Controllability of nonlocal second order impulsive …...neutral stochastic functional integro-differential equations with delay and Poisson jumps Diem Dang Huan1,2* and Hongjun Gao3

Huan & Gao, Cogent Engineering (2015), 2: 1065585http://dx.doi.org/10.1080/23311916.2015.1065585

SYSTEMS & CONTROL | RESEARCH ARTICLE

Controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with delay and Poisson jumpsDiem Dang Huan1,2* and Hongjun Gao3

Abstract: The current paper is concerned with the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps in Hilbert spaces. Using the theory of a strongly continuous cosine family of bounded linear operators, stochastic analysis theory and with the help of the Banach fixed point theorem, we derive a new set of suf-ficient conditions for the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps. Finally, an application to the stochastic nonlinear wave equation with infinite delay and Poisson jumps is given.

Subjects: Non-Linear Systems; Probability Theory & Applications; Stochastic Models & Processes

Keywords: controllability; impulsive neutral stochastic integro-differential equations; Poisson jumps; cosine functions of operators; infinite delay; Banach fixed point theorem

2010 Mathematics Subject classifications: 34A37; 93B05; 93E03; 60H20; 34K50

1. IntroductionAs one of the fundamental concepts in mathematical control theory, controllability plays an impor-tant role both in deterministic and stochastic control problems such as stabilization of unstable systems by feedback control. It is well known that controllability of deterministic equation is widely

*Corresponding author: Diem Dang Huan, Faculty of Basic Sciences, Bacgiang Agriculture and Forestry University, Bacgiang 21000, Vietnam; Vietnam National University, Hanoi, 144 Xuan Thuy Street, Cau Giay, Hanoi 10000, Vietnam E-mail: [email protected]

Reviewing editor:James Lam, University of Hong Kong, Hong Kong

Additional information is available at the end of the article

ABOUT THE AUTHORSDiem Dang Huan was born in Bacgiang, Vietnam, on 13 July 1980. He received his BS and MS degrees in Mathematics and Theory of Probability and Statistics from University of Science—Vietnam National University, Hanoi, in 2004 and 2008, respectively. From 2004 to August 2010, he has been employed at Bacgiang Agriculture and Forestry University. After he got a scholarship from the Vietnamese Government in August 2010, he started his PhD study in Applied Mathematic group in the Institute of Mathematics, School of Mathematical Science, in Nanjing Normal University, China. His research interests include stochastic functional differential equations, stochastic partial differential equations, and theory control of dynamical systems.

Hongjun Gao, professor, speciality in stochastic partial differential equations and its dynamics.

PUBLIC INTEREST STATEMENTControllability plays an important role in the analysis and design of control systems. Roughly speaking, controllability generally means that it is possible to steer dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It is well known that stochastic control theory is stochastic generalization of the classic control theory. In this paper, we study the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps and our results can complement the earlier publications in the existing literature.

Received: 08 March 2015Accepted: 20 June 2015Published: 30 July 2015

© 2015 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.

Page 1 of 16

Diem Dang Huan

http://crossmark.crossref.org/dialog/?doi=10.1080/23311916.2015.1065585&domain=pdf&date_stamp=2015-07-30mailto:[email protected]://

of 16


used in many fields of science and technology, say, physics and engineering (e.g. see Ahmed, 2014a; Balachandran & Dauer, 2002; Coron, 2007; Curtain & Zwart, 1995; Zabczyk, 1992, and the references therein). Stochastic control theory is stochastic generalization of the classic control theory. The the-ory of controllability of differential equations in infinite dimensional spaces has been extensively studied in the literature, and the details can be found in various papers and monographs (Ahmed, 2014b; Astrom, 1970; Balachandran & Dauer, 2002; Karthikeyan & Balachandran, 2013; Yang, 2001; Zabczyk, 1991, and the references therein). Any control system is said to be controllable if every state corresponding to this process can be affected or controlled in a respective time by some con-trol signals. If the system cannot be controlled completely, then different types of controllability can be defined such as approximate, null, local null, and local approximate null controllabilities. On this matter, we refer the reader to Ahmed (2014c), Chang (2007), Karthikeyan and Balachandran (2009), Ntouyas and ÓRegan (2009), Sakthivel, Mahmudov, and Lee (2009), and the references therein.

The theory of impulsive differential equations as much as neutral differential equations has been emerging as an important area of investigations in recent years, stimulated by their numerous applica-tions to problems in physics, mechanics, electrical engineering, medicine biology, ecology, and so on. The impulsive differential systems can be used to model processes which are subject to abrupt changes, and which cannot be described by the classical differential systems (Lakshmikantham, Baǐnov, & Simeonov, 1989). Partial neutral integro-differential equation with infinite delay has been used for modeling the evolution of physical systems, in which the response of the system depends not only on the current state, but also on the past history of the system, for instance, for the description of heat conduction in materials with fading memory, we refer the reader to the papers of Gurtin and Pipkin (1968), Nunziato (1971), and the references therein related to this matter. Besides, noise or stochastic perturbation is unavoidable and omnipresent in nature as well as in man-made systems. Therefore, it is of great significance to import the stochastic effects into the investigation of impulsive neutral differential equations. As the generalization of the classic impulsive neutral differential equations, impulsive neutral stochastic integro-differential differential equations with infinite delays have attracted the researchers’ great interest. On the existence and the controllability for these equations, we refer the reader to (e.g. see Chang, 2007; Chang, Anguraj, & Arjunan, 2008; Karthikeyan & Balachandran, 2009, 2013; Park, Balachandran, & Annapoorani, 2009; Park, Balasubramaniam, & Kumaresan, 2007; Shen & Sun, 2012; Yan & Yan, 2013, and the references therein).

Recently, Park, Balachandran, and Arthi (2009) investigated the controllability of impulsive neutral integro-differential systems with infinite delay in Banach spaces using Schauder-type fixed point theo-rem. Arthi and Balachandran (2012) established the controllability of damped second-order impulsive neutral functional differential systems with infinite delay by means of the Sadovskii fixed point theorem combined with a noncompact condition on the cosine family of operators. Very recently, also using Sadovskii’s fixed point theorem, Muthukumar and Rajivganthi (2013) proved sufficient conditions for the approximate controllability of fractional order neutral stochastic integro-differential systems with nonlo-cal conditions and infinite delay.

