π, π R are C -funcion, for each i =, n (iv The funcion η, ρ C(R, R uch ha for =, m, = l(η (d <, M J I = l(ρ J (d < We define he funcion g, f :J H H, σ:j J, γ:j H H, and I, I : H, =, m by g(, ψ, V ψ(ξ = J δ (, ξ, θp (ψ(θ(ξdθ + V ψ(ξ, f (, ψ, V ψ(ξ = J δ (, ξ, θg (ψ(θ(ξdθ + V ψ(ξ, σ(,, ψ(ξ = J δ(θψ(θ(ξdθ, γ(, ψ(ξ, v =ψ(ξv, I (, ψ(ξ = J η ( ψ(θ(ξd, =, m, I (, ψ(ξ = J ρ ( ψ(θ(ξd, =, m, Page 4 of 6

15 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ where V ψ(ξ = b ( θp (ψ(θ(ξdθd, V ψ(ξ = J b ( θg (ψ(θ(ξdθd J Then, he yem (4 can be wrien in he abrac form a he yem ( Furher, we can impoe ome uiable condiion on he above-defined funcion a hoe in he aumpion (H (H Therefore, by Theorem 3, we can conclude ha he yem (4 i conrollable on J T 5 Concluion In hi paper, we have udied he conrollabiliy for a cla of nonlocal econd-order impulive neural ochaic funcional inegro-differenial equaion wih infinie delay and Poion jump in Hilber pace, which i new and allow u o develop he conrollabiliy of he econd-order ochaic parial differenial equaion Uing he Banach fixed poin heorem combined wih heorie of a rongly coninuou coine family of bounded linear operaor, and ochaic analyi heory, he conrollabiliy of nonlocal econd-order impulive neural ochaic funcional inegro-differenial equaion wih infinie delay and Poion jump i obained In addiion, an applicaion i provided o illurae he effecivene of he conrollabiliy reul obained The reul in our paper exend and improve he correponding one announced by Arhi and Balachandran (, Balaubramaniam and Muhuumar (9, Muhuumar and Rajivganhi (3, Par, Balachandran, and Annapoorani (9, Par, Balachandran, and Arhi (9, Travi and Webb (978, and ome oher reul Funding Thi wor wa uppored by he China NSF [gran number 758 Auhor deail Diem Dang Huan, huandd@bafueduvn Hongjun Gao 3 gaohj@njnueducn Faculy of Baic Science, Bacgiang Agriculure and Forery Univeriy, Bacgiang, Vienam Vienam Naional Univeriy, Hanoi, 44 Xuan Thuy Sree, Cau Giay, Hanoi, Vienam 3 School of Mahemaical Science, Nanjing Normal Univeriy, Nanjing, 3 PR China Ciaion informaion Cie hi aricle a: Conrollabiliy of nonlocal econd-order impulive neural ochaic funcional inegro-differenial equaion wih delay and Poion jump, Diem Dang Huan & Hongjun Gao, Cogen Engineering (5, : Reference Ahmed, H M (4a Non-linear fracional inegro-differenial yem wih non-local condiion IMA Journal of Mahemaical Conrol doi:93/imamci/dnu49 Ahmed, H M (4b Conrollabiliy of impulive neural ochaic differenial equaion wih fracional Brownian moion IMA Journal of Mahemaical Conrol doi:93/ imamci/dnu9 Ahmed, H M (4c Approximae conrollabiliy of impulive neural ochaic differenial equaion wih fracional Brownian moion in a Hilber pace Advance in Difference Equaion, 3, Arhi, G, & Balachandran, K ( Conrollabiliy of damped econd-order impulive neural funcional differenial yem wih infinie delay Journal of Opimizaion Theory and Applicaion, 5, Arom, K J (97 Inroducion o ochaic conrol heory New Yor, NY: Academic Pre Balachandran, K, & Dauer, J P ( Conrollabiliy of nonlinear ochaic yem in Banach pace Journal of Opimizaion Theory and Applicaion, 5, 7 8 Balaubramaniam, P, & Dauer, J P ( Conrollabiliy of emilinear ochaic delay evoluion equaion in Hilber pace Inernaional Journal of Mahemaic and Mahemaical Science, 3, Balaubramaniam, P, & Muhuumar, P (9 Approximae conrollabiliy of