APPROXIMATE CONTROLLABILITY OF SECOND-ORDER IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY ∗

In this paper we study a kind of second-order impulsive stochastic differential equations with state-dependent delay in a real separable Hilbert space. Some sufficient conditions for the approximate controllability of this system are formulated and proved under the assumption that the corresponding deterministic linear system is approximately controllable. The results concerning the existence and approximate controllability of mild solutions have been addressed by using strongly continuous cosine families of operators and the contraction mapping principle. At last, an example is given to illustrate the theory.


Introduction
The stochastic differential equations (SDEs) in both finite dimensional and infinite dimensional spaces have been extensively studied.It has played an important role in many ways such as option pricing, forecast of the growth of population and so on.For first-order SDEs, some qualitative properties such as existence, controllability and stability have been investigated in several papers [5,12,19,25].In setting of second-order systems, it is advantages to treat second-order abstract differential equations directly rather than convert them to first-order systems.The secondorder SDEs are the precise model in continuous time to account for integrated processes that can be made stationary.For instance, it is useful for engineers to model mechanical vibrations or charge on a capacitor or condenser subjected to white noise excitation through second-order SDEs.Recently, the problem of controllability for second-order SDEs has received considerable attention.Mahmudov and Mckibben [14] focused on the approximate controllability problem for the class of abstract neutral semi-linear stochastic evolution equations in a real separable Hilbert space.Based on the theory of strongly continuous cosine families and Sadovskii fixed point theorem, Parthasarathy and Arjunan [17] obtained the controllability results of second-order impulsive stochastic differential and neutral differential systems with state-dependent delay.By the means of the Leray-Schauder Alternative fixed point theorem, Arthi et al. [2] proved the existence and controllability results for second-order impulsive stochastic evolution systems with state-dependent delay.The approximate controllability of a class of secondorder neutral stochastic differential equations with infinite delay and Poisson jumps was considered by Muthukumar and Rajivganthi [16].Das et al. [6] studied the existence and uniqueness of mild solution and approximate controllability for the second-order stochastic neutral partial differential equation with state-dependent delay.
However, in above mentioned papers, the authors didn't consider the damped term x ′ (•) in defining the exact and approximate controllability of the systems.It is not coincide with the definition of the controllability, because apart from x(t), x ′ (t) is also a state variable of a second-order system.Kang et al. [11] studied the exact controllability for the second-order differential inclusion in Banach spaces.With the help of a fixed point theorem for condensing maps due to Martelli [15], the authors found a control u(•) in L 2 (J, U ) such that the solution satisfies x(b) = x 1 and x ′ (b) = y 1 .Afterwards Balachandran and Kim [3] made some remarks on the paper [11] and indicated that the result of [11] is true only for finite dimensional Banach spaces.Very recently, after taking into account the damped term x ′ (t) in defining the approximate controllability of the second-order abstract system, Li and Ma [13] established a new set of sufficient conditions for the approximate controllability of second-order impulsive functional differential system with infinite delay in Banach spaces.Inspired by the above mentioned works [2,13,17], the main purpose of this paper is to investigate the approximate controllability for the following second-order impulsive stochastic differential equations with statedependent delay d[x ′ (t)] = [Ax(t) + Bu(t)]dt + f (t, x ρ(t,xt) , x ′ (t))dω(t), t ∈ J = [0, b], t ̸ = t k , where A is the infinitesimal generator of a strongly continuous cosine family of bounded linear operator {C(t)} t∈R on a Hilbert space H.The state variable x(•) takes the values in H with the inner product ⟨•, •⟩ and the norm ∥ • ∥.The control function u(•) takes values in L F 2 (J, U ) of admissible control functions for a separable Hilbert space U and B is a bounded linear operator from U into H.Let K be another separable Hilbert space with inner product ⟨•, •⟩ K and norm ∥•∥ K .Suppose {ω(t)} t≥0 is a given K-valued Brownian motion or Wiener process with a finite trace nuclear covariance operator Q ≥ 0.Moreover, L(K, H) denotes the space of all bounded linear operators from K into H endowed with the same norm ∥ • ∥, simply L(H) if K = H.For t ∈ J, x t represents the function x t : (−∞, 0] → H defined by x t (θ) = x(t + θ), −∞ < θ ≤ 0 which belongs to some abstract phase space B defined axiomatically.Assume that f : , 2 are appropriate functions and will be specified later.Moreover, represents the jump in the state x at time t k .Similarly x ′ (t + k ) and x ′ (t − k ) denote, respectively, the right and left limits of x ′ (t) at t = t k .The rest of this paper is organized as follows.In Section 2, we recall some essential facts.In Section 3, we derive the existence of mild solution to problem (1.1).In section 4, we present the approximate controllability result.In Section 5, an example is provided to illustrate our results.We end this article with conclusion in Section 6.