By contrast, there has not been very much research on the controllability of second-order impul-sive neutral stochastic functional differential equations with infinite delays, or in other words, the litera-ture about controllability of second-order impulsive neutral stochastic functional differential equations with infinite delays is very scarce. To be more precise, Balasubramaniam and Muthukumar (2009) dis-cussed on approximate controllability of second-order stochastic distributed implicit functional dif-ferential systems with infinite delay. Mahmudova and McKibben (2006) established the results concerning the global existence, uniqueness, approximate, and exact controllability of mild solutions for a class of abstract second-order damped McKean–Vlasov stochastic evolution equations in a real separable Hilbert space. More recently, using Holder’s inequality, stochastic analysis, and fixed point strategy, Sakthivel, Ren, and Mahmudov (2010) considered sufficient conditions for the approximate controllability of nonlinear second-order stochastic infinite dimensional dynamical systems with impul-sive effects. And Muthukumar and Balasubramaniam (2010) investigated sufficient conditions for

of 16


the approximate controllability of a class of second-order damped McKean–Vlasov stochastic evolu-tion equations in a real separable Hilbert space.

On the other hand, in recent years, stochastic partial differential equations with Poisson jumps have gained much attention since Poisson jumps not only exist widely, but also can be used to study many phenomena in real lives. Therefore, it is necessary to consider the Poisson jumps into the sto-chastic systems. For instance, Luo and Liu (2008) studied the existence and uniqueness of mild solu-tions to stochastic partial functional differential equations with Markovian switching and Poisson jumps using the Lyapunov–Razumikhin technique. Ren, Zhou, and Chen (2011) investigated the exist-ence, uniqueness, and stability of mild solutions for a class of time-dependent stochastic evolution equations with Poisson jumps. More specifically, just recently, there is an article on the complete control-lability of stochastic evolution equations with jumps in a separable Hilbert space discussed by Sakthivel and Ren (2011) and in reference Ren, Dai, and Sakthivel (2013), Ren et al. studied the approximate con-trollability of stochastic differential systems driven by Teugels martingales associated with a Lévy pro-cess. For more details about the stochastic partial differential equations with Poisson jumps, one can see a recent monograph of Peszat and Zabczyk (2007) as well as papers of Cao (2005), Marinelli & Rockner (2010), Rockner and Zhang (2007), and the references therein.

To the best of our knowledge, there is no work reported on nonlocal second-order impulsive neu-tral stochastic functional integro-differential equations with infinite delay and Poisson jumps. To close the gap, motivated by the above works, the purpose of this paper is to study the controllability of nonlo-cal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps in Hilbert spaces. More precisely, we consider the following form:

where 0 < t1< t

2< ⋯ < tn < T, n ∈ ℕ; x(⋅) is a stochastic process taking values in a real separable

Hilbert space ℍ; A:D(A) ⊂ ℍ → ℍ is the infinitesimal generator of a strongly continuous cosine fam-ily on ℍ. The history xt: J0 → ℍ, xt(�) = x(t + �) for t ≥ 0, belongs to the phase space , which will be described in Section 2. Assume that the mappings f ,g: J × × ℍ → ℍ, �: J × J × → 0

2,

�i : J × J × → ℍ, i = 1, 2, I1k , I2k : → ℍ, k = 1,m, q:n → , and � : J × ℍ × → ℍ are appropri-ate functions to be specified later. The control function u(⋅) takes values in L2(J,U) of admissible control functions for a separable Hilbert space U and B is a bounded linear operator from U into ℍ. Furthermore, let 0 = t

0< t

1< ⋯ < tm < tm+1 = T be prefixed points, and Δx(tk) = x(t

+

k ) − x(t−

k ) represents the jump of the function x at time tk with Ik, determining the size of the jump, where x(t

+

k ) and x(t−k ) represent the right and left limits of x(t) at t = tk, respectively. Similarly x

�(t+k ) and x�(t−k )

denote, respectively, the right and left limits of x�(t) at tk. Let �(t) ∈ 2(Ω,) and x1(t) be ℍ-valued t-measurable random variables independent of the Wiener process {w(t)} and the Poisson point process p(⋅) with a finite second moment.

The main techniques used in this paper include the Banach contraction principle and the theories of a strongly continuous cosine family of bounded linear operators.

The structure of this paper is as follows: in Section 2, we briefly present some basic notations, preliminaries, and assumptions. The main results in Section 3 are devoted to study the controllability for the system (1.1) with their proofs. An example is given in Section 4 to illustrate the theory. In Section 5, concluding remarks are given.

(1.1)

⎧⎪⎪⎪⎨⎪⎪⎪⎩

d�x�(t) − g

�t, xt, � t0 �1(t, s, xs)ds�� = �Ax(t) + f �t, xt, � t0 �2(t, s, xs)ds� + Bu(t)�dt

+ � t−∞

�(t, s, xs)dw(s) + � ��t, x(t−), v�Ñ(dt,dv), tk ≠ t ∈ J: = [0, T],Δx(tk) = I

1

k (xtk), k = {1,⋯ ,m} = : 1,m,

Δx�(tk) = I2

k (xtk), k = 1,m,

x�(0) = x1∈ ℍ,

x(0) − q(xt1

, xt2

,⋯ , xtn) = x

0= � ∈ , for a.e. s ∈ J

0: = (−∞, 0],

of 16


2. PreliminariesIn this section, we briefly recall some basic definitions and results for stochastic equations in infinite dimensions and cosine families of operators. For more details on this section, we refer the reader to Da Prato and Zabczyk (1992), Fattorini (1985), Protter (2004), and Travis and Webb (1978).

Let (ℍ, ‖ ⋅ ‖ℍ, ⟨⋅, ⋅⟩

ℍ) and (�, ‖ ⋅ ‖

�, ⟨⋅, ⋅⟩

�) denote two real separable Hilbert spaces, with their

vectors, norms, and their inner products, respectively. We denote by (𝕂;ℍ) the set of all linear bounded operators from � into ℍ, which is equipped with the usual operator norm ‖ ⋅ ‖. In this paper, we use the symbol ‖ ⋅ ‖ to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. Let (Ω, , � = {t}t≥0,P) be a complete filtered probability space satisfying the usual condition (i.e. it is right continuous and

0 contains all P-null sets). Let w = (w(t))t≥0

be a Q-Wiener process defined on the probability space (Ω, , � ,P) with the covariance operator Q such that Tr(Q) < ∞. We assume that there exists a complete orthonormal system {ek}k≥1 in �, a bounded sequence of nonnegative real numbers �k such that Qek = �kek, k = 1, 2,… , and a sequence of independent Brownian motions {�k}k≥1 such that

Let 02=

2(Q

1

2𝕂;ℍ) be the space of all Hilbert–Schmidt operators from Q1

2� into ℍ with the inner product ⟨Ψ,�⟩0

2

= Tr[ΨQ�∗], where �∗ is the adjoint of the operator �. Let p = p(t), t ∈ Dp (the domain of p(t)) be a stationary t-Poisson point process taking its value in a measurable space ( ,�( )) with a �-finite intensity measure �(dv) by N(dt,dv) the Poisson counting measure associated with p, that is,

for any measurable set ∈ �(� − {0}), which denotes the Borel �-field of (� − {0}). Let

be the compensated Poisson measure that is independent of w(t). Denote by 2(J × ;ℍ) the space of all predictable mappings � : J × → ℍ for which

We may then define the ℍ-valued stochastic integral � t0� �(t, v)Ñ(dt,dv), which is a centered square-

integrable martingale. For the construction of this kind of integral, we can refer to Protter (2004).