econd-order ochaic diribued implici funcional differenial yem wih infinie delay Journal of Opimizaion Theory and Applicaion, 43, 5 44 Balaubramaniam, P, & Nouya, S K (6 Conrollabiliy for neural ochaic funcional differenial incluion wih infinie delay in abrac pace Journal of Mahemaical Analyi, 34, 6 76 Cao, G (5 Sochaic differenial evoluion equaion in infinie dimenional (MS hei, Huazhong Univeriy of Science and Technology, Wuhan Carimichel, N, & Quinn, M D (984 Fixed poin mehod in nonlinear conrol (Lecure Noe in Conrol and Informaion Sociey, Vol 75 Berlin: Springer Chang, Y K (7 Conrollabiliy of impulive funcional differenial yem wih infinie delay in Banach pace Chao, Solion & Fracal, 33, 6 69 Chang, Y K, Anguraj, A, & Arjunan, M M (8 Exience reul for impulive neural funcional differenial equaion wih infinie delay Nonlinear Analyi: Hybrid Syem,, 9 8 Coron, J M (7 Conrol and nonlineariy, mahemaical urvey and monograph (Vol 36 Providence, RI: American Mahemaical Sociey Curain, R F, & Zwar, H J (995 An inroducion o infinie dimenional linear yem heory Berlin: Springer Da Prao, G, & Zabczy, J (99 Sochaic equaion in infinie dimenion (Vol 44 Cambridge: Cambridge Univeriy Pre Faorini, H O (985 Second order linear differenial equaion in Banach pace (Norh-Holland mahemaic udie, Vol 8 Amerdam: Norh-Holland Gurin, M E, & Pipin, A C (968 A general heory of hea conducion wih finie wave peed Archive for Raional Mechanic and Analyi, 3, 3 6 Hino, Y, Muraami, S, & Naio, T (99 Funcional differenial equaion wih infinie delay (Lecure noe in Page 5 of 6

16 Huan & Gao, Cogen Engineering (5, : hp://dxdoiorg/8/ mahemaic, Vol 473 Berlin: Springer-Verlag Karhieyan, S, & Balachandran, K (9 Conrollabiliy of nonlinear ochaic neural impulive yem Nonlinear Analyi, 3, Karhieyan, S, & Balachandran, K (3 On conrollabiliy for a cla of ochaic impulive yem wih delay in conrol Inernaional Journal of Syem Science, 44, Lahmianham, V, Baǐnov, D, & Simeonov, P S (989 Theory of impulive differenial equaion (Serie in modern applied mahemaic, Vol 6 Teanec, NJ: World Scienific Publihing Luo, J, & Liu, K (8 Sabiliy of infinie dimenional ochaic evoluion wih memory and Marovian jump Sochaic Procee and heir Applicaion, 8, Mahmudova, N I, & McKibben, M A (6 Abrac econdorder damped McKean--Vlaov ochaic evoluion equaion Sochaic Procee and heir Applicaion, 4, Marinelli, C, & Röcner, M ( Well-poedne and aympoic behavior for ochaic reacion--diffuion equaion wih muliplicaive Poion noie Elecronic Journal of Probabiliy, 5, Muhuumar, P, & Balaubramaniam, P ( Approximae conrollabiliy of econd-order damped McKean--Vlaov ochaic evoluion equaion Compuer & Mahemaic wih Applicaion, 6, Muhuumar, P, & Rajivganhi, C (3 Approximae conrollabiliy of fracional order neural ochaic inegro-differenial yem wih nonlocal condiion and infinie delay Taiwanee Journal of Mahemaic, 7, Nouya, S K (9 D ÓRegan; Some remar on conrollabiliy of evoluion equaion in Banach pace, Elecronic Journal of Qualiaive Theory of Differenial Equaion, 9, 6 Nunziao, J W (97 On hea conducion in maerial wih memory Quarerly of Applied Mahemaic, 9, 87 4 Par, J Y, Balachandran, K, & Annapoorani, N (9 Exience reul for impulive neural funcional inegrodifferenial equaion wih infinie delay Nonlinear Analyi, 7, Par, J Y, Balachandran, K, & Arhi, G (9 Conrollabiliy of impulive neural inegrodifferenial yem