Approximate Controllability of Semilinear Stochastic System with State Delay

Abstract

The objective of this paper is to present some sufficient conditions for approximate controllability of semilinear stochastic system with state delay. Sufficient conditions are obtained by separating the given semilinear system into two systems namely a semilinear deterministic system and a linear stochastic system. To prove our results, the Schauder fixed-point theorem is applied. At the end, an example is given to show the effectiveness of the result.

1 Introduction

Controllability concepts play a vital role in deterministic control theory. It is well known that controllability of deterministic equation is widely used in many fields of science and technology. But in many practical problems such as fluctuating stock prices or physical system subject to thermal fluctuations, population dynamics, etc., some randomness appear, so the system should be modelled stochastic form.

In setting of deterministic systems: Kalman [1] introduced the concept of controllability for finite-dimensional deterministic linear control systems. Then Barnett [2] and Curtain [3] introduced the concepts of deterministic control theory in finite and infinite-dimensional spaces. Naito [4] established sufficient conditions for approximate controllability of deterministic semilinear control system dominated by the linear part using Schauder’s fixed-point theorem. In [5, 6], Wang extended the results of [4] and established sufficient conditions for delayed deterministic semilinear systems using same Schauder’s fixed-point theorem. In [7] author provided more applications of Schauder’s fixed-point theorem in nonlinear controllability problems.

In setting of stochastic systems: In [8, 9] Mahmudov established some results for controllability of linear stochastic systems in finite-dimensional and infinite-dimensional spaces, respectively. Sukavanam et al. in [10] obtained some sufficient conditions for s-controllability of an abstract first-order semilinear control system using Schauder’s fixed-point theorem. Recently, Anurag et al. [11] obtained some sufficient conditions for approximate controllability of retarded semilinear stochastic system with nonlocal conditions using Banach fixed-point theorem.

The present paper is generalized form of the system taken in [10]. In this paper system is taken with finite delay in state which is not discussed up to now in the literature in best of my knowledge. The technique is adopted similar to discussed in [10, 12] with suitable modifications.

Let X and U be the Hilbert spaces and $Z=L_2[0,b;X]$, $Z_h=L_2[-h,b;X]$, $0<h<b$, and $Y=L_2[0,b;U]$ be function spaces. $\mathbf{R }^k$ denotes k-dimensional real Euclidean space. Let $(\varOmega ,\zeta ,P)$ be the probability space with a probability measure P on $\varOmega$ and a filtration $\{\zeta _t|t\in [0,b]\}$ generated by a Wiener Process $\{\omega (s):0\le s\le t\}$.

We consider the semilinear stochastic control system of the form:

$$\begin{aligned} dx(t)= & {} [Ax(t)+Bu(t)+f(t,x_t)]dt+d\omega (t),\;t>0\nonumber \\ x(t)= & {} \xi (t),\quad t\in [-h,0] \end{aligned}$$

(1)

where the state function $x\in Z;$ $A:D(A)\subseteq X\rightarrow X$ is a closed linear operator which generates a strongly continuous semigroup S(t); $B:Y\rightarrow Z$ is a bounded linear operator; function $f:[0,b]\times X\rightarrow X$ is a nonlinear operator such that, f is measurable with respect to t, for all $x\in Z$ and continuous with respect to x for almost all $t\in [0,b]$; $x_t\in L_2([-h,0],X)=\mathbf{C }\;(let)$-valued stochastic processes and defined as $x_t(s)=\{x(t+s)|-h\le s\le 0|\}$; Control u(t) takes values in U for each $t\in [0,b]$.

By splitting the system (1), we get the following pair of coupled systems

$$\begin{aligned} \frac{dy(t)}{dt}= & {} [Ay(t)+Bv(t)+f(t,(y+z)_t];\quad 0\le t\le b\nonumber \\ y(t)= & {} \psi (t),\quad t\in [-h,0] \end{aligned}$$

(2)

and

$$\begin{aligned} dz(t)= & {} [Az(t)+Bw(t)]dt+d\omega (t);\quad 0\le t\le b\nonumber \\ z(t)= & {} \xi (t)-\psi (t),\quad t\in [-h,0] \end{aligned}$$

(3)

The system represented by (3) is linear stochastic system and for each realization z(t) of system (3), the system given by (2) is a deterministic system. Thus the solution y(t) of the semilinear system (2) depends on the solution z(t) of linear stochastic system (3). The functions v and w are Y-valued control function, such that $u=v\,+\,w$.

