Approximate controllability of impulsive neutral stochastic differentialequations with fractional Brownian motion in a Hilbert space

Abstract

Approximate controllability for impulsive neutral stochastic functionaldifferential equations with finite delay and fractional Brownian motion in aHilbert space are studied. The results are obtained by using semigroup theory,stochastic analysis, and Banach’s fixed point theorem. Finally, an exampleis given to illustrate the application of our result.

MSC: 60H15, 60G22, 93B05, 34A37.

1 Introduction

The impulsive differential systems are used to describe processes which are subjectedto abrupt changes at certain moments. The impulsive effects exist widely in thedifferent areas of the real world such as mechanics, electronics,telecommunications, neural networks, finance and economics, etc. (see [1–7]). On the other hand, it is well known that the stochastic control theoryis a stochastic generalization of classical control theory. As one of thefundamental concepts in mathematical control theory, controllability plays animportant role both in deterministic and stochastic control theory. Controllabilitygenerally means that it is possible to steer a dynamical control system from anarbitrary initial state to an arbitrary final state using the set of admissiblecontrols (see [8–17]). Moreover, the approximate controllability means that the system can besteered to arbitrary small neighborhood of final state. Approximate controllablesystems are more prevalent and very often approximate controllability is completelyadequate in applications (see [18–24]).

The purpose of this paper is to investigate the approximate controllability problemfor the class of impulsive neutral stochastic functional differential equations withfinite delay and fractional Brownian motion in a Hilbert space of the form

$$\{\begin{array}{l}d[x(t)+g(t,x(t-r(t)))]=[Ax(t)+f(t,x(t-\nu (t)))+Bu(t)]dt+\sigma (t)d{B}^H(t),\\ 0\le t\le T,t\ne t_k,\\ \mathrm{\Delta }x{|}_{t=t_k}=I_k(x(t_k^{-})),k=1,2,\dots ,m,\\ x(t)=\phi (t),-\tau \le t\le 0,\end{array}$$

(1.1)

where A is the infinitesimal generator of an analytic semigroup of boundedlinear operators, $S{(t)}_{t\ge 0}$, in a Hilbert space X,$B^H$ is a fractional Brownian motion on a real andseparable Hilbert space Y, the initial data $\phi \in C([-\tau ,0],L^2(\mathrm{\Omega },X))$ and the control function $u(\cdot )$ is given in $L^2([0,T],U)$, the Hilbert space of admissible control functionswith U a Hilbert space. The symbol B stands for a bounded linearfrom U into X. The functions $r,\nu :[0,+\mathrm{\infty })\to [0,\tau ]$ ($\tau >0$) are continuous, $\mathrm{\Delta }x{|}_{t=t_k}=I_k(x(t_k^{-}))$, where $x(t_k^{+})$ and $x(t_k^{-})$ represent the right and left limits of$x(t)$ at $t=t_k$, respectively, and $f,g:[0,+\mathrm{\infty })\times X\to X$, $\sigma :[0,+\mathrm{\infty })\to L_2^0(Y,X)$, are appropriate Lipschitz type functions. Here$R_T:=C([-\tau ,T],L^2(\mathrm{\Omega },X))$ be the Banach space of all continuous functionsξ from $[-\tau ,T]$ into $L^2(\mathrm{\Omega },X)$, equipped with the supremum norm${\parallel \xi \parallel }_{R_T}={sup}_{y\in [-\tau ,T]}{(E{\parallel \xi (y)\parallel }^2)}^{1/2}$.

2 Fractional Brownian motion

Fix a time interval $[0,T]$ and let $(\mathrm{\Omega },F,P)$ be a complete probability space.

Suppose that $\{{\beta }^H(t),t\in [0,T]\}$ is the one-dimensional fractional Brownian motionwith Hurst parameter $H\in (1/2,1)$. That is, ${\beta }^H$ is a centered Gaussian process with covariancefunction $R_H(s,t)=\frac{1}{2}(t^{2H}+s^{2H}-{|t-s|}^{2H})$ (see [25]).

Moreover, ${\beta }^H$ has the following Wiener integral representation:

$${\beta }^H(t)={\int }_0^t{K}_H(t,s)d\beta (s),$$

where $\beta =\{\beta (t),t\in [0,T]\}$ is a Wiener process, and $K_H(t,s)$ is the kernel given by

$$K_H(t,s)=c_H{s}^{\frac{1}{2}-H}{\int }_s^t{(u-s)}^{H-\frac{3}{2}}{u}^{H-\frac{1}{2}}du$$

for $s<t$, where $c_H=\sqrt{\frac{H(2H-1)}{\beta (2-2H,H-\frac{1}{2})}}$ and

$$\beta (p,q)={\int }_0^1{t}^{p-1}{(1-t)}^{q-1},p>0,q>0.$$

We put $K_H(t,s)=0$ if $t\le s$.