wih infinie delay in Banach pace Nonlinear Analyi: Hybrid Syem, 3, Par, J Y, Balaubramaniam, P, & Kumarean, N (7 Conrollabiliy for neural ochaic funcional inegrodifferenial infinie delay yem in abrac pace Numerical Funcional Analyi and Opimizaion, 8, Peza, S, & Zabczy, J (7 Sochaic parial differenial equaion wih lévy noie Cambridge: Cambridge Univeriy Pre Proer, P E (4 Sochaic inegraion and differenial equaion (nd ed New Yor, NY: Springer Ren, Y, Dai, H L, & Sahivel, R (3 Approximae conrollabiliy of ochaic differenial yem driven by a lévy proce Inernaional Journal of Conrol, 86, Ren, Y, Zhou, Q, & Chen, L ( Exience, uniquene and abiliy of mild oluion for ime-dependen ochaic evoluion equaion wih Poion jump and infinie delay Journal of Opimizaion Theory and Applicaion, 49, Röcner, M, & Zhang, T (7 Sochaic evoluion equaion of jump ype: Exience, uniquene and large deviaion principle Poenial Analyi, 6, Sahivel, R, Mahmudov, N I, & Lee, S-G (9 Conrollabiliy of non-linear impulive ochaic yem Inernaional Journal of Conrol, 8, 8 87 Sahivel, R, & Ren, Y ( Complee conrollabiliy of ochaic evoluion equaion wih jump Repor on Mahemaical Phyic, 68, Sahivel, R, Ren, Y, & Mahmudov, N I ( Approximae conrollabiliy of econd-order ochaic differenial equaion wih impulive effec Modern Phyic Leer B, 4, Shen, L J, & Sun, J T ( Approximae conrollabiliy of ochaic impulive funcional yem wih infinie delay Auomaica, 48, Travi, C C, & Webb, G F (977 Compacne, regulariy, and uniform coninuiy properie of rongly coninuou coine familie Houon Journal of Mahemaic, 3, Travi, C C, & Webb, G F (978 Coine familie and abrac nonlinear econd order differenial equaion Aca Mahemaica Academiae Scieniarum Hungaricae, 3, Travi, C C, & Webb, G F (987 Second oder differenial equaion in Banach pace In Proceedind Inernaional Sympoium on Nonlinear Equaion in Abrac Space (pp New Yor, NY: Academic Pre Yan, Z M, & Yan, X X (3 Exience of oluion for impulive parial ochaic neural inegrodifferenial equaion wih ae-dependen delay Collecanea Mahemaica, 64, 35 5 Yang, T ( Impulive yem and conrol: Theory and applicaion Berlin: Springer Zabczy, J (99 Conrollabiliy of ochaic linear yem Syem & Conrol Leer,, 5 3 Zabczy, J (99 Mahemaical conrol heory Bael: Birhauer 5 The Auhor( Thi open acce aricle i diribued under a Creaive Common Aribuion (CC-BY 4 licene You are free o: Share copy and rediribue he maerial in any medium or forma Adap remix, ranform, and build upon he maerial for any purpoe, even commercially The licenor canno revoe hee freedom a long a you follow he licene erm Under he following erm: Aribuion You mu give appropriae credi, provide a lin o he licene, and indicae if change were made You may do o in any reaonable manner, bu no in any way ha ugge he licenor endore you or your ue No addiional rericion You may no apply legal erm or echnological meaure ha legally reric oher from doing anyhing he licene permi Page 6 of 6

Research Article Existence and Uniqueness of Solutions for a Class of Nonlinear Stochastic Differential Equations

Research Article Existence and Uniqueness of Solutions for a Class of Nonlinear Stochastic Differential Equations Hindawi Publihing Corporaion Abrac and Applied Analyi Volume 03, Aricle ID 56809, 7 page hp://dx.doi.org/0.55/03/56809 Reearch Aricle Exience and Uniquene of Soluion for a Cla of Nonlinear Sochaic Differenial