It can be easily seen that, the solution x(t) of the semilinear stochastic system (1) is given by $y(t)+z(t)$ where y(t) and z(t) are the solutions of the systems (2) and (3), respectively.

2 Preliminaries

In this section, some definitions are discussed which will be used in proof of main results.

The mild solution of the systems (1) can be written as

$$\begin{aligned} x(t)=\left\{ \begin{array}{l}S(t)\xi (0)+\displaystyle \int _{0}^{t}S(t-s)\{Bu(s)+f(s,x_s)\}ds+\displaystyle \int _{0}^{t}S(t-s)d\omega (s),\;t>0\\ {} \xi (t)\quad \quad \quad -h\le t\le 0\end{array}\right. \end{aligned}$$

(4)

the mild solution of the semilinear system (2) can be written as

$$\begin{aligned} y(t)=\left\{ \begin{array}{l}S(t)\psi (0)+\displaystyle \int _{0}^{t}S(t-s)\{Bv(s)+f(s,(y+z)_s\}ds,\;t>0\\ \psi (t)\quad \quad \quad -h\le t\le 0\end{array}\right. \end{aligned}$$

(5)

and the mild solution of the linear stochastic system (3) can be written as

$$\begin{aligned} z(t)=\left\{ \begin{array}{l}S(t)(\xi (0)-\psi (0))+\displaystyle \int _{0}^{t}S(t-s)Bw(s)ds+\displaystyle \int _{0}^{t}S(t-s)d\omega (s),\;t>0\\ \xi (t)-\psi (t)\quad \quad \quad -h\le t\le 0\end{array}\right. \end{aligned}$$

(6)

Consider the linear system corresponding to the system (2) given by

$$\begin{aligned} \frac{dp(t)}{dt}= & {} Ap(t)+Br(t),\quad t>0\nonumber \\ p(t)= & {} \psi (t)\quad t\in [-h,0] \end{aligned}$$

(7)

The mild solution of the above linear system is expressed as

$$\begin{aligned} p(t)=\left\{ \begin{array}{l}S(t)\psi (0)+\displaystyle \int _{0}^{t}S(t-s)Br(s)ds\; t > 0\\ \psi (t)\quad \quad \quad -h\le t\le 0\end{array}\right. \end{aligned}$$

(8)

Definition 1

The set given by $K_T(f) =\{x(T)\in X{:}\,x\in Z_h\}$ where x is a mild solution of (1) corresponding to control $u\in Y$ is called Reachable set of the system (1).

Definition 2

The system (1) is said to be approximately controllable if $K_T(f)$ is dense in X, means $\overline{K_T(f)}=X$.

3 Basic Assumptions

In this section, some basic conditions and lemmas are assumed and discussed for obtaining the main results. Throughout this paper D(A), R(A), and $N_0(A)$ denote the domain, range, and null space of operator A, respectively.

The following conditions are assumed:

$(H_1)$ For every $p\in Z$ there exists a $q\in \overline{R(B)}$ such that $Lp=Lq$ where the operator $L:Z\rightarrow X$ is defined as

$$\begin{aligned} Lx=\int _{0}^{b}S(b-s)x(s)ds \end{aligned}$$

$(H_2)$ The semigroup $\{S(t), t\ge 0\}$ generated by A is compact on X and there is a constant $M\ge 0$ such that $||S(t)||\le M$.