We will denote by ζ the reproducing kernel Hilbert space of the fBm. Infact ζ is the closure of set of indicator functions$\{1_{[0,t]},t\in [0,T]\}$ with respect to the scalar product${〈1_{[0,t]},1_{[0,s]}〉}_{\zeta }=R_H(t,s)$.

The mapping $1_{[0,t]}\to {\beta }^H(t)$ can be extended to an isometry from ζonto the first Wiener chaos and we will denote by ${\beta }^H(\phi )$ the image of φ under this isometry.

We recall that for $\psi ,\phi \in \zeta$ their scalar product in ζ is given by

$${〈\psi ,\phi 〉}_{\zeta }=H(2H-1){\int }_0^T{\int }_0^T\psi (s)\phi (t){|t-s|}^{2H-2}dsdt.$$

Let us consider the operator $K^{\ast }$ from ζ to $L^2([0,T])$ defined by

$$\left(K_H^{\ast }\phi \right)(s)={\int }_S^T\phi (r)\frac{\partial K}{\partial r}(r,s)dr.$$

Moreover, for any $\phi \in \zeta$, we have

$${\beta }^H(\phi )={\int }_0^T\left(K_H^{\ast }\phi \right)(t)d\beta (t).$$

Let X and Y be two real, separable Hilbert spaces and let$L(Y,X)$ be the space of bounded linear operators fromY to X. For the sake of convenience, we shall use the samenotation to denote the norms in X, Y and $L(Y,X)$. Let $Q\in L(Y,Y)$ be an operator defined by $Q{e}_n={\lambda }_n{e}_n$ with finite trace $trQ={\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n<\mathrm{\infty }$, where ${\lambda }_n\ge 0$ ($n=1,2,\dots$) are non-negative real numbers and$\{e_n\}$ ($n=1,2,\dots$) is a complete orthonormal basis in Y.

We define the infinite-dimensional fBm on Y with covariance Q as

$$B^H(t)=B_Q^H(t)=\sum _{n=1}^{\mathrm{\infty }}\sqrt{{\lambda }_n}{e}_n{\beta }_n^H(t),$$

where ${\beta }_n^H$ are real, independent fBm’s. TheY-valued process is Gaussian, starts from 0, has mean zero and covariance:

$$E〈B^H(t),x〉〈B^H(s),y〉=R(s,t)〈Q(x),y〉\text{for all }x,y\in Y\text{ and }t,s\in [0,T].$$

In order to define Wiener integrals with respect to the Q-fBm, we introducethe space $L_2^0:=L_2^0(Y,X)$ of all Q-Hilbert Schmidt operators$\psi :Y\to X$. We recall that $\psi \in L(Y,X)$ is called a Q-Hilbert-Schmidt operator, if

$${\parallel \psi \parallel }_{L_2^0}^2:=\sum _{n=1}^{\mathrm{\infty }}{\parallel {\sqrt{\lambda }}_n\psi e_n\parallel }^2<\mathrm{\infty }$$

and that the space $L_2^0$ equipped with the inner product${〈\phi ,\psi 〉}_{L_2^0}={\sum }_{n=1}^{\mathrm{\infty }}〈\phi e_n,\psi e_n〉$ is a separable Hilbert space.

Let $\varphi (s)$; $s\in [0,T]$ be a function with values in $L_2^0(Y,X)$, the Wiener integral of ϕ with respectto $B^H$ is defined by

$$\begin{array}{rl}{\int }_0^t\varphi (s)d{B}^H(s)& =\sum _{n=1}^{\mathrm{\infty }}{\int }_0^t\sqrt{\lambda }\varphi (s)e_{n}d{\beta }_n^H\\ =\sum _{n=1}^{\mathrm{\infty }}{\int }_0^t\sqrt{\lambda }{K}^{\ast }(\varphi e_n)(s)d{\beta }_n(s),\end{array}$$

(2.1)

where ${\beta }_n$ is the standard Brownian motion.