The collection of all strongly measurable, square-integrable ℍ-valued random variables, denoted by

2(Ω,ℍ), is a Banach space equipped with norm ‖x‖

2

=�E‖x‖2� 12. Let �(J,

2(Ω,ℍ)) be the Banach

space of all continuous maps from J to 2(Ω,ℍ), satisfying the condition supt∈J E‖x(t)‖2 < ∞. An

important subspace is given by 02(Ω,ℍ) = {f ∈

2(Ω,ℍ): f is

0-measurable}. Further, let

𝔽2(0, T;ℍ) =

�g: J × Ω → ℍ:g(⋅) is 𝔽 -progressively measurable andE

� �J‖g(t)‖2

ℍdt�< ∞

�.

Next, to be able to access controllability for the system (1.1), we need to introduce the theory of cosine functions of operators and the second-order abstract Cauchy problem.

⟨w(t), e⟩�=

∞�k=1

√�k⟨ek, e⟩��k(t), e ∈ �, t ≥ 0.

N(t, ) = ∑s∈Dp , s≤t

� (p(s))

Ñ(dt,dv): = N(dt,dv) − �(dv)dt

�t

0 � E‖𝛾(t, v)‖2

ℍ𝜆(dv)dt < ∞.

of 16


Definition 2.1 (1) The one-parameter family {C(t)}t∈ℝ ⊂ (ℍ) is said to be a strongly continuous cosine family if the following hold:

(i) C(0) = I, I is the identity operators in ℍ;

(ii) C(t)x is continuous in t on ℝ for any x ∈ ℍ; and

(iii) C(t + s) + C(t − s) = 2C(t)C(s) for all t, s ∈ ℝ.

(2) The corresponding strongly continuous sine family {S(t)}t∈ℝ ⊂ (ℍ), associated to the given strongly continuous cosine family {C(t)}t∈ℝ ⊂ (ℍ) is defined by

(3) The infinitesimal generator A:ℍ → ℍ of {C(t)}t∈ℝ ⊂ (ℍ) is given by

It is well known that the infinitesimal generator A is a closed, densely defined operator on ℍ, and the following properties hold (see Travis & Webb, 1978).

Proposition 2.1 Suppose that A is the infinitesimal generator of a cosine family of operators {C(t)}t∈ℝ. Then, the following hold:

(i) There exist a pair of constants MA ≥ 1 and � ≥ 0 such that ‖C(t)‖ ≤ MAe��t�, and hence ‖S(t)‖ ≤ MAe��t�;

(ii) A ∫ rsS(u)xdu = [C(r) − C(s)]x, for all 0 ≤ s ≤ r < ∞; and

(iii) There exists N ≥ 1 such that ‖S(s) − S(r)‖ ≤ N�� rs e��s�ds��, 0 ≤ s ≤ r < ∞.Thanks to the Proposition 2.1 and the uniform boundedness principle that we see a direct consequence that both {C(t)}t∈J and {S(t)}t∈J are uniformly bounded by M̃ = MAe

�|T|.

The existence of solutions for the second-order linear abstract Cauchy problem

where h: J → ℍ is an integrable function that has been discussed in Travis and Webb (1977). Similarly, the existence of solutions of the semilinear second-order abstract Cauchy problem has been treated in Travis and Webb (1978).

Definition 2.2 The function x(⋅) given by

is called a mild solution of (2.1), and that when z ∈ ℍ, x(⋅) is continuously differentiable and

S(t)x = ∫t

0

C(s)xds, t ∈ ℝ, x ∈ ℍ.

Ax =d2

dt2C(t)x

|||t=0, for all x ∈ D(A) = {x ∈ ℍ:C(⋅) ∈ �2(ℝ,ℍ)}.

(2.1)

{x

��

(t) = Ax(t) + h(t), t ∈ J,

x(0) = z, x�(0) = w,

x(t) = C(t)z + S(t)w + ∫t

0

S(t − s)h(s)ds, t ∈ J,

x�(t) = AS(t)z + C(t)w + ∫t

0

C(t − s)h(s)ds, t ∈ J.

of 16


For additional details about cosine function theory, we refer the reader to Travis and Webb (1977, 1978).

Since the system (1.1) has impulsive effects, the phase space used in Balasubramaniam and Ntouyas (2006) and Park et al. (2007) cannot be applied to these systems. So, we need to introduce an abstract phase space , as follows:

Assume that l:J0→ (0,+∞) is a continuous function with l

0= ∫

J0

l(t)dt < ∞. For any a > 0, we define

: = �� :J0→ ℍ:(E‖�(�)‖2) 12 is a bounded and measurable function on [−a, 0] and

∫J0

l(s) sup𝜃∈[s,0](E‖𝜓(𝜃)‖2)

1

2 ds < +∞�

.

If is endowed with the norm

then, it is clear that (, ‖ ⋅ ‖) is a Banach space (Hino, Murakami, & Naito, 1991).Let JT = (−∞, T]. We consider the space

T : ={x:JT → ℍ such that xk ∈ �(Jk,ℍ) and there exist x(t

−

k ) and x(t+

k ) with

x(t−k ) = x(t+

k ), x(0) − q(xt1 , xt2 ,⋯ , xtn ) = � ∈ , k = 1,m}

,

where xk is the restriction of x to Jk = (tk, tk+1], k = 1,m. Set ‖ ⋅ ‖T be a seminorm in T defined by

Now, we recall the following useful lemma that appeared in Chang (2007).

Lemma 2.1 (Chang, 2007) Assume that x ∈ T, then for t ∈ J, xt ∈ . Moreover,

Next, we give the definition of mild solution for (1.1).

Definition 2.3 An t-adapted càdlàg stochastic process x:JT → ℍ is called a mild solution of (1.1) on JT if x(0) − q(xt1 , xt2 ,⋯ , xtn ) = x0 = � ∈ and x�(0) = x1 ∈ ℍ, satisfying �, x1, q ∈ 02(Ω,ℍ); the func-tions C(t − s)g(s, xs, ∫ s0 �1(s, �, x� )d�) and S(t − s)f (s, xs, ∫ s0 �2(s, �, x� )d�) are integrable on [0, T) such that the following conditions hold:

(i) {xt:t ∈ J} is a -valued stochastic process;(ii) For arbitrary t ∈ J, x(t) satisfies the following integral equation:

‖�‖ = �J0

l(s) sup�∈[s,0]

(E‖�(�)‖2) 12 ds, ∀� ∈ ,

‖x‖T = ‖�‖ + sups∈J

(E‖x(s)‖2) 12 , x ∈ T .

l0

�E‖x(t)‖2� 12 ≤ ‖xt‖ ≤ ‖x0‖ + l0 sup

s∈[0,t]

(E‖x(s)‖2) 12 .