Preliminaries
In this section, we review some concepts, notations and properties necessary to establish our results.Let (Ω, F, P ) be a complete probability space furnished with a complete family of right continuous increasing sub σ-algebras {F t , t ∈ J} satisfying F t ⊂ F. An H-valued random variable is an F-measurable function x(t) : Ω → H and a collection of random variables S = {x(t, ω) : Ω → H | t ∈ J} is called a stochastic process.Usually we write x(t) instead of x(t, ω) and x(t) : J → H in the space of S. Let α n (t)(n = 1, 2, • • • ) be a sequence of real-valued independent one-dimensional standard Brownian motions over (Ω, F, P ).Set where is the σ-algebra generated by ω and F b = F. Let ξ ∈ L(K, H) and define ) with respect to the topology induced by the norm ∥ • ∥ Q , where ∥ξ∥ 2 Q = ⟨ξ, ξ⟩ is a Hilbert space with the above norm topology.The collection of all strongly-measurable, square-integrable H-valued random variables, denoted by L 2 (Ω, H), which is a Banach space when endowed with the norm ∥x∥ L2 = (E∥x∥ 2 ) 1 2 , where the expectation E is defined by Ex = . For more details reader may refer the reference [18].
The theory of cosine functions of operator plays an essential role in investigating the existence and controllability of mild solutions.Next we introduce the following definition.
Definition 2.1 (see [20,21]).A one parameter family {C(t)} t∈R , of bounded linear operators defined on a Banach space H is called a strongly continuous cosine family The strongly continuous sine family {S(t)} t∈R , associated to the given strongly continuous cosine family {C(t)} t∈R , is defined by Moreover, M and N are positive constants such that ∥C(t)∥ ≤ M and ∥S(t)∥ ≤ N for every t ∈ J.
The infinitesimal generator of a strongly continuous cosine family {C(t)} t∈R is the operator A : H → H defined by where  [22]: 3) The existence of solutions of the second order abstract Cauchy problem where h : J → H is an integral function, has been discussed in [20].While, the existence of solutions for semilinear second order abstract Cauchy problem has been studied in [21].In addition, the solution of (2.4) is introduced in [21] as follows.
When t ∈ J, the function x(•) given by is called a mild solution of (2.4), and that when ϵ 1 ∈ E the function x(•) is continuously differentiable and In what follows, we put t 0 = 0, t m+1 = b and a function x : [σ, τ ] → H is said to be a normalized piecewise continuous function on [σ, τ ] if x is piecewise continuous and left continuous on (σ, τ ].We denote by PC([σ, τ ], H) the space formed by the normalized piecewise continuous, F t -adapted measurable process from [σ, τ ] into H.In particular, we introduce the space PC formed by all F tadapted measurable, H-valued stochastic process x : For x ∈ PC,we denote the function x(t + k ) for t = t k .A normalized piecewise continuous function x : [σ, τ ] → H is said to be normalized piecewise smooth on [σ, τ ] if x is continuously differentiable except on a finite set S, the left derivative exists on (σ, τ ] and the right derivative exists on [σ, τ ).In this case, we present by x ′ (t) the left derivative at t ∈ (σ, τ ] and by x ′ (σ) the right derivative at σ.We denote by PC 1 ([σ, τ ], H) the space of normalized piecewise smooth functions from [σ, τ ] into H and by PC 1 the space of F t -adapted measurable, H-valued stochastic process x : J → H such that x(•) is piecewise smooth.Obviously, PC 1 is also a Banach space with the norm ∥x∥ In this paper, the phase space (B, ∥ • ∥ B ) denotes a seminormed linear space of F 0 -measurable functions mapping from (−∞, 0] into H and such that the following axioms hold (Hale and Kato [9]). ( (B) The space B is complete.
The next result is a consequence of the phase space axioms.

Lemma 2.1 ( [25]
).Let x : (−∞, b] → H be an F t -adapted measurable process such that the F 0 -adapted process where In order to define the solution of the system(1.1),we consider the space h1 and B ′ h2 , and they are defined by