Lemma 2.1 (see [26])

If$\psi :[0,T]\to L_2^0(Y,X)$satisfies${\int }_0^T{\parallel \psi (s)\parallel }_{L_2^0}^2<\mathrm{\infty }$then the above sum in (2.1) is well defined as X-valued random variable and we have

$$E{\parallel {\int }_0^t\psi (s)d{B}^H(s)\parallel }^2\le 2H{t}^{2H-1}{\int }_0^t{\parallel \psi (s)\parallel }_{L_2^0}^{2}ds.$$

3 Approximate controllability

Let $A:D(A)\to X$ be the infinitesimal generator of an analyticsemigroup, ${(S(t))}_{t\ge 0}$, of bounded linear operators on X. It iswell known that there exist $M\ge 1$ and $\lambda \in R$ such that $\parallel S(t)\parallel \le M{e}^{\lambda t}$ for every $t\ge 0$.

If ${(S(t))}_{t\ge 0}$ is uniformly bounded and analytic semigroup such that$0\in \rho (A)$, where $\rho (A)$ is the resolvent set of A, then it ispossible to define the fractional power ${(-A)}^{\alpha }$ for $0<\alpha \le 1$, as a closed linear operator on its domain$D{(-A)}^{\alpha }$. Furthermore, the subspace $D{(-A)}^{\alpha }$ is dense in X and the expression${\parallel h\parallel }_{\alpha }=\parallel {(-A)}^{\alpha }h\parallel$ defines a norm in $D{(-A)}^{\alpha }$. If $X_{\alpha }$ represents the space $D{(-A)}^{\alpha }$ endowed with the norm ${\parallel \cdot \parallel }_{\alpha }$, then the following properties are well known.

Lemma 3.1 ([27])

• (1) Let $0<\alpha \le 1$, then $X_{\alpha }$ is a Banach space.

• (2) If $0<\beta \le \alpha$, then the injection $X_{\alpha }\hookrightarrow X_{\beta }$ is continuous.

• (3) For every $0<\alpha \le 1$ there exists $M_{\alpha }>0$ such that

  $$\parallel {(-A)}^{\alpha }S(t)\parallel \le M_{\alpha }{t}^{-\alpha }{e}^{-\lambda t},t>0,\lambda >0.$$

Now, we present the mild solution of the problem (1.1):

Definition 3.1 An X-valued process $\{x(t),t\in [-\tau ,T]\}$ is called a mild solution of equation (1.1) if

• (i) $x(\cdot )\in C([-\tau ,T],L^2(\mathrm{\Omega },X))$,

• (ii) $x(t)=\phi (t)$, $-\tau \le t\le 0$,

• (iii) for arbitrary $t\in [0,T]$, we have

  $$\begin{array}{rl}x(t)=& S(t)[\phi (0)+g(0,\phi (-r(0)))]-g(t,x(t-r(t)))\\ -{\int }_0^{t}AS(t-s)g(s,x(s-r(s)))ds\\ +{\int }_0^{t}S(t-s)f(s,x(s-\nu (s)))ds+{\int }_0^{t}S(t-s)Bu(s)ds\\ +{\int }_0^{t}S(t-s)\sigma (s)d{B}^H(s)+\sum _{0<t_k<t}S(t-t_k)I_k\left(x\left(t_k^{-}\right)\right).\end{array}$$

  (3.1)

In this paper, we will make the following assumptions.

(H1) The operator A is the infinitesimal generator of an analytic semigroup,${(S(t))}_{t\ge 0}$, consisting of bounded linear operators onX. Furthermore, there exist constants M and$M_{1-\beta }$ such that for every $t\in [0,T]$ the inequalities $\parallel S(t)\parallel \le M$ and $t^{1-\beta }\parallel {(-A)}^{1-\beta }S(t)\parallel \le M_{1-\beta }$ hold.

(H2) There exist finite positive constants $C_i=C_i(T)$, $i=1,2$, such that the function $f:[0,+\mathrm{\infty })\times X\to X$ satisfies the following Lipschitz conditions: for all$t\in [0,T]$ and $x,y\in X$ the inequalities $\parallel f(t,x)-f(t,y)\parallel \le C_1\parallel x-y\parallel$ and ${\parallel f(t,x)\parallel }^2\le C_2^2(1+{\parallel x\parallel }^2)$ are valid.

(H3) The function g is $X_{\beta }$-valued, and there exist constants$\frac{1}{2}<\beta <1$, $C_i=C_i(T)$, $i=3,4$, such that for all $t\in [0,T]$ and $x,y\in X$ the following inequalities are satisfied:

• (i) $\parallel {(-A)}^{\beta }g(t,x)-{(-A)}^{\beta }g(t,y)\parallel \le C_3\parallel x-y\parallel$;

• (ii) ${\parallel {(-A)}^{\beta }g(t,x)\parallel }^2\le C_4^2(1+{\parallel x\parallel }^2)$;

• (iii) $C_3\parallel {(-A)}^{-\beta }\parallel <1$.

(H4) The function ${(-A)}^{\beta }g$ is continuous in the quadratic mean sense: for all$x\in C([0,T],L^2(\mathrm{\Omega },X))$, the equality

$$\underset{t\to s}{lim}E{\parallel {(-A)}^{\beta }g(t,x(t))-{(-A)}^{\beta }g(s,x(s))\parallel }^2=0$$

is true.

(H5) The function $\sigma :[0,\mathrm{\infty })\to L_2^0(Y,X)$ satisfies ${\int }_0^T{\parallel \sigma (s)\parallel }_{L_2^0}^{2}ds<\mathrm{\infty }$.

(H6) The functions $I_k:X\to X$ are continuous and there exist finite positiveconstants $C_i=C_i(T)$, $i=5,6$, such that for all $t\in [0,T]$ and $x,y\in X$ the inequalities $\parallel I_k(x(t))-I_k(y(t))\parallel \le C_5\parallel x-y\parallel$ and ${\parallel I_k(x(t))\parallel }^2\le C_6^2(1+{\parallel x\parallel }^2)$ are valid.

In order to study the approximate controllability for the system (1.1), we introducethe following linear differential system:

$$\{\begin{array}{l}\frac{dx(t)}{dt}=Ax(t)+Bu(t),t\in [0,T],\\ x(0)=x_0.\end{array}$$

(3.2)

The controllability operator associated with (3.2) is defined by

$${\mathrm{\Gamma }}_0^T={\int }_0^{T}S(T-s)B{B}^{\ast }{S}^{\ast }(T-s)ds,$$

where $B^{\ast }$ and $S^{\ast }$ denote the adjoint of B and S,respectively.

Let $x(T;\phi ,u)$ be the state value of (1.1) at terminal stateT, corresponding to the control u and the initial valueφ. Denote by $R(T,\phi )=\{x(T;\phi ,u):u\in L^2([0,T],U)\}$ the reachable set of system (1.1) at terminal timeT, its closure in X is denoted by $\overline{R(T,\phi )}$.

Definition 3.2 The system (1.1) is said to be approximately controllable onthe interval $[0,T]$ if $\overline{R(T,\phi )}=L^2(\mathrm{\Omega },X)$.

Lemma 3.2 (see [19])

The linear control system (3.2) is approximately controllable on$[0,T]$if and only if$z{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}\to 0$strongly as$z\to 0^{+}$.

Lemma 3.3 For any ${\overline{x}}_T\in L^2(\mathrm{\Omega },X)$ there exists $\overline{\phi }\in L^2(\mathrm{\Omega };L^2([0,T];L_2^0))$ such that

$${\overline{x}}_T=E{\overline{x}}_T+{\int }_0^T\overline{\phi }(s)d{B}^H(s).$$

Now for any$\delta >0$and${\overline{x}}_T\in L^2(\mathrm{\Omega },X)$, we define the control function in the followingform:

$$\begin{array}{rl}{u}^{\delta }(t,x)=& B^{\ast }{S}^{\ast }(T-t){(zI+{\mathrm{\Gamma }}_0^T)}^{-1}\\ \times \{E{\overline{x}}_T-S(T)[\phi (0)-g(0,\phi (-r(0)))]+g(T,x(T))\}\\ +B^{\ast }{S}^{\ast }(T-t){\int }_0^t{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}\overline{\phi }(s)d{B}^H(s)\\ -B^{\ast }{S}^{\ast }(T-t){\int }_0^t{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}AS(T-s)g(s,x(s-r(s)))ds\\ -B^{\ast }{S}^{\ast }(T-t){\int }_0^t{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}S(T-s)f(s,x(s-\nu (s)))ds\\ -B^{\ast }{S}^{\ast }(T-t){\int }_0^t{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}S(T-s)\sigma (s)d{B}^H(s)\\ -B^{\ast }{S}^{\ast }(T-t)\sum _{0<t_k<t}{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}S(T-t_k)I_k\left(x\left(t_k^{-}\right)\right).\end{array}$$