$(H_3)$ f(t, x) satisfies Lipschitz continuity on Z. i.e

$$\begin{aligned} ||f(t,x_1)-f(t,x_2)||\le l_p||x_1-x_2||,\quad l_p>0 \end{aligned}$$

$(H_4)$ f(t, x) satisfies linear growth condition, that is,

$$\begin{aligned} ||f(t,x)||\le a_1+b_1||x||, \end{aligned}$$

where $a_1$ and $b_1$ are constants.

$(H_5)$ $Mbb_1(1+c)<1$

where the constants b and $b_1$ appear in the above conditions. The constant c is defined in Lemma 1.

Let $G:N_0^{\bot }(L)\rightarrow \overline{R(B)}$ be an operator defined as follows:

$$\begin{aligned} Ga=a_0 \end{aligned}$$

where $a\in N_0^{\bot }(L)$ and $a_0$ is the unique minimum norm element in the set $\{a+N_0(L)\}\bigcap \overline{R(B)}\}$ satisfying the following condition

$$\begin{aligned} ||Ga||=||a_0||=\min \bigg [||e||:e\in \{a+N_0(L)\}\bigcap \overline{R(B)}\}\bigg ] \end{aligned}$$

(9)

The operator G is well defined, linear, and continuous (see [4], Lemma 1). From continuity of G, it follows that $||Ga||\le c||a||_Z$, for some constant $c\ge 0$.

Since $Z=N_0(L)+\overline{R(B)}$ as is evident from condition (H1), any element $z\in Z$ can be expressed as

$$\begin{aligned} z=n+q:n\in N_0(L), q\in \overline{R(B)} \end{aligned}$$

Lemma 1

In [12], for $z\in Z$ and $n\in N_0(L)$, the following inequality holds

$$\begin{aligned} ||n||_Z\le (1+c)||z||_Z \end{aligned}$$

(10)

where c is such that $||G||\le c$.

Let us introduce some operators in the following way:

$K:Z\rightarrow Z$ defined by

$$\begin{aligned} (Kz)(t)=\int _{0}^{t}S(t-s)z(s)ds \end{aligned}$$

Now, let $M_0$ be the subspace of $Z_h$ (see [11]) such that

$$\begin{aligned} M_0={\left\{ \begin{array}{ll}m\in Z_h:m(t)=(Kn)(t),\;n\in N_0(L) &{}0\le t\le b\\ m(t)=0,&{}\ -h\le t\le 0\end{array}\right. } \end{aligned}$$

It can be noted that $m(b)=0$, for all $m\in M_0$.

For each solution p(t) of the system (7) with control r and for each realization z(t) of the system (3), define the random operator ${f_{p}}\,:\,\overline{M_0}\rightarrow M_0$ as

$$\begin{aligned} f_p=\left\{ \begin{array}{l} Kn, \quad 0<t<b\\ 0,\quad \quad \ -h\le t\le 0\end{array}\right. \end{aligned}$$

(11)

where n is given by the unique decomposition

$$\begin{aligned} F(p+z+m)=n+q{:}\,n\in N_0(L),\;q\in \overline{R(B)}, \end{aligned}$$

(12)

where $F{:}\,L_2([0,b],\mathbf{C })\rightarrow X$ given by

$$\begin{aligned} (Fx)(t)=f(t,x_t(.));\quad \quad 0\le t\le b \end{aligned}$$

It is easy to see that F satisfies Lipschitz continuity $(H_3)$ and linear growth conditions $(H_4)$.

4 Main Results

In this section, approximate controllability of systems (2), (3) is proved. System (1) is splitted in systems (2), (3), so if systems (2), (3) are approximately controllable then system (1) is also approximately controllable.

The linear system (7) corresponding to system (2) is approximately controllable under the condition $(H_1)$ (see [5]).

For approximate controllability of (3)

$$\begin{aligned} dz(t)= & {} [Az(t)+Bw(t)]dt+d\omega (t);\quad 0\le t\le b\nonumber \\ z(t)= & {} \xi (t)-\psi (t),\quad t\in [-h,0] \end{aligned}$$

(13)

The mild solution of above system is

$$\begin{aligned} z(t)=\left\{ \begin{array}{l}S(t)(\xi (0)-\psi (0))+\displaystyle \int _{0}^{t}S(t-s)Bw(s)ds+\displaystyle \int _{0}^{t}S(t-s)d\omega (s),\;t>0\\ \xi (t)-\psi (t)\quad \quad \quad \quad -h\le t\le 0\end{array}\right. \end{aligned}$$

(14)

Define the operator $L_0^b:L_2[0,b;U]\rightarrow L_2[\varOmega ,\zeta _t,X]$, the controllability operator $\varPi _s^b:L_2[\varOmega ,\zeta _t,X]\rightarrow L_2[\varOmega ,\zeta _t,X]$ associated with (14), and the controllability operator $\varGamma _s^b:X\rightarrow X$ associated with the corresponding deterministic system of (14) as

$$\begin{aligned} L_0^b=\int _0^bS(b-s)Bw(s)ds\end{aligned}$$

(15)

$$\begin{aligned} \varPi _s^b\{.\}=\int _s^bS(b-t)BB^*S^*(b-t)\mathbf{E }\{.|\zeta _t\}dt\end{aligned}$$

(16)

$$\begin{aligned} \varGamma _s^b=\int _s^bS(b-t)BB^*S^*(b-t)dt \end{aligned}$$

(17)

It is easy to see that the operators $L_0^b,\varPi _s^b,\varGamma _s^b$ are linear-bounded operators, and the adjoint $(L_0^b)^*:L_2[\varOmega ,\zeta _t,X]\rightarrow L_2[0,b;U]$ of $L_0^b$ is defined by

$$\begin{aligned} (L_0^b)^*=B^*S^*(b-t){\mathbf{E }}\{z|\zeta _t\} \varPi _0^b=L_0^b(L_0^b)^*. \end{aligned}$$

Before studying the approximate controllability of system (3), let us first investigate the relation between $\varPi _s^b$ and $\varGamma _s^b$; $s\le r<b$ and resolvent operator $R(\lambda ,\varPi _s^b)=(\lambda I+\varPi _s^b)^{-1}$ and $R(\lambda ,\varGamma _r^b)=(\lambda I+\varGamma _r^b)^{-1}$, $s\le r<b$ for $\lambda >0$, respectively.

Lemma 2

For every $z\in L_2[\varOmega ,\zeta _t,X]$ there exists $\varphi (.)\in L_2^\zeta (0,b;\mathbf{L }(\mathbf{R }^k,X))$ such that

• 1. $\mathbf{E }\{z|\zeta _t\}=\mathbf{E }\{z\}+\int _0^t\varphi (s)d\omega (s)$,

• 2. $\varPi _s^bz=\varGamma _s^b\mathbf{E }z+\int _s^b\varGamma _r^b\varphi (r)d\omega (r)$,

• 3. $R(\lambda ,\varPi _s^b)z=R(\lambda ,\varGamma _s^b)\mathbf{E }\{z|\zeta _t\}+\int _s^b\varGamma _r^b\varphi (r)d\omega (r)$.

Proof

The proof is straightforward adaption of the proof of [10, Lemma 2.3]. $\square$

Theorem 1

The control system (3) is approximately controllable on [0, b] if and only if one of the following conditions holds.

• 1. $\varPi _0^b>0.$

• 2. $\lambda R(\lambda ,\varPi _0^b)$ converges to the zero operator as $\lambda \rightarrow 0^+$ in the strong operator topology.

• 3. $\lambda R(\lambda ,\varPi _0^b)$ converges to the zero operator as $\lambda \rightarrow 0^+$ in the weak operator topology.

Proof

The proof is straightforward adaption of the proof of [9, Theorem 2]. $\square$

Lemma 3

Under the conditions $(H_2), (H_4)$, and $(H_5)$, the operator $f_p$ has a fixed point $m_0\in M_0$ for each realization z(t) of the system (3).