(2.2)

x(t) =C(t)[𝜑(0) + q(xt1, xt

2,⋯ , xtn

)(0)] + S(t)[x1− g(0, x

0, 0)]

+ �t

0

C(t − s)g(s, xs, �s

0

𝜎1(s, 𝜏, x

𝜏)d𝜏)ds

+ �t

0

S(t − s)f (s, xs, �s

0

𝜎2(s, 𝜏, x

𝜏)d𝜏)ds + �

t

0

S(t − s)Bu(s)ds

+ �t

0

S(t − s) �s

−∞

𝜎(s, 𝜏, x𝜏)dw(𝜏)ds + �

t

0

S(t − s) � 𝛾(t, x(t−), v

)�N(dt,dv)

+∑0

of 16


(iii) Δx(tk) = I1

k (xtk), Δx�(tk) = I

2

k (xtk), k = 1,m.

Definition 2.4 The system (1.1) is said to be controllable on the interval JT, if for every initial stochas-tic process � ∈ defined on J

0, x�(0) = x

1∈ ℍ and y

1∈ ℍ; there exists a stochastic control u ∈ L2(J,U)

which is adapted to the filtration {t}t∈J such that the solution x(⋅) of the system (1.1) satisfies x(T) = y

1, where y

1 and T are the preassigned terminal state and time, respectively.

To prove our main results, we list the following basic assumptions of this paper.

(H1) There exists positive constants MC, MS, and M�1

such that for all t, s ∈ J,x, y ∈

(H2) The function g:J × × ℍ → ℍ is continuous and there exists a positive constant Mg such that for all t ∈ J,x

1, x

2∈ , y

1, y

2∈

2(Ω,ℍ)

(H3) For each (t, s) ∈ J × J, the function �2:J × J × → ℍ is continuous and there exists a positive

constant M�2

such that for all t, s ∈ J,x, y ∈

(H4) The function f :J × × ℍ → ℍ is continuous and there exists a positive constant Mf such that for all t ∈ J,x

1, x

2∈ , y

1, y

2∈

2(Ω,ℍ)

(H5) The functions I1k, I2

k ∈ �(,ℍ), k = 1,m and there exist positive constants MI1k ,MI1k, MI2k, andMI2k

such that for all x, y ∈

(H6) For each � ∈ , h(t) = limc→∞ ∫ 0−c �(t, s,�)dw(s) exists and continuous. Further, there exists a positive constant Mh such that

(H7) The function �:J × J × → (𝕂,ℍ) is continuous and there exists positive constants M�,M

�

such that for all s, t ∈ J and x, y ∈

(H8) The function q:n → is continuous and there exist positive constants Mq,Mq such that for all x, y ∈ , t ∈ J

0

‖C(t)‖2 ≤ MC , ‖S(t)‖2 ≤ MS;

E��

t

0

[�1(t, s, x) − �

1(t, s, y)]ds

��2 ≤ M

�1

‖x − y‖2.

E‖g(t, x1, y

1) − g(t, x

2, y

2)‖2 ≤ Mg(‖x1 − x2‖2 + E‖y1 − y2‖2).

E��

t

0

[�2(t, s, x) − �

2(t, s, y)]ds

��2 ≤ M

�2

‖x − y‖2.

E‖f (t, x1, y

1) − f (t, x

2, y

2)‖2 ≤ Mf (‖x1 − x2�2 + E‖y1 − y2‖2).

E‖I1k (x)‖2 ≤ MI1k , E‖I2k (x)‖2 ≤ MI2k ;E‖I1k (x) − I1k (y)‖2 ≤ MI1k‖x − y‖2, E‖I2k (x) − I2k (y)‖2 ≤ MI2k‖x − y‖2.

E‖h(t)‖2 ≤ Mh.

E‖�(t, s, x)‖202

≤ M�;

E‖�(t, s, x) − �(t, s, y)‖202

≤ M�‖x − y‖2.

E‖q(xt1

, xt2

,⋯ , xtn)(t)‖2 ≤ Mq;

of 16


(H9) The linear operator W:L2(J,U) → L2(Ω,ℍ) defined by

has an induced inverse W−1 which takes values in L2(J,U)∕KerW (see Carimichel & Quinn, 1984) and there exist two positive constants MB and MW such that

(H10) The function � :J × ℍ × → ℍ is a Borel measurable function and satisfies the Lipschitz continuity condition, the linear growth condition, and there exists positive constants M

�,M

� such

that for any x, y ∈ 𝔽2(0, T;ℍ), t ∈ J

3. Main resultsIn this section, we shall investigate the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps in Hilbert spaces.

The main result of this section is the following theorem.

Theorem 3.1 Assume that the assumptions (H1) − −(H10) hold. If Ξ < 1 and Θ < 1, then the system (1.1) is controllable on JT, where

Proof Using the assumption (H9), for an arbitrary function x(⋅), we define the control process

E‖q(xt1

, xt2

,⋯ , xtn)(t) − q(yt

1

, yt2

,⋯ , ytn)(t)‖2 ≤ Mq‖x − y‖2.

Wu = ∫J S(T − s)Bu(s)ds

‖B‖2 ≤ MB and ‖W−1‖2 ≤ MW .

E

��t

0 � ‖�(t, x(s−), v)‖2

ℍ�(dv)ds

�∨ E

��t

0 � ‖�(t, x(s−), v)‖4

ℍ�(dv)ds

� 12

≤ M�E �

t

0

�1 + ‖x(s)‖2

ℍ

�ds;

E

��t

0 � ‖�(t, x(s−), v) − �(t, y(s−), v)‖2

ℍ�(dv)ds

�

∨E��t

0 � ‖�(t, x(s−), v) − �(t, y(s−), v)‖4

ℍ�(dv)ds

� 12 ≤ M

�E �

t

0

‖x(s) − y(s)‖2ℍds.

Ξ: = 32(1 + 9T2MBMSMW

){2l20T2[MCMg(1 + 2M�

1) +MSMf (1 + 2M�

2)]+ TM

�C̃

},

Θ: =

{98l2

0T2MBMCMSMWMq + 14l

2

0

(1 + 7T2MBMSMW

)[T2MCMg

(1 +M

�1

)

+ T2MSMf(1 +M

�2

)+ T3MSM�Tr(Q) +

TM�C̃

2l20

+mMC

m∑k=1

MI1k+mMS

m∑k=1

MI2k

]}.

of 16


We transform (1.1) into a fixed point problem. Consider the operator Π:T → T defined by

In what follows, we shall show that using the control uTx(⋅), the operator Π has a fixed point, which is then a mild solution for system (1.1).

Clearly, Πx(T) = y1.

For � ∈ , we defined �̃ by

then �̃ ∈ T.Set x(t) = z(t) + �̃(t), t ∈ JT. It is easy to see that x satisfies (2.2) if and only if z satisfies z0 = 0, x�(0) = x

1= z�(0) = z

1 and

(3.1)

uTx(t) =W−1{y1− C(T)[𝜑(0) + q(xt

1, xt

2,⋯ , xtn

)(0)] − S(T)[x1− g(0, x

0, 0)]

− �T

0

C(T − s)g(s, xs, �s

0

𝜎1(s, 𝜏, x

𝜏)d𝜏)ds −

∑0

of 16


where uTz+�̃(t) is obtained from (3.1) by replacing xt = zt + �̃t.

Let 0T = {y ∈ T :y0 = 0 ∈ }. For any y ∈0T, we have

and thus (0T , ‖ ⋅ ‖T) is a Banach space. Set

then Br ⊆ 0T is uniformly bounded, and for u ∈ Br, by Lemma 2.1, we have

Define the map Π:0T → 0T defined by Πz(t) = 0, for t ∈ J0 and

Obviously, the operator Π has a fixed point which is equivalent to prove that Π has a fixed point. Note that, by our assumptions, we infer that all the functions involved in the operator are continuous, therefore Π is continuous.

Let z, z ∈ 0T. From (3.1), by our assumptions, Hölder’s inequality, the Doob martingale inequality, and the Burkholder–Davis–Gundy inequality for pure jump stochastic integral in Hilbert space (see Luo & Liu, 2008), Lemma 2.1, and in view of (3.2), for t ∈ J, we obtain the following estimates.

and

‖y‖T = ‖y0‖ + sups∈J

(E‖y(s)‖2) 12 = sups∈J

(E‖y(s)‖2) 12 ,

Br = {y ∈ 0T :‖y‖2T ≤ r} for some r ≥ 0,

(3.2)

‖zt + �𝜑t‖2 ≤ 2(‖zt‖2 + ‖�𝜑t‖2)≤ 4�l2

0sups∈[0,t]

(E‖z(s)‖2 + ‖z0‖2 + l20 sup

s∈[0,t]

(E‖�𝜑(s)‖2 + ‖�𝜑0‖2

�

≤ 4l20

�r + 2MC

�E‖𝜑(0)‖2 +Mq

��+ 4‖�𝜑‖2

: = r⋆.

Πz(t) =S(t)[z1− g(0, �𝜑

0, 0)] + �

t

0

C(t − s)g(s, zs + �𝜑s, �s

0

𝜎1(s, 𝜏, z

𝜏+ �𝜑

𝜏)d𝜏)ds

+ �t

0

S(t − s)f (s, zs + �𝜑s, �s

0

𝜎2(s, 𝜏, z

𝜏+ �𝜑

𝜏)d𝜏)ds + �

t

0

S(t − s)BuTz+�𝜑(s)ds

+ �t

0

S(t − s)[h(s) + �

s

0

𝜎(s, 𝜏, z𝜏+ �𝜑

𝜏)dw(𝜏)

]ds

+ �t

0

S(t − s) � 𝛾(t, z(t−) + �𝜑(t−), v

)�N(dt,dv)

+∑0

of 16


where �C > 0 is a positive constant and

Lemma 3.1 Under the assumptions of Theorem 3.1, there exists r > 0 such that Π(Br) ⊆ Br.

Proof If this property is false, then for each r > 0, there exists a function zr(⋅) ∈ Br, but Π(zr) ∉ Br, i.e.

‖Π(zr)(t)‖2 > r for some t ∈ J. However, by our assumptions, Hölder’s inequality and the Burkholder–Davis–Gundy inequality, we have

where

Dividing both sides of (3.3) by r and noting that

and taking the limit as r → ∞, we obtain

which contradicts our assumption. Thus, for some positive number r , Π(Br) ⊆ Br. This completes the proof of Lemma 3.1.

Lemma 3.2 Under the assumptions of Theorem 3.1, Π:0T → 0T is a contraction mapping.Proof Let z, z ∈ 0T. Then, by our assumptions, Hölder’s inequality, Burkholder–Davis–Gundy’s inequal-

E‖uTz+�̃(t) − uTz+�̃(t)‖2

≤ 14l20MW

�MCMq + T

2MCMg�1 +M

�1

�+ T2MSMf

�1 +M

�2

�

+ T3MSM�Tr(Q) +TM

�C̃

2l20

+mMC

m�k=1

MI1k+mMS

m�k=1

MI2k

�sups∈J

E‖z(t) − z(t)‖2,

C1: = T sup

(t,s)∈J×J

�21(t, s, 0), C

2: = sup

t∈J

‖g(t, 0, 0)‖2,

C3: = T sup

(t,s)∈J×J

�22(t, s, 0), C

4: = sup

t∈J

‖f (t, 0, 0)‖2.

(3.3)

r < E‖Π(zr)(t)‖2

≤ 8�2MS

�E‖x

1‖2 + 2(Mg‖�𝜑‖2 + C2)

�+ 2T2MC

�Mg

�[1 + 2M

𝜎1]r⋆ + 2C

1

�+ C

2

�

+ 2T2MS

�Mf

�[1 + 2M

𝜎2]r⋆ + 2C

3

�+ C

4

�+ T2MSMB�

+ 2T2MS(Mh + TTr(Q)M𝜎) + TM𝛾�C(1 +

r⋆

l20

) +mMC

m�k=1

MI1k+mMS

m�k=1

MI2k

�,

≤ M⋆⋆ + 8(1 + 9T2MBMSMW)�2T2

�MCMg(1 + 2M𝜎

1) +MSMf (1 + 2M𝜎

2)�+TM

𝛾�C

l20

�r⋆,

M⋆⋆

: = 72(1 + 9T2MBMSMW)�E‖y

1‖2 + 2MC(E‖𝜑(0)‖2 +Mq)

�+ 8(1 + 9T2MBMSMW)

×

�2MS

�E‖x

1‖2 + 2(Mg‖�𝜑‖2 + C2)

�+ 2T2MC(2MgC1 + C2) + 2T

2MS(2Mf C3 + C4)

+ 2T2MS(Mh + TTr(Q)M𝜎) + TM𝛾�C +mMC

m�k=1

MI1k+mMS

m�k=1

MI2k

�.

r⋆ = 4l20

�r + 2MC

�E‖𝜑(0)‖2 +Mq

��+ 4‖�𝜑‖2

r→∞��→ ∞

1 ≤ Ξ

of 16


ity, Lemma 2.1, and since ‖z0‖2 = 0 and ‖z0‖2 = 0, for each t ∈ J, we see that

Taking the supremum over t, we obtain

By our assumption, we conclude that Π is a contraction on 0T. Thus, we have completed the proof of Lemma 3.2.

