Complete controllability of semi-linear stochastic system with delay

Abstract

This paper deals with the complete controllability of semilinear stochastic system with delay under the assumption that the corresponding linear system is completely controllable. The control function for this system is suitably constructed by using the controllability operator. With this control function, the sufficient conditions for the complete controllability of the proposed problem in finite dimensional are established. The results are obtained by using Banach fixed point theorem. Finally, one example is provided to illustrate the application of the obtained results.

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 medelled stochastic form.

In setting of deterministic systems: Kalman [1] introduced the concept of controllability for finite dimensional deterministic linear control systems. The basic concepts of control theory in finite dimensional spaces has been introduced in [2]. In [3] Naito established sufficient conditions for approximate controllability of deterministic semilinear control system dominated by the linear part using Schuder’s fixed point theorem. Balachandran et al. [4] obtained results for controllability of nonlinear systems in Banach spaces. In [5, 6] Wang extended the results of [3] and established sufficient conditions for delayed deterministic semilinear systems using Schauder’s fixed point theorem and concept of fundamental solution. In [7, 8] Sukavanam et al. obtained the results for approximate controllability of a delayed semilinear control system with growing nonlinear term using Schauder’s fixed point theorem.

In setting of stochastic systems: in [9] Bashirov et al. provides some concepts for controllability of linear stochastic systems. Using these concepts Mahmudov [10] established sufficient conditions for controllability of linear stochastic systems in Hilbert spaces. In [11–15] Klamka obtained some sufficient conditions for controllablity of delay linear systems in finite dimensional using Rank theorem. In [16, 17] Mahmudov obtained results for controllability of semilinear stochastic systems using Banach fixed point theorem. Shen et al. [18] extended the results of [12] in infinite dimensional using technique of [16] and obtained sufficient conditions for Relative controllability of stochastic nonlinear systems with delay in control. In [19] Sukavanam et al. obtained some results for stochastic controllability of an abstract first order semilinear control system using Schauder’s fixed point theorem. Recently Shukla et al. [20] obtained some sufficient conditions for approximate controllability of retarded semilinear stochastic system with non local conditions using Banach fixed point theorem. However in best of our knowledge, there is no result on complete controllability of semilinear stochastic system with delay as treated in this paper. In this paper results are obtained in $L_2$ norm and delay is considered in both state and control term simultaneously which in not previously discussed up to now in the literature.

In this paper we adopt the following notations:

• (i) $(\Omega ,\digamma ,P)$:Let $\digamma$ be the $\sigma$ algebra generated by $\Omega \subset R^n$ and $P:\digamma \rightarrow [0,1]$ be the probability measure on $\digamma$. Then the triple $(\Omega ,\digamma ,P)$ is called a probability space.

• (ii) Let $\omega$ be the Wiener process and $\{\digamma _t|t\in [0,T]\}$ is the filtration generated by $\{\omega (s):0\le s\le t\}$.

• (iii) $L_2(\Omega ,\digamma _T,R^n)$=the Hilbert space of all $\digamma _T$-measurable square integrable variables with values in $R^n$.

• (iv) $L_P^\digamma ([0,T],R^n)$ is the Banach space of all $p$-integrable and $\digamma _t$-measurable processes with values in $R^n$,   for $p\ge 2$.

• (v) $X_2$=the Banach space of all square integrable and $\digamma _t$-adapted processes $\varphi (t)$ with norm

  $$\begin{aligned}||\varphi ||^2=\left( \int _{-h}^{t}\mathbf E ||\varphi (t)||^2\right) ,\quad where\quad \mathbf E \quad is\quad Expected \quad value.\end{aligned}$$

• (vi) $\mathbb {L}(X,Y)$ is the space of all linear bounded operators from a Banach space $X$ in to a Banach space $Y$.

• (vii) Let set of admissible controls is $U_{ad}=L_2^\digamma ([0,T],R^m)$. $\square$

Complete controllability of linear stochastic system with control delay

$$\begin{aligned} dx(t)=[Ax(t)+B_0u(t)+B_1u(t-h)]dt+\sigma d\omega (t) \end{aligned}$$

(1.1)

given the initial condition as a random function

$$\begin{aligned} x(0)=x_0\in L_2(\Omega ,\digamma _T,\mathbb {R}^n)\quad and\quad u(t)=0\quad for\quad t\in [-h,0) \end{aligned}$$

has been studied by various authors under Supremum norm. (see Klamka [12] and the references there in).