Proof

From the compactness of S(t) the integral operator K is compact and hence $f_p$ is compact for each p, (see [1]). Now let $||m||\le \tilde{r}$. Then from the condition $(H_4)$ and from the inequality (10) and (12), we have

$$\begin{aligned} ||f_p(m)||^2&\le \bigg |\bigg |\int _0^tS(t-s)n(s)ds\bigg |\bigg |^2\nonumber \\&\le \int _0^b\bigg |\bigg |\int _0^tS(t-s)n(s)ds\bigg |\bigg |^2dt\nonumber \\&\le M^2b^2(1+c)^2||F(p+z+m||^2_Z\nonumber \\&\le M^2b^2(1+c)^2\{a_1+b_1||p+z+m||_Z\}^2\nonumber \\&\le M^2b^2(1+c)^2\{a_1+b_1||p+z||+b_1\tilde{r}\}^2 \end{aligned}$$

(18)

Using Schauder’s fixed-point theorem, it is clear from the compactness of $f_p$ and (18) that $f_p$ has a fixed point in $M_0$ in a ball of radius $\tilde{r}>0$, if

$$\begin{aligned} \tilde{r}>\frac{Mb(1+c)(a_1+b_1||p+z||)}{1-Mb(1+c)b_1} \end{aligned}$$

Thus $f_p(m_0)=m_0$

The approximate controllability of the semilinear system (2) is proved in following manner using the above lemma.

Lemma 4

For each realization z(t) of the system (3), the semilinear control system (2) is approximately controllable under the conditions $(H_1)$–$(H_4)$.

Proof

From the Eq. (12), we have

$$\begin{aligned} F(p+z+m)=n+q \end{aligned}$$

Operating K on both the sides at $m=m_0$ (fixed point of $f_p$) and using (11), we get

$$\begin{aligned} {\textit{KF}}(p+z+m_0)= & {} {\textit{Kn}}+{\textit{Kq}}\\= & {} m_0+{\textit{Kq}} \end{aligned}$$

Adding p on both sides, we get

$$\begin{aligned} p+{\textit{KF}}(p+z+m_0)=p+m_0+{\textit{Kq}} \end{aligned}$$

Let $p+m_0=y^*$, then the above equation is equivalent to

$$\begin{aligned} p+KF(y^*+z)=y^*+Kq \end{aligned}$$

Since, from the Eq. (8)

$$\begin{aligned} p=S(t)\psi (0)+{\textit{KBr}} \end{aligned}$$

we have

$$\begin{aligned} S(t)\psi (0)+KBr+KF(y^*+z)=y^*+Kq\\ S(t)\psi (0)+K(Br-q)+KF(y^*+z)=y^* \end{aligned}$$

Thus, it follows that $y^*(t)$ is a solution of the semilinear system

$$\begin{aligned} \frac{dy^*(t)}{dt}= & {} Ay^*(t)+f(t,(y^*+z)_t)+Br(t)-q(t),\nonumber \\ y^*(0)= & {} \psi (0) \end{aligned}$$

(19)

with control $(Br-q)$.

Moreover, since $y^*(t)=p(t)+m_0(t)$, it follows that

$$\begin{aligned} y^*(b)=p(b)+m_0(b), \end{aligned}$$

as $m_0(b)=0$ it follows that

$$\begin{aligned} y^*(b)=p(b) \end{aligned}$$

(20)

From the Eqs. (19) and (20), it is clear that the reachable set of (19) is a superset of the reachable set of the system (7), which is dense in X.

Further $q\in \overline{R(B)}$ implies that for any given $\varepsilon _1>0$, there exists $v_1\in Y$ such that $||q-Bv_1||\le \varepsilon _1$.