Existence results
In this section, we study the existence of mild solutions for the impulsive stochastic differential system (1.1).We present the definition of mild solutions for the system firstly.
satisfies the following integral equation: In this paper, we assume that ρ : In the following, we give the following hypotheses firstly.(H 1 ) For each 0 ≤ t < b, the operator α(αI + Γ b t ) −1 → 0 in the strong operator topology as α → 0 + , where the controllability operator Γ b t , associated with (1.1) is defined as H) is a continuous function and there exist positive constants k 1 and k 2 such that H → H are continuous and there exist positive constants and For ϕ ∈ B, we define ϕ by if and only if x satisfies x 0 = 0, and It is also easy to verify that x ′ satisfies ) is a Banach space.
Let Z = B ′′ h1 × PC 1 be the space It is now shown, ( x, z) ∈ Z implies x ∈ B ′ h2 .On the space Z, we define the nonlinear operator Φ( x, z) = (Φ 1 ( x, z), Φ 2 ( x, z)), where and The continuity and well definition of Φ follow directly from the assumptions.Next, we will show that the operator Φ has a fixed point.
where r is a positive constant.For ( x, z) ∈ Q, by Lemma 2.1 and x 0 = 0, we can obtain the following estimates: and So, we can obtain ( For ( x, z) ∈ Q, by taking expectation on (3.1), then from (H 2 )-(H 3 ), we have From condition (H 2 ) and above mentioned estimates, we can find that Then by using (H 3 ) and above mentioned estimates, we get and Similarly, from the expressions of (I 1 ), (I 3 ) and (I 4 ), we have Thus, for ( x, z) ∈ Q, we have where , and substitute r 1 and r 2 into this inequality, which is equivalent to (3.6) Then there exists r 2 such that (3.6) holds if, Similarly, taking expectation on (3.2), where . Similarly, substitute r 1 and r 2 into this inequality, we abtain Next, we show that Φ is a contraction mapping on Q.Let ( x, z), ( v, w) ∈ Q, then we get From the condition (H 2 ), we get we obtain Similarly, we can obtain and .
h2 is a mild solution of (1.1).This completes the proof.

Approximate controllability
In this section, we compare approximate controllability of the semilinear system (1.1) with approximate controllability of the associated linear system.For this reason, we consider the linear system with initial condition 2) are said to be approximately controllable on The following result has been established by Fattorini [7] and Triggiani [23,24].We introduce the sets We are on the position to give the approximate controllability of (4.1)-(4.2),that is Theorem 4.1.(see [7,23,24]) (i) Systems (4.1)-(4.2) are approximately controllable on J if, and only if, 2) are approximately controllable on J.
(iii) If BU 0 is dense in BU and system (4.1)-(4.2) are approximately controllable on J, then Sp{A n BU 0 : n ≥ 0} is dense in H.
Next, we discuss the approximate controllability of the semilinear system (1.1).Before stating and proving our main result, we give the definition of approximate controllability firstly.Definition 4.2.System (1.1) is said to be approximately controllable on and x(•, ϕ, ζ, u) is a mild solution of (1.1).Now, under the above conditions, we prove the following approximately controllable theorem.Theorem 4.2.Assume that BU 0 is dense in BU and the conditions (H 1 )-(H 4 ) are satisfied.If systems (4.1)-(4.2) are approximately controllable on J, then system (1.1) is approximately controllable on J.
This implies that z ∈ R(f, ϕ, ζ).Because z was arbitrarily chosen, this completes the proof.

Example
In this section, we consider an application of the theory developed in the previous section.Let H = L 2 ([0, π]).We choose the space B = PC 0 × L 2 (g, H) constructed in [10], which satisfies the axioms (A) and (B) with H = 1, M (t) = ϱ(−t)   [8] for details).
Consider the impulsive stochastic partial differential equation where we assume ϕ ∈ B, with the identifications ϕ(τ ), is continuously differentiable, and α(t) is a one-dimensional standard Brownian motion.We assume that the function q can be expressed in the form q = ∞ ∑ n=1 e −n 2 q n z n , where Assume that the following conditions hold: (i) c(t) is measurable and continuous with finite The problem (5.1) can be modeled as the abstract impulsive Cauchy problem 2) are approximately controllable on J. Then the operator α(αI + Γ b t ) −1 → 0 in the strong operator topology as α → 0 + (see [4,23,24]).So assumption (H 1 ) is satisfied.
Under these conditions, where K 2 = max{1, K 2 f } and which implies that the function f satisfies the following condition Let The next proposition is a consequence of Theorem 4.2.

Conclusion
In this paper, we discussed approximate controllability results for the second-order impulsive stochastic differential equations with state-dependent delay by using phase space axioms.We have proved the result without compactness of family of cosine operators.Several explicit sufficient conditions of such systems have been established by utilizing the fixed point strategy.Finally, an example is illustrated for the effectiveness of the approximate controllability results.
For the future research, we will consider the control problem for second-order stochastic functional differential equations with infinite delay in L p space by using fundamental solution theory.By this manner, we will show some interesting results for the system involving a linear term (non-uniformly bounded), which has a wide practical background such as some heat conduction models with fading memory.

1 2 2 π
for t ≥ 0. Let A : H → H be an operator defined by Af = f ′′ with domainD(A) = {f ∈ H : f and f ′ are absolutely continuous, f ′′ ∈ H, f (0) = f (π) = 0}.It is well known that A is the infinitesimal generator of a strongly continuous cosine family of operators on H.The spectrum of A consists of the eigenvalues −n 2 for n ∈ N , which associated eigenvectors z n (ξ) = √ sin(nξ).Furthermore, the set {z n , n ∈ N } is an orthonormal basis of H and the following properties hold: