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

Abstract: The current paper is concerned with the controllability of nonlocal secondorder 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 sufficient 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.


Introduction
As one of the fundamental concepts in mathematical control theory, controllability plays an important 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 ABOUT THE AUTHORS Diem 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, 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 STATEMENT
Controllability 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.
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 theory 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 control 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 andÓRegan (2009), Sakthivel, Mahmudov, andLee (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 applications 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, 2009Park, 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 theorem. 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 nonlocal conditions and infinite delay.
By contrast, there has not been very much research on the controllability of second-order impulsive neutral stochastic functional differential equations with infinite delays, or in other words, the literature 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) discussed on approximate controllability of second-order stochastic distributed implicit functional differential 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 impulsive effects. And Muthukumar and Balasubramaniam (2010) investigated sufficient conditions for the approximate controllability of a class of second-order damped McKean-Vlasov stochastic evolution 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 stochastic systems. For instance, Luo and Liu (2008) studied the existence and uniqueness of mild solutions to stochastic partial functional differential equations with Markovian switching and Poisson jumps using the Lyapunov-Razumikhin technique. Ren, Zhou, and Chen (2011) investigated the existence, 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 controllability 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 controllability of stochastic differential systems driven by Teugels martingales associated with a Lévy process. 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 neutral 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 nonlocal 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 < t 1 < t 2 < ⋯ < t n < 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 family on ℍ. The history x t : J 0 → ℍ, x t ( ) = x(t + ) for t ≥ 0, belongs to the phase space , which will be described in Section 2. Assume that the mappings f , g: represents the jump of the function x at time t k with I k , 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 = t k , respectively. Similarly x � (t + k ) and x � (t − k ) denote, respectively, the right and left limits of x � (t) at t k . Let (t) ∈  2 (Ω, ) and x 1 (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.

Preliminaries
In 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 ( ;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 {e k } k≥1 in , a bounded sequence of nonnegative real numbers k such that Qe k = k e k , k = 1, 2, … , and a sequence of independent Brownian motions { k } k≥1 such that ) 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 : 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 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.
(2) The corresponding strongly continuous sine family {S(t)} t∈ℝ ⊂ (ℍ), associated to the given strongly continuous cosine family 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). 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 For additional details about cosine function theory, we refer the reader to 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: is a bounded and measurable function on [−a, 0] and If  is endowed with the norm then, it is clear that (, ‖ ⋅ ‖  ) is a Banach space (Hino, Murakami, & Naito, 1991).
We consider the space where x k is the restriction of x to J k = (t k , t k+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, x t ∈ . Moreover, Next, we give the definition of mild solution for (1.1).

S(t − s)Bu(s)ds
Definition 2.4 The system (1.1) is said to be controllable on the interval J T , if for every initial stochastic process ∈  defined on J 0 , x � (0) = x 1 ∈ ℍ and y 1 ∈ ℍ; there exists a stochastic control u ∈ L 2 (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.
(H9) The linear operator W:L 2 (J, U) → L 2 (Ω, ℍ) defined by has an induced inverse W −1 which takes values in L 2 (J, U)∕KerW (see Carimichel & Quinn, 1984) and there exist two positive constants M B and M W 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 ∈  2 (0, T;ℍ), t ∈ J

Main results
In 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 J T , where Proof Using the assumption (H9), for an arbitrary function x(⋅), we define the control process We transform (1.1) into a fixed point problem. Consider the operator Π: In what follows, we shall show that using the control u T x (⋅), the operator Π has a fixed point, which is then a mild solution for system (1.1).
For ∈ , we defined ̃ by It is easy to see that x satisfies (2.2) if and only if z satisfies z 0 = 0, where u T z+̃ (t) is obtained from (3.1) by replacing x t = z t +̃ t .
Let  0 T = {y ∈  T :y 0 = 0 ∈ }. For any y ∈ 0 T , we have and thus ( 0 T , ‖ ⋅ ‖ T ) is a Banach space. Set then B r ⊆  0 T is uniformly bounded, and for u ∈ B r , by Lemma 2.1, we have Define the map Π: 0 T →  0 T defined by Πz(t) = 0, for t ∈ J 0 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 ∈  0 T . 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. Proof If this property is false, then for each r > 0, there exists a function z r (⋅) ∈ B r , but Π(z r ) ∉ B r , i.e. ‖Π(z r )(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(B r ) ⊆ B r . This completes the proof of Lemma 3.1.
Lemma 3.2 Under the assumptions of Theorem 3.1, Π: 0 T →  0 T is a contraction mapping.
Proof Let z, z ∈  0 T . Then, by our assumptions, Hölder's inequality, Burkholder-Davis-Gundy's inequal- ity, Lemma 2.1, and since ‖z 0 ‖ 2  = 0 and ‖z 0 ‖ 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  0 T . 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(⋅) ∈  0 T 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 J T . 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 Ξ ,Θ such that and If � Ξ < 1 and � Θ < 1, then the system (3.4) is controllable on J T .

Application
In this section, the established previous results are applied to study the controllability of the stochastic 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 < ⋯ < t n < T, n ∈ ℕ; 0 = t 0 < t 1 < ⋯ < t m < t m+1 < T are prefixed numbers, and ∈ .

Assume that
To rewrite (4.1) into the abstract from of (1.1), we consider the space ℍ = L 2 ([0, ]) with the norm ‖ ⋅ ‖. Let e n ( ): = √ 2 sin n , n = 1, 2, 3, … denote the completed orthogonal basics in ℍ and (t) = ∑ ∞ n=1 √ n n (t)e n , t ≥ 0, n > 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).   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 J T .

Conclusion
In 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 spaces, 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 continuous cosine family of bounded linear operators, and stochastic analysis theory, the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps is obtained. In addition, an application is provided to illustrate the effectiveness 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), Annapoorani (2009), Park, Balachandran, andArthi (2009), Travis and Webb (1978), and some other results.