On the other hand, by Banach fixed point theorem, there exists a unique fixed point x(⋅) ∈ 0T such that (Πx)(t) = x(t). This fixed point is then the mild solution of the system (1.1). Clearly, x(T) = (Πx)(T) = y

1.

Thus, the system (1.1) is controllable on JT. The proof for Theorem 3.1 is thus complete.

Now, let us consider a special case for the system (1.1).

If �(t, x(t−), v

) ≡ 0, the system (1.1) becomes the following nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay without Poisson jumps:

Corollary 3.1 Assume that all assumptions of Theorem 3.1 hold except that (H11) and Ξ,Θ replaced by Ξ̂, Θ̂ such that

and

If �Ξ < 1 and �Θ < 1, then the system (3.4) is controllable on JT.

E‖(Πz)(t) − (Πz)(t)‖2

≤ 14l20

�T2MCMg

�1 +M

�1

�+ T2MSMf

�1 +M

�2

�+ T3MSM�Tr(Q) +

TM�C̃

2l20

+mMC

m�k=1

MI1k+mMS

m�k=1

MI2k

�sups∈J

E‖z(t) − z(t)‖2 + 7T2MSMBE‖uTz+�̃(t) − uTz+�̃(t)‖2

≤�98l2

0T2MBMCMSMWMq + 14l

2

0

�1 + 7T2MBMSMW

�

×�T2MCMg

�1 +M

�1

�+ T2MSMf

�1 +M

�2

�+ T3MSM�Tr(Q) +

TM�C̃

2l20

+mMC

m�k=1

MI1k+mMS

m�k=1

MI2k

��sups∈J

E‖z(t) − z(t)‖2.

‖(Πz) − (Πz)‖2T ≤ Θ‖z − z‖2T .

(3.4)

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

d�x�(t) − g

�t, xt, � t0 �1(t, s, xs)ds�� = �Ax(t) + f �t, xt, � t0 �2(t, s, xs)ds� + Bu(t)�dt

+ � t−∞

�(t, s, xs)dw(s), tk ≠ t ∈ J: = [0, T],Δx(tk) = I

1

k (xtk), k = {1,⋯ ,m} = :1,m,

Δx�(tk) = I2

k (xtk), k = 1,m,

x�(0) = x1∈ ℍ,

x(0) − q(xt1, xt

2,⋯ , xtn

) = x0= � ∈ , for a.e.s ∈ J

0: = (−∞, 0],

Ξ̂: = 56l20T2(1 + 8T2MBMSMW

)[MCMg(1 + 2M�

1) +MSMf (1 + 2M�

2)],

Θ̂: =

{72l2

0T2MBMCMSMWMq + 12l

2

0

(1 + 6T2MBMSMW

)[T2MCMg

(1 +M

�1

)

+ T2MSMf(1 +M

�2

)+ T3MSM�Tr(Q) +mMC

m∑k=1

MI1k+mMS

m∑k=1

MI2k

]}.

of 16


4. ApplicationIn this section, the established previous results are applied to study the controllability of the stochas-tic nonlinear wave equation with infinite delay and Poisson jumps. Specifically, we consider the following controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps of the form:

where �(t) is a standard one-dimensional Wiener process in ℍ, defined on a stochastic basis (Ω, ,P); = {v ∈ ℝ:0 < ‖v‖

ℝ≤ a, a > 0}; 0 < t

1< t

2< ⋯ < tn < T, n ∈ ℕ;

0 = t0< t

1< ⋯ < tm < tm+1 < T are prefixed numbers, and � ∈ .

Let p = p(t), t ∈ Dp be a �-valued �-finite stationary Poisson point process (independent of �(t)) on a complete probability space with the usual condition (Ω, , (t)t⩾0,P). Let Ñ(ds,dv): = N(ds,dv) − �(dv)ds, with the characteristic measure �(dv) on ∈ �(� − {0}). Assume that

To rewrite (4.1) into the abstract from of (1.1), we consider the space ℍ = L2([0,�]) with the norm ‖ ⋅ ‖. Let en(�): =

√2

�sinn�, n = 1, 2, 3,… denote the completed orthogonal basics in ℍ and

�(t) =∑∞

n=1

√�n�n(t)en, t ≥ 0,𝜆n > 0, where {�n(t)}n≥0 are one-dimensional standard Brownian

motions mutually independent on a usual complete probability space (Ω, , (t)t⩾0,P).Defined A:ℍ → ℍ by A = �

2

��2, with domain D(A) = ℍ2([0,�]) ∩ ℍ1

0([0,�]), where

and

Then,

(see Travis & Webb, 1987, Example 5.1). Using (4.2), one can easily verify that the operators C(t) defined by

(4.1)

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

�

�t

��

�ty(t, �) − � t

−∞�1(t, �, s − t, )P

1(y(s, �))ds − � t

0� s−∞b1(s − �)P

2(y(�, �))d�ds

�

=�

�2

��2y(t, �) + � t

−∞�2(t, �, s − t, )G

1(y(s, �))ds + � t

0� s−∞b2(s − �)G

2(y(�, �))d�ds

+b(�)u(t)�dt + � t

−∞�(s − t)y(t, �)d�(s) + � y(t−, �)vÑ(dt,dv), tk ≠ t ∈ J, � ∈ [0,�],

Δy(tk)(�) = � tk−∞ �k(tk − s)y(s, �)ds, k = 1,m, � ∈ [0,�],Δy�(tk)(�) = � tk−∞ �k(tk − s)y(s, �)ds, k = 1,m, � ∈ [0,�],y(t, 0) = y(t,�) = 0, t ∈ J,�

�ty(0, �) = x

1(�), � ∈ [0,�],

y(t, �) −∑n

i=1 � �0 pi(�, � )y(ti , � )d� = �(t, �), t ∈ J0, � ∈ [0,�],

� v2𝜆(dv) < ∞ and � v

4𝜆(dv) < ∞.

ℍ1

0([0,�]) = {w ∈ L2([0,�]):

�w

�z∈ L2([0,�]),w(0) = w(�) = 0}

ℍ2([0,�]) = {w ∈ L2([0,�]):

�w

�z,�2w

�z2∈ L2([0,�])}.

(4.2)Ax = −∞�n=1

n2⟨x, en⟩en, x ∈ D(A),

C(t)x =

∞�n=1

cos(nt)⟨x, en⟩en, t ∈ ℝ,

of 16


from a cosine function on ℍ, with associated sine function

It is clear that (see Travis & Webb, 1977), for all x ∈ ℍ, t ∈ ℝ,C(⋅)x and S(⋅)x are periodic functions with ‖C(t)‖ ≤ 1 and ‖S(t)‖ ≤ 1. Thus, (H1) is true.

Now, we give a special -space. Let l(s) = e2s, s ≤ 0, then l0= ∫

J0

l(s)ds = 12 and define

It follows from Hino et al. (1991) that (, ‖ ⋅ ‖) is a Banach space. Hence, for (t,�) ∈ J × , where �(�)x = �(�, x), (�, x) ∈ J

0× [0,�]. Let y(t)(�) = y(t, �).

To study the system (4.1), we assume that the following conditions hold:

(i) Let B ∈ (ℝ,ℍ) be defined as

(ii) The linear operator W: L2(J,U) → ℍ defined by

is a bounded linear operator but not necessarily one-to-one. Let KerW = {u ∈ L2(J,U):Wu = 0} be null space of W and [KerW]⟂ be its orthogonal complement in L2(J,U). Let W∗:[KerW]⟂ → Range(W) be the restriction of W to [KerW]⟂, W∗ is necessarily one-to-one operator. The inverse mapping theo-rem says that (W∗)−1 is bounded since [KerW]⟂ and Range(W) are Banach spaces. Since the inverse operator W−1 is bounded and takes values in L2(J,U)∕KerW, the assumption (H9) is satisfied.

(iii) The functions pi :[0,�] × 0,�] → ℝ are �2-functions, for each i = 1,n.

(iv) The functions �k, �k ∈ �(ℝ,ℝ) such that for k = 1,m,

We define the functions g, f :J × × ℍ → ℍ, �:J × J × → 02, � :J × ℍ × → ℍ, and I1k , I2k : → ℍ,

k = 1,m by

S(t)x =

∞�n=1

sin(nt)

n⟨x, en⟩en, t ∈ ℝ.

‖�‖ = �J0

l(s) sup�∈[s,0]

(E‖�(�)‖2) 12 ds, ∀� ∈ .

Bu(�) = b(�)u, 0 ≤ � ≤ �, u ∈ ℝ, b(�) ∈ L2([0,�]).

Wu = ∫J S(T − s)b(�)u(s)ds

MI1k= ∫J

0

l(s)𝜂2k (s)ds < ∞, MI2k= ∫J

0

l(s)𝜌2k (s)ds < ∞.

g(t,� ,V1�)(�) = ∫J

0

�1(t, �, �)P

1(�(�)(�))d� + V

1�(�),

f (t,� ,V2�)(�) = ∫J

0

�2(t, �, �)G

1(�(�)(�))d� + V

2�(�),

�(t, s,�)(�) = ∫J0

�(�)�(�)(�)d�, �(t,�(�), v) = �(�)v,

I1k (t,�)(�) = ∫J0

�k(−s)�(�)(�)ds, k = 1,m,

I2k (t,�)(�) = ∫J0

�k(−s)�(�)(�)ds, k = 1,m,

of 16


where

Then, the system (4.1) can be written in the abstract form as the system (1.1). Further, we can impose some suitable conditions on the above-defined functions as those in the assumptions (H1) − −(H10). Therefore, by Theorem 3.1, we can conclude that the system (4.1) is controllable on JT.

5. ConclusionIn this paper, we have studied the controllability for a class of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps in Hilbert spac-es, which is new and allows us to develop the controllability of the second-order stochastic partial differential equations. Using the Banach fixed point theorem combined with theories of a strongly con-tinuous cosine family of bounded linear operators, and stochastic analysis theory, the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infi-nite delay and Poisson jumps is obtained. In addition, an application is provided to illustrate the effective-ness of the controllability results obtained. The results in our paper extend and improve the corresponding ones announced by Arthi and Balachandran (2012), Balasubramaniam and Muthukumar (2009), Muthukumar and Rajivganthi (2013), Park, Balachandran, and Annapoorani (2009), Park, Balachandran, and Arthi (2009), Travis and Webb (1978), and some other results.

V1�(�) = ∫

t

0 ∫J0

b1(s − �)P

2(�(�)(�))d�ds, V

2�(�) = ∫

t

0 ∫J0

b2(s − �)G

2(�(�)(�))d�ds.

FundingThis work was supported by the China NSF [grant number 11171158].

Author detailsDiem Dang Huan1,2

E-mail: [email protected] Gao3

E-mail: [email protected] Faculty of Basic Sciences, Bacgiang Agriculture and Forestry University, Bacgiang, 21000 Vietnam.

2 Vietnam National University, Hanoi, 144 Xuan Thuy Street, Cau Giay, Hanoi, 10000 Vietnam.

3 School of Mathematical Science, Nanjing Normal University, Nanjing, 210023 P.R. China.

Citation informationCite this article as: Controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with delay and Poisson jumps, Diem Dang Huan & Hongjun Gao, Cogent Engineering (2015), 2: 1065585.

ReferencesAhmed, H. M. (2014a). Non-linear fractional integro-differential

systems with non-local conditions. IMA Journal of Mathematical Control. doi:10.1093/imamci/dnu049

Ahmed, H. M. (2014b). Controllability of impulsive neutral stochastic differential equations with fractional Brownian motion. IMA Journal of Mathematical Control. doi:10.1093/imamci/dnu019

Ahmed, H. M. (2014c). Approximate controllability of impulsive neutral stochastic differential equations with fractional Brownian motion in a Hilbert space. Advances in Difference Equations, 113, 1–11.

Arthi, G., & Balachandran, K. (2012). Controllability of damped second-order impulsive neutral functional differential systems with infinite delay. Journal of Optimization Theory and Applications, 152, 799–813.

Astrom, K. J. (1970). Introduction to stochastic control theory. New York, NY: Academic Press.

Balachandran, K., & Dauer, J. P. (2002). Controllability of nonlinear stochastic systems in Banach spaces. Journal of

Optimization Theory and Applications, 115, 7–28.Balasubramaniam, P., & Dauer, J. P. (2002). Controllability

of semilinear stochastic delay evolution equations in Hilbert spaces. International Journal of Mathematics and Mathematical Sciences, 31, 157–166.

Balasubramaniam, P., & Muthukumar, P. (2009). Approximate controllability of second-order stochastic distributed implicit functional differential systems with infinite delay. Journal of Optimization Theory and Applications, 143, 225–244.

Balasubramaniam, P., & Ntouyas, S. K. (2006). Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space. Journal of Mathematical Analysis, 324, 161–176.

Cao, G. (2005). Stochastic differential evolution equations in infinite dimensional (MS thesis), Huazhong University of Science and Technology, Wuhan.

Carimichel, N., & Quinn, M. D. (1984). Fixed point methods in nonlinear control (Lecture Notes in Control and Information Society, Vol. 75). Berlin: Springer.

Chang, Y. K. (2007). Controllability of impulsive functional differential systems with infinite delay in Banach spaces. Chaos, Solitons & Fractals, 33, 1601–1609.