The problem of controllability of semi-linear stochastic system

$$\begin{aligned} \begin{array}{c} \textstyle dx(t)=[Ax(t)+Bu(t)+f(t,x(t))]dt+\sigma (t,x(t))d\omega (t)\\ \textstyle x(0)=x_0 \in R^n \end{array} \end{aligned}$$

has been studied by various authors under Supremum norm. (see Mahmudov [16] and Sukavanam [8])

In this paper we examine the complete controllability of the following semi-linear stochastic system with delay in $L_2$ norm:

$$\begin{aligned} dx(t)&= (Ax(t)+B_0u(t)+B_1u(t-h)+f(t,x_t))dt+\sigma (t,x_t) d\omega (t)\quad \text{ for } \text{ t } \in (0,T]\end{aligned}$$

(1.2)

$$\begin{aligned} x(t)&= \psi (t),\quad for\; t\in [-h,0],\quad x(0)=\psi (0)=x_0(let)\end{aligned}$$

(1.3)

$$\begin{aligned} u(t)&= 0,\quad for\;\in [-h,0] \end{aligned}$$

(1.4)

where the state $x(t)\in L_2(\Omega ,\digamma _t,R^n)=X$ and the control $u(t)\in L_2^\digamma ([0,T],R^m)=U$, $A$ is $n\times n$ constant matrix, $B_0$ and $B_1$ are an $n\times m$ constant matrices .$x_t\in L^2([-h,0],R^n)$-valued stochastic processes and defined as $x_t(s)=\{x(t+s)|-h\le s\le 0|\}$ and $\psi =\{\psi (s)|-h\le s\le 0\}\in L^2([-h,0],R^n)$-valued stochastic processes and $h\ge 0$ is the upper bound for the time delay.Moreover, the functions $f(.,.),\sigma (.,.)$ are defined as $\sigma :[0,T]\times L^2([-h,0],R^n)\rightarrow R^{n\times n}$, $f:[0,T]\times L^2([-h,0], R^n)\rightarrow R^{n}$ are nonlinear functions and $\omega$ is a $n$-dimensional Wiener process.

2 Preliminaries

It is well known [8, 16] that for given initial conditions $(1.3),(1.4)$, any admissible control $u\in U_{ad}$, for $t\in [-h,T]$ and suitable nonlinear functions $f(t,x_t)$ and $\sigma (t,x_t)$ there exists unique solution $x(t;x_0,u)\in L_2(\Omega ,\digamma _t,R^n)$ of the semilinear stochastic differential state equation (1.2) which can be represented in the following integral form

$$\begin{aligned} x(t;x_0,u)=\left\{ \begin{array}{l} \textstyle exp(At)x_0+\int _{0}^{t}exp(A(t-s))(B_0u(s)+B_1u(s-h)+f(s,x_s))ds\\ \textstyle \quad +\int _{0}^{t}exp(A(t-s))\sigma (s,x_s)d\omega (s)\quad { for}\quad t>0\\ \textstyle \psi (t) \quad { for}\quad t\in [-h,0] \end{array}\right. \end{aligned}$$

(2.1)

Since $u(t)=0$ for $t\in [-h,0]$ therefore solution for $t\in [0,h]$ has the following form

$$\begin{aligned} x(t;x_0,u)&= exp(At)x_0+\int _{0}^{t}\exp (A(t-s))(B_0u(s)+f(s,x_s)ds\nonumber \\&\quad +\int _{0}^{t}exp(A(t-s))\sigma (s,x_s)d\omega (s) \end{aligned}$$

(2.2)

for $t>h$ we have

$$\begin{aligned} x(t;x_0,u)&= exp(At)x_0+\int _{0}^{t}exp(A(t-s))(B_0u(s)+f(s,x_s))ds\\&\quad +\int _{0}^{t-h}exp(A(t-s-h))B_1u(s)ds \\&\quad +\displaystyle \int _{0}^{t}exp(A(t-s))\sigma (s,x_s)d\omega (s) \end{aligned}$$

or equivalent

$$\begin{aligned} x(t;x_0,u)&= exp(At)x_0+\int _{0}^{t-h}(exp(A(t-s))B_0+exp(A(t-s-h))B_1)u(s)ds\nonumber \\&\quad +\int _{t-h}^{t}exp(A(t-s))B_0u(s)ds\nonumber \\&\quad +\int _{0}^{t}exp(A(t-s))f(s,x_s)ds+\int _{0}^{t}exp(A(t-s))\sigma (s,x_s)d\omega (s) \end{aligned}$$

(2.3)

for $T>h$, let us introduce the following operators and sets.