Lemma 3.4 There exists a positive real constant$M_C$such that, for all$x,y\in R_T$, we have

$$E{\parallel u^{\delta }(t,x)-u^{\delta }(t,y)\parallel }^2\le \frac{M_C}{z^2}{\int }_0^{t}E{\parallel x(s)-y(s)\parallel }^{2}ds,$$

(3.3)

$$E{\parallel u^{\delta }(t,x)\parallel }^2\le \frac{M_C}{z^2}(1+{\int }_0^{t}E{\parallel x(s)\parallel }^{2}ds).$$

(3.4)

Proof The proof of this lemma similar to the proof of the Lemma 2.5(see [28]). □

Theorem 3.1 Assume assumptions (H1)-(H6) are satisfied. Then, forall$T>0$, the system (1.1) has a mild solutionon$[-\tau ,T]$.

Proof Fix $T>0$ and let us consider ${\mathrm{\Upsilon }}_T=\{x\in R_T:x(s)=\phi (s),\text{ for }s\in [-\tau ,0]\}$.

${\mathrm{\Upsilon }}_T$ is a closed subset of $R_T$ provided with the norm ${\parallel \cdot \parallel }_{R_T}$. For any $\delta >0$, consider the operator ${\mathrm{\Pi }}_{\delta }$ on $R_T$ defined as follows:

$$({\mathrm{\Pi }}_{\delta }x)(t)=\{\begin{array}{l}\phi (t),t\in [-\tau ,0],\\ S(t)[\phi (0)+g(0,\phi (-r(0)))]-g(t,x(t-r(t)))\\ -{\int }_0^{t}AS(t-s)g(s,x(s-r(s)))ds\\ +{\int }_0^{t}S(t-s)f(s,x(s-\nu (s)))ds+{\int }_0^{t}S(t-s)B{u}^{\delta }(s,x)ds\\ +{\int }_0^{t}S(t-s)\sigma (s)d{B}^H(s)+{\sum }_{0<t_k<t}S(t-t_k)I_k(x(t_k^{-})),t\in [0,T].\end{array}$$

(3.5)

It will be shown that, for all $\delta >0$, the operator ${\mathrm{\Pi }}_{\delta }$ has a fixed point. This fixed point is then asolution of equation (1.1). To prove this result, we divide the subsequent proofinto two steps.

Step 1: For arbitrary $x\in {\mathrm{\Upsilon }}_T$, let us prove that $t\to {\mathrm{\Pi }}_{\delta }(x)(t)$ is continuous on the interval $[0,T]$ in the $L^2(\mathrm{\Omega },X)$-sense.

Let $0<t<t+h<T$, where $t,t+h\in [0,T]\mathrm{\setminus }\{t_1,t_2,\dots ,t_m\}$, and let $|h|$ be sufficiently small. Then for any fixed$x\in {\mathrm{\Upsilon }}_T$, it follows from Holder’s inequality and theassumptions on the theorem that