Chang, Y. K., Anguraj, A., & Arjunan, M. M. (2008). Existence results for impulsive neutral functional differential equations with infinite delay. Nonlinear Analysis: Hybrid Systems, 2, 209–218.

Coron, J. M. (2007). Control and nonlinearity, mathematical surveys and monographs (Vol. 136). Providence, RI: American Mathematical Society.

Curtain, R. F., & Zwart, H. J. (1995). An introduction to infinite dimensional linear systems theory. Berlin: Springer.

Da Prato, G., & Zabczyk, J. (1992). Stochastic equations in infinite dimensions (Vol. 44). Cambridge: Cambridge University Press.

Fattorini, H. O. (1985). Second order linear differential equations in Banach spaces (North-Holland mathematics studies, Vol. 108). Amsterdam: North-Holland.

Gurtin, M. E., & Pipkin, A. C. (1968). A general theory of heat conduction with finite wave speed. Archive for Rational Mechanics and Analysis, 31, 113–126.

Hino, Y., Murakami, S., & Naito, T. (1991). Functional differential equations with infinite delay (Lecture notes in

mailto:[email protected]:[email protected]://dx.doi.org/10.1093/imamci/dnu049http://dx.doi.org/10.1093/imamci/dnu019http://dx.doi.org/10.1093/imamci/dnu019

of 16


© 2015 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.You are free to: Share — copy and redistribute the material in any medium or format Adapt — remix, transform, and build upon the material for any purpose, even commercially.The licensor cannot revoke these freedoms as long as you follow the license terms.

Under the following terms:Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.

mathematics, Vol. 1473). Berlin: Springer-Verlag.Karthikeyan, S., & Balachandran, K. (2009). Controllability of

nonlinear stochastic neutral impulsive systems. Nonlinear Analysis, 3, 266–276.

Karthikeyan, S., & Balachandran, K. (2013). On controllability for a class of stochastic impulsive systems with delays in control. International Journal of Systems Science, 44, 67–76.

Lakshmikantham, V., Baǐnov, D., & Simeonov, P. S. (1989). Theory of impulsive differential equations (Series in modern applied mathematics, Vol. 6). Teaneck, NJ: World Scientific Publishing.

Luo, J., & Liu, K. (2008). Stability of infinite dimensional stochastic evolution with memory and Markovian jumps. Stochastic Processes and their Applications, 118, 864–895.

Mahmudova, N. I., & McKibben, M. A. (2006). Abstract second-order damped McKean--Vlasov stochastic evolution equations. Stochastic Processes and their Applications, 24, 303–328.

Marinelli, C., & Röckner, M. (2010). Well-posedness and asymptotic behavior for stochastic reaction--diffusion equations with multiplicative Poisson noise. Electronic Journal of Probability, 15, 1528–1555.

Muthukumar, P., & Balasubramaniam, P. (2010). Approximate controllability of second-order damped McKean--Vlasov stochastic evolution equations. Computers & Mathematics with Applications, 60, 2788–2796.

Muthukumar, P., & Rajivganthi, C. (2013). Approximate controllability of fractional order neutral stochastic integro-differential systems with nonlocal conditions and infinite delay. Taiwanese Journal of Mathematics, 17, 1693–1713.

Ntouyas, S. K. (2009). D. ÓRegan; Some remarks on controllability of evolution equations in Banach spaces, Electronic Journal of Qualitative Theory of. Differential Equations, 29, 1–6.

Nunziato, J. W. (1971). On heat conduction in materials with memory. Quarterly of Applied Mathematics, 29, 187–204.

Park, J. Y., Balachandran, K., & Annapoorani, N. (2009). Existence results for impulsive neutral functional integrodifferential equations with infinite delay. Nonlinear Analysis, 71, 3152–3162.

Park, J. Y., Balachandran, K., & Arthi, G. (2009). Controllability of impulsive neutral integrodifferential systems with infinite delay in Banach spaces. Nonlinear Analysis: Hybrid Systems, 3, 184–194.

Park, J. Y., Balasubramaniam, P., & Kumaresan, N. (2007). Controllability for neutral stochastic functional integrodifferential infinite delay systems in abstract space. Numerical Functional Analysis and Optimization, 28, 1369–1386.

Peszat, S., & Zabczyk, J. (2007.). Stochastic partial differential equations with lévy noise. Cambridge: Cambridge

University Press. Protter, P. E. (2004). Stochastic integration and differential

equations (2nd ed.). New York, NY: Springer.Ren, Y., Dai, H. L., & Sakthivel, R. (2013). Approximate

controllability of stochastic differential systems driven by a lévy process. International Journal of Control, 86, 1158–1164.

Ren, Y., Zhou, Q., & Chen, L. (2011). Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with Poisson jumps and infinite delay. Journal of Optimization Theory and Applications, 149, 315–331.

Röckner, M., & Zhang, T. (2007). Stochastic evolution equation of jump type: Existence, uniqueness and large deviation principles. Potential Analysis, 26, 255–279.

Sakthivel, R., Mahmudov, N. I., & Lee, S.-G. (2009). Controllability of non-linear impulsive stochastic systems. International Journal of Control, 82, 801–807.

Sakthivel, R., & Ren, Y. (2011). Complete controllability of stochastic evolution equations with jumps. Reports on Mathematical Physics, 68, 163–174.

Sakthivel, R., Ren, Y., & Mahmudov, N. I. (2010). Approximate controllability of second-order stochastic differential equations with impulsive effects. Modern Physics Letters B, 24, 1559–1572.

Shen, L. J., & Sun, J. T. (2012). Approximate controllability of stochastic impulsive functional systems with infinite delay. Automatica, 48, 2705–2709.

Travis, C. C., & Webb, G. F. (1977). Compactness, regularity, and uniform continuity properties of strongly continuous cosine families. Houston Journal of Mathematics, 3, 555–567.

Travis, C. C., & Webb, G. F. (1978). Cosine families and abstract nonlinear second order differential equations. Acta Mathematica Academiae Scientiarum Hungaricae, 32, 76–96.

Travis, C. C., & Webb, G. F. (1987). Second oder differential equations in Banach spaces. In Proceedinds International Symposium on Nonlinear Equations in Abstract Spaces (pp. 331–361). New York, NY: Academic Press.

Yan, Z. M., & Yan, X. X. (2013). Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay. Collectanea Mathematica, 64, 235–250.

Yang, T. (2001). Impulsive systems and control: Theory and applications. Berlin: Springer.

Zabczyk, J. (1991). Controllability of stochastic linear systems. Systems & Control Letters, 1, 25–31.

Zabczyk, J. (1992). Mathematical control theory. Basel: Birkhauser.

1. Introduction2. Preliminaries3. Main results4. Application5. ConclusionReferences

Documents

Controllability of nonlocal second order impulsive …...neutral stochastic functional integro-differential equations with delay and Poisson jumps Diem Dang Huan1,2* and Hongjun Gao3