Now consider the equation

$$\begin{aligned} \frac{dy(t)}{dt}= & {} Ay(t)+f(t,(y+z)_t)+B(r(t)-v_1(t)),\nonumber \\ y(0)= & {} \psi (0) \end{aligned}$$

(21)

Let y(t) be the solution of the system (21), corresponding to control $v=r-v_1$. Then $||y^*(b)-y(b)||$ can be made arbitrary small by choosing a suitable $v_1$, which implies that the reachable set of the system (21) is dense in the reachable set of the system (19), which in turn is dense in X. This proves that the system (2) is approximately controllable. $\square$

5 Example

Consider the stochastic control system with delay governed by the semilinear heat equation

$$\begin{aligned} \partial y(t,x)=\bigg [\frac{\partial ^2y(t,x)}{\partial x^2}+Bu(t,x)+f(t,y(t+v,x))\bigg ]\partial t+\partial \omega (t)\nonumber \\ {\text {for}} \quad 0<t<\tau ;\;v\in [-h,0];\quad 0<x<\pi \nonumber \\ {\text {with conditions}}\quad y(t,0)=y(t,\pi )=0,\quad 0\le t\le \tau \nonumber \\ y(t,x)=\xi (t,x),\quad \quad -h\le t\le 0,\;0\le x\le \pi \end{aligned}$$

(22)

The system (22) can be written in the abstract form (1), by setting $X=L_2(0,\pi )$ and $A=\frac{d^2}{dx^2}$, with domain consisting of all $y\in X$ with $\left( \frac{d^2y}{dx^2}\right) \in X$ and $y(0)=0=y(\pi )$. Take $\phi _(x)=(2/\pi )^{1/2}sin(nx),\; 0\le x\le \pi ,\;n=1,2,3,...$, then $\{\phi _n(x)\}$ is an orthonormal basis for X and $\phi _n$ ia an eigenfunction corresponding to the eigenvalue $\lambda _n=-n^2$ of the operator A, $n=1,2,3,...$. Then the $C_0$-semigroup T(t) generated by A has $e^{\lambda _nt}$ as the eigenvalues and $\phi _n$ as their corresponding eigenfunctions.

Define an infinite-dimensional space U by

$$\begin{aligned} U=\bigg \{u:u=\sum _{n=2}^{\infty }u_n\phi _n\;{\text {with}}\;\sum _{n=2}^{\infty }u_n^2<\infty \bigg \} \end{aligned}$$

The norm defined by

$$\begin{aligned} ||u||_U=\bigg (\sum _{n=2}^{\infty }u_n^2\bigg )^{1/2} \end{aligned}$$

$\xi (t,x)$ is known function.

Let B be a continuous linear operator from U to X defined as

$$\begin{aligned} Bu=2u_2\phi _1+\sum _{n=2}^{\infty }u_n\phi _n,\quad u=\sum _{n=2}^{\infty }u_n\phi _n\in U \end{aligned}$$

The nonlinear operator f is assumed to satisfy conditions $(H_3)$ and $(H_4)$.

The approximate controllability of the corresponding semilinear deterministic heat equation of (22) was considered by Naito [4] and proved under the conditions $(H_1)$–$(H_4)$. Here approximate controllability of the stochastic semilinear heat control system (22) is considered.

The system (22) can be associated with two control systems under the initial and boundary conditions, as given below

$$\begin{aligned} \frac{\partial y(t,x)}{\partial t}= & {} \frac{\partial ^2y(t,x)}{\partial x^2}+y(t-h,x)\nonumber \\&+Bv(t,x)+f(t,y(t-h,x)+z(t-h,x))\;t\in [0,b]\;x\in [0,\pi ]\qquad \quad \end{aligned}$$

(23)

$$\begin{aligned} y(t,x)= & {} \xi (t,x),\quad \quad -h\le t\le 0,\;0\le x\le \pi \nonumber \\ \partial z(t,x)= & {} \bigg [\frac{\partial ^2z(t,x)}{\partial x^2}+z(t-h,x)+Bw(t)\bigg ]\partial t+\partial \omega (t) \end{aligned}$$

(24)

The system (24) is a linear stochastic system and for each realization z(t) of the system (24), the system (23) is a deterministic system.

From Lemma 4 and using the conditions $(H_1)$–$(H_4)$, it is clear that for each realization z(t) of the system (24), the system (23) is approximately controllable. The linear stochastic system (24) is approximately controllable from Lemma 3 corresponding to (23) and linear system corresponding to system (23) is approximately controllable from [4].