$$\begin{array}{l}E{\parallel ({\mathrm{\Pi }}_{\delta }x)(t+h)-({\mathrm{\Pi }}_{\delta }x)(t)\parallel }^2\\ \le 8\{E{\parallel (S(t+h)-S(t))(\phi (0)+g(0,\phi (-r(0))))\parallel }^2\\ +E{\parallel g(t+h,x(t+h-r(t)))-g(t,x(t-r(t)))\parallel }^2\\ +E{\parallel {\int }_0^{t}A(S(t+h-s)-S(t-s))g(s,x(s-r(s)))ds\parallel }^2\\ +E{\parallel {\int }_t^{t+h}{(-A)}^{1-\beta }S(t+h-s){(-A)}^{\beta }g(s,x(s-r(s)))ds\parallel }^2\\ +E{\parallel {\int }_0^t(S(t+h-s)-S(t-s))B{u}^{\delta }(s,x)ds\parallel }^2+E{\parallel {\int }_t^{t+h}S(t+h-s)B{u}^{\delta }(s,x)ds\parallel }^2\\ +E{\parallel {\int }_0^t(S(t+h-s)-S(t-s))\sigma (s)d{B}^H(s)\parallel }^2+E{\parallel {\int }_t^{t+h}S(t+h-s)\sigma (s)d{B}^H(s)\parallel }^2\\ +\sum _{0<t_k<t}E{\parallel (S(t+h-t_k)-S(t-t_k))I_k\left(x\left(t_k^{-}\right)\right)\parallel }^2\\ +\sum _{t<t_k<t+h}E{\parallel S(t+h-t_k)I_k\left(x\left(t_k^{-}\right)\right)\parallel }^2\}\\ \le 8\{E{\parallel (S(t+h)-S(t))(\phi (0)+g(0,\phi (-r(0))))\parallel }^2\\ +E{\parallel g(t+h,x(t+h-r(t)))-g(t,x(t-r(t)))\parallel }^2\\ +t{\int }_0^{t}E{\parallel A(S(t+h-s)-S(t-s))g(s,x(s-r(s)))\parallel }^{2}ds\\ +\frac{C_4^2{M}_{1-\beta }^2}{2\beta -1}{(t+h-t)}^{2\beta -1}{\int }_t^{t+h}(1+E{\parallel x\parallel }^2)ds\\ +t{\int }_0^{t}E{\parallel (S(t+h-s)-S(t-s))B{u}^{\delta }(s,x)\parallel }^{2}ds\\ +(t+h-t)M^2{\parallel B\parallel }^2{\int }_t^{t+h}E{\parallel u^{\delta }(s,x)\parallel }^{2}ds\\ +2H{t}^{2H-1}{\int }_0^{t}E{\parallel (S(t+h-s)-S(t-s))\sigma (s)\parallel }_{L_2^0}^{2}ds\\ +2H{(t+h-t)}^{2H-1}{M}^2{\int }_t^{t+h}E{\parallel \sigma (s)\parallel }_{L_2^0}^{2}ds\\ +\sum _{0<t_k<t}E\left({\parallel (S(t+h-t_k)-S(t-t_k))\parallel }^2{\parallel I_k\left(x\left(t_k^{-}\right)\right)\parallel }^2\right)+M^2\sum _{t<t_k<t+h}E{\parallel I_k\left(x\left(t_k^{-}\right)\right)\parallel }^2\}.\end{array}$$

Hence using the strong continuity of $S(t)$ and Lebesgue’s dominated convergence theorem,we conclude that the right-hand side of the above inequalities tends to zero as$h\to 0$. Thus we conclude ${\mathrm{\Pi }}_{\delta }(x)(t)$ is continuous from the right in$[0,T]$. A similar argument shows that it is also continuousfrom the left in $(0,T]$. Thus ${\mathrm{\Pi }}_{\delta }(x)(t)$ is continuous on $[0,T]$ in the $L^2$-sense.

Step 2: Now, we are going to show that ${\mathrm{\Pi }}_{\delta }$ is a contraction mapping in ${\mathrm{\Upsilon }}_{T_1}$ with some $T_1\le T$ to be specified latter. Let $x,y\in {\mathrm{\Upsilon }}_T$, we obtain for any fixed $t\in [0,T]$

$$\begin{array}{r}{\parallel {\mathrm{\Pi }}_{\delta }(x)(t)-{\mathrm{\Pi }}_{\delta }(y)(t)\parallel }^2\\ \le 5{\parallel g(t,x(t-r(t))-g(t,y(t-r(t))\parallel }^2+5{\parallel {\int }_0^{t}S(t-s)B[u^{\delta }(s,x)-u^{\delta }(s,y)]ds\parallel }^2\\ +5{\parallel {\int }_0^t{(-A)}^{1-\beta }S(t-s)({(-A)}^{\beta }g(s,x(s-r(s))-{(-A)}^{\beta }g(s,y(s-r(s))))ds\parallel }^2\\ +5{\parallel {\int }_0^{t}S(t-s)(f(s,x(s-\nu (s)))-f(s,y(s-\nu (s))))ds\parallel }^2\\ +5{\parallel \sum _{0<t_k<t}S(t-t_k)(I_k\left(x\left(t_k^{-}\right)\right)-I_k\left(y\left(t_k^{-}\right)\right))\parallel }^2.\end{array}$$

By the Lipschitz property of ${(-A)}^{\beta }g$ and f combined with Holder’sinequality, we obtain

$$\begin{array}{rl}E{\parallel {\mathrm{\Pi }}_{\delta }(x)(t)-{\mathrm{\Pi }}_{\delta }(y)(t)\parallel }^2\le & 5E{\parallel x(t-r(t))-y(t-r(t))\parallel }^2\\ +\frac{5M^2{\parallel B\parallel }^2{M}_C}{z^2}{\int }_0^{t}E{\parallel x(s)-y(s)\parallel }^{2}ds\\ +5C_3^2{M}_{1-\beta }^2\frac{T^{2\beta -1}}{2\beta -1}{\int }_0^{t}E{\parallel x(s-r(s))-y(s-r(s))\parallel }^{2}ds\\ +5T{C}_1^2{M}^2{\int }_0^{t}E{\parallel x(s-\nu (s))-y(s-\nu (s))\parallel }^{2}ds\\ +5m^2{M}^2{C}_5^{2}E{\parallel x(t)-y(t)\parallel }^2.\end{array}$$

Hence

$$\underset{t\in [-\tau ,T]}{sup}E{\parallel {\mathrm{\Pi }}_{\delta }(x)(t)-{\mathrm{\Pi }}_{\delta }(y)(t)\parallel }^2\le \gamma (T)\underset{t\in [-\tau ,T]}{sup}E{\parallel x(t)-y(t)\parallel }^2,$$

where

$$\gamma (T)=5[1+\frac{M^2{\parallel B\parallel }^2{M}_C}{z^2}T+\frac{C_3^2{M}_{1-\beta }^2}{2\beta -1}{T}^{2\beta }+M^2{C}_1^2{T}^2+m^2{M}^2{C}_5^2].$$

Then there exists $0<T_1\le T$ such that $0<\gamma (T_1)<1$ and ${\mathrm{\Pi }}_{\delta }$ is a contraction mapping on $S_{T_1}$ and therefore has a unique fixed point, which is amild solution of equation (1.1) on $[-\tau ,T_1]$. This procedure can be repeated in order to extendthe solution to the entire interval $[-\tau ,T]$ in finitely many steps. This completes theproof. □

Theorem 3.2 Assume that (H1)-(H6) are satisfied. Further, if thefunctions f and g are uniformly bounded, and$S(t)$is compact, then the system (1.1) is approximatelycontrollable on$[0,T]$.

Proof Let $x_{\delta }$ be a fixed point of ${\mathrm{\Pi }}_{\delta }$. By using the stochastic Fubini theorem, it caneasily be seen that

$$\begin{array}{rl}{x}_{\delta }(T)=& {\overline{x}}_T-z{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}\{E{\overline{x}}_T-S(T)[\phi (0)-g(0,\phi (-r(0)))]\\ +g(T,x_{\delta }(T))+{\int }_0^T\overline{\phi }(s)d{B}^H(s)\}\\ +z{\int }_0^T{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}AS(T-s)g(s,x_{\delta }(s-r(s)))ds\\ +z{\int }_0^T{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}S(T-s)f(s,x_{\delta }(s-\nu (s)))ds\\ +z{\int }_0^T{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}S(T-s)\sigma (s)d{B}^H(s)\\ +\sum _{0<t_k<T}z{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}S(T-t_k)I_k\left(x_{\delta }\left(t_k^{-}\right)\right).\end{array}$$

It follows from the assumption on f and g that there exists$\overline{D}>0$ such that

$${\parallel f(s,x_{\delta }(s-\nu (s)))\parallel }^2+{\parallel g(s,x_{\delta }(s-r(s)))\parallel }^2\le \overline{D}$$

(3.6)

for all $(s,\omega )\in [0,T]\times \mathrm{\Omega }$. Then there is a subsequence still denoted by$\{f(s,x_{\delta }(s-\nu (s))),g(s,x_{\delta }(s-r(s)))\}$ which converges weakly to, say,$\{f(s),g(s)\}$ in $X\times L_2^0$.

From the above equation, we have

$$\begin{array}{r}E{\parallel x_{\delta }(T)-{\overline{x}}_T\parallel }^2\\ \le 7E\parallel z{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}\{{\overline{x}}_T-S(T)[\phi (0)-g(0,\phi (-r(0)))]\\ +g(T,x_{\delta }(T))+{\int }_0^T\overline{\phi }(s)d{B}^H(s)\}{\parallel }^2\\ +7E{\left({\int }_0^T\parallel z{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}\parallel \parallel AS(T-s)[g(s,x_{\delta }(s-r(s)))-g(s)]\parallel ds\right)}^2\\ +7E{\left({\int }_0^T\parallel z{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}AS(T-s)g(s)\parallel ds\right)}^2\\ +7E{\left({\int }_0^T\parallel z{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}\parallel \parallel S(T-s)f(s,x_{\delta }(s-\nu (s)))-f(s)\parallel ds\right)}^2\\ +7E{\left({\int }_0^T\parallel z{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}S(T-s)f(s)\parallel ds\right)}^2\\ +14H{T}^{2H-1}{\int }_0^T{\parallel z{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}S(T-s)\sigma (s)\parallel }_{L_2^0}^{2}ds\\ +7E{\parallel \sum _{0<t_k<T}z{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}S(T-t_k)I_k\left(x_{\delta }\left(t_k^{-}\right)\right)\parallel }^2.\end{array}$$

On the other hand, by Lemma 3.2, the operator $z{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}\to 0$ strongly as $z\to 0^{+}$ for all $0\le s\le T$, and, moreover, $\parallel z{(zI+{\mathrm{\Gamma }}_0^T)}^{-1}\parallel \le 1$. Thus, by the Lebesgue dominated convergence theoremthe compactness of $S(t)$ implies that $E{\parallel x_{\delta }(T)-{\overline{x}}_T\parallel }^2\to 0$ as $z\to 0^{+}$. This gives the approximate controllability of(1.1). □

4 Example

In this section, we present an example to illustrate our main result.

Let us consider the following stochastic control partial neutral functionaldifferential equation with finite variable delays driven by a fractional Brownianmotion:

$$\{\begin{array}{l}d[x(t,\xi )-G(t,x(t-r(t)),\xi )]=[\frac{{\partial }^{2}x(t,\xi )}{\partial {\xi }^2}+F(t,x(t-\nu (t)),\xi )+\mu (t,\xi )]dt\\ +\sigma (t)d{B}^H(t),0\le \xi \le \pi ,0\le t\le T,t\ne t_k,\\ x(t,0)=x(t,\pi )=0,t\ge 0,\\ x(t_k^{+},\xi )-x(t_k^{-},\xi )=I_k(x(t_k^{-},\xi )),k=1,2,\dots ,m,\\ x(t,\xi )=\phi (t,\xi ),t\in [-\tau ,0],0\le \xi \le \pi ,\end{array}$$

(4.1)

where $B^H$ is a fractional Brownian motion and$F,G:R^{+}\times R\to R$ are continuous functions.

To study this system, let $X=Y=U=L^2([0,\pi ],R)$ and let the operator $A:D(A)\subset X\to X$ be given by $Ay=y^{″}$ with

$$D(A)=\{y\in X:y^{″}\in X,y(0)=y(\pi )=0\}.$$

It is well known that A is the infinitesimal generator of an analyticsemigroup ${\{T(t)\}}_{t\ge 0}$ on X. Furthermore, A has discretespectrum with eigenvalues $-n^2$, $n\in N$ and the corresponding normalized eigenfunctions aregiven by

$$e_n=\sqrt{\frac{2}{\pi }}sinnx,n=1,2,\dots .$$

In addition ${(e_n)}_{n\in N}$ is a complete orthonormal basis in X and

$$T(t)x=\sum _{n=1}^{\mathrm{\infty }}{e}^{-n^{2}t}{〈x,e_n〉}_{e_n}$$

for $x\in X$ and $t\ge 0$. It follows from this representation that$T(t)$ is compact for every $t>0$ and that $\parallel T(t)\parallel \le e^{-t}$ for every $t\ge 0$.

In order to define the operator $Q:Y\to R$, we choose a sequence ${\{{\lambda }_n\}}_{n\in N}\subset R^{+}$, set $Q{e}_n={\lambda }_n{e}_n$, and assume that

$$tr(Q)=\sum _{n=1}^{\mathrm{\infty }}\sqrt{{\lambda }_n}<\mathrm{\infty }.$$

Define the fractional Brownian motion in Y by

$$B^H(t)=\sum _{n=1}^{\mathrm{\infty }}\sqrt{{\lambda }_n}{\beta }^H(t)e_n,$$

where $H\in (\frac{1}{2},1)$ and ${\{{\beta }_n^H\}}_{n\in N}$ is a sequence of one-dimensional fractional Brownianmotions mutually independent.

Define $x(t)(\cdot )=x(t,\cdot )$, $f(t,x)(\cdot )=F(t,x(\cdot ))$, and $g(t,x)(\cdot )=G(t,x(\cdot ))$. Define the bounded operator $B:U\to X$ by $Bu(t)(\xi )=\mu (t,\xi )$, $0\le \xi \le \pi$, $u\in U$. Therefore, with the above choice, the system (4.1)can be written into the abstract form (1.1) and all conditions of Theorem 3.2are satisfied. Thus by Theorem 3.2, the stochastic partial neutral functionaldifferential equation with finite variable delays driven by a fractional Brownianmotion is approximately controllable on $[0,\pi ]$.