$L_T\in \mathbb {L}((L_2^\digamma ([0,T]),R^m),L_2(\Omega ,\digamma _T,R^n))$, defined by

$$\begin{aligned} L_Tu&= \int _{0}^{T-h}(exp(A(T-s))B_0+exp(A(T-s-h))B_1)u(s)ds\\&\quad +\int _{T-h}^{T}exp(A(T-s))B_0u(s)ds \end{aligned}$$

Then it can be seen that the adjoint operator $L_T^*\in L_2(\Omega ,\digamma _T,R^n)\rightarrow L_2^\digamma ([0,T]),R^m)$ is given by

$$\begin{aligned} L_T^*z=\left\{ \begin{array}{l} \textstyle (B_0^*exp(A^*(T-t))+B_1^*exp(A^*(T-t-h)))E\{z|\digamma _t\}\quad \text{ for } \text{ t } \in [0,T-h]\\ \textstyle B_0^*exp(A^*(T-t))E\{z|\digamma _t\}\quad \text{ for } \text{ t } \in (T-h,T]\end{array}\right. \end{aligned}$$

The set of all states reachable in time $T$ from initial state $x(0)=x_0\in L_2(\Omega ,\digamma _T,R^n)$, using admissible controls is defined as

$$\begin{aligned} R_T(U_{ad})&= \{x(T;x_0,u)\in L_2(\Omega ,\digamma _T,R^n):u\in U_{ad}\} \\ where\quad x(T;x_0,u)&= exp(At)x_0 +L_Tu+\int _{0}^{T}exp (A(T-s))(f(s,x_s)ds+\sigma (s,x_s) d\omega (s)) \end{aligned}$$

Now we introduce the linear controllability operator $\Pi _0^T \in \mathbb {L}(L_2(\Omega ,\digamma _T,R^n), L_2(\Omega ,\digamma _T, R^n))$ as follows:

$$\begin{aligned} \Pi _0^T \{.\}&= L_T(L_T)^*\{.\}\\&= \int _{0}^{T-h}(exp(A(T-t))B_0B_0^*exp(A^*(T-t))+exp(A(T-t-h))\\&\quad \times \, B_1B_1^*exp(A^*(T-t-h)))E\{.|\digamma _t\}dt\\&\quad +\int _{T-h}^{T}(exp(A(T-t))B_0B_0^*exp(A(T-t)))E\{.|\digamma _t\}dt \end{aligned}$$

The corresponding controllability $n\times n$ matrix for deterministic model is:

$$\begin{aligned} \Gamma _s^T&= L_T(s) L_T^*(s)\\&= \int _{s}^{T-h}(exp(A(T-t))B_0B_0^*exp(A^*(T-t))+exp(A(T-t-h))\\&\quad \times \, B_1B_1^*exp(A^*(T-t-h)))dt\\&\quad +\int _{T-h}^{T}exp(A(T-t))B_0B_0^*exp(A^*(T-t))dt \end{aligned}$$

Definition 2.1

A control system is said to be completely controllable in the interval $I=[0,T]$ if for every initial state $x_0$ and desired final state $x_1$, there exists a control $u(t)$ such that the solution $x(t)$ of the system corresponding to this control u satisfies $x(T)=x_1$.

Remark 2.1

For dynamical system (1.2) it is possible to define many different concepts of controllability. Using this admissible controls in [21, 22] Klamka obtained complete controllability with constrained admissible controls of nonlinear systems. It is generally assumed that the control values are in a convex and closed cone with vertex at zero, or in a cone with nonempty interior. Klamka obtained sufficient conditions for constrained exact local controllability using the generalized open mapping theorem. Let $U_0\subset U$ be a closed convex cone with nonempty interior. The set of admissible controls for the system (1.2) is given by $U_{ad}=L_\infty ([0,T],U_0)$ (for more detail see [21, 22]).

In this paper some sufficient conditions for complete controllability with unconstrained admissible controls of system (1.2) is obtained. Unconstrained admissible control for the system (1.2) in this paper is defined in notation (vii).

3 Main results

Lemma 1

Assume that the operator $\Pi _0^T$ is invertible. Then for arbitrary $x_T\in L_2(\Omega ,\digamma _T, R^n)$, $f(.,.)\in L_2^\digamma ([0,T],R^n)$,$\sigma (.,.)\in (L_2^\digamma ([0,T],R^{n\times n}$), the control

$$\begin{aligned} u(t)=\left\{ \begin{array}{l} \textstyle B_0^*exp(A^*(T-t))\\ \textstyle \,\,\times E\bigg \{(\Pi _0^T)^{-1}\left( x_T-exp(AT)x_0- \int _{0}^{T-h} exp(A(T-s))(f(s,x_s)ds+\sigma (s,x_s)d\omega (s))\right) |\digamma _t\bigg \} \quad {\textit{for}} \ t\in [0,h]\\ \textstyle (B_0^*exp(A^*(T-t))+B_1^*exp(A^*(T-h-t)))\\ \textstyle \,\,\times E\bigg \{(\Pi _0^T)^{-1}\left( x_T-exp(AT)x_0- \int _{T-h}^{T}exp(A(T-s))(f(s,x_s)ds+\sigma (s,x_s) d\omega (s))\right) |\digamma _t\bigg \} \quad {\textit{for}} \ t\in (h,T] \end{array}\right. \end{aligned}$$

transfers the system (2.1) from $x_0\in R^n$ to $x_T$ at time $T$ and

$$\begin{aligned} x(t)&= \exp (At)x_0+\Pi _{0}^{t}[exp(A^*(T-t))(\Pi _{0}^{T})^{-1}\times (x_T-exp(AT)x_0\nonumber \\&\quad -\int _{0}^{T}exp(A(T-r))f(r,x_r)dr-\int _{0}^{T}exp(A(T-r))\sigma (r,x_r)d\omega (r))]\nonumber \\&\quad +\int _{0}^{t}exp(A(t-s))f(s,x_s)ds+\int _{0}^{t}exp(A(t-s))\sigma (s,x_s)d\omega (s) \end{aligned}$$

(3.1)

provided the solution of (3.1) exists.

Proof By substituting $u(t)$ in (2.2) and (2.3), we can easily obtain the following

For $t\in [0,h]$

$$\begin{aligned} x(t;x_0,u)&= exp(At)x_0+\int _{0}^{t}exp(A(t-s))B_0B_0^*exp(A^*(t-s))\\&\quad \times \, E\bigg \{(\Pi _0^T)^{-1} \bigg (x_T-exp(AT)x_0-\displaystyle \int _{0}^{T-h}exp(A(T-s))(f(s,x_s)ds\\&\quad +\,\sigma (s,x_s)d\omega (s))\bigg ) |\digamma _s\bigg \} ds+\int _{0}^{t}exp(A(t-s))f(s,x_s)ds\\&\quad +\int _{0}^{t}exp(A(t-s))\sigma (s,x_s) d\omega (s) \end{aligned}$$

For $t\in (h,T]$

$$\begin{aligned} x(t;x_0,u)&= exp(At)x_0+\int _{0}^{t-h}(exp(A(t-s))B_0B_0^*exp(A^*(t-s))\\&\quad +\, exp(A(T-h-s))B_1B_1^*exp(A^*(T-h-s)))\\&\quad \times \, E\bigg \{(\Pi _0^T)^{-1}\left( x_T-exp(AT)x_0-\displaystyle \int _{T-h}^{T}exp(A(T-s))(f(s,x_s)ds\right. \\&\quad +\left. \sigma (s,x_s)d\omega (s))\right) |\digamma _s\bigg \} ds+\int _{t-h}^{t}exp(A(t-s))B_0B_0^*exp(A^*(T-s)) \\&\quad \times \, E\bigg \{(\Pi _0^T)^{-1}\left( x_T-exp(AT)x_0-\displaystyle \int _{0}^{T-h}exp(A(T-s))(f(s,x_s)ds\right. \\&\quad \left. +\,\sigma (s)d\omega (s,x_s))\right) |\digamma _s\bigg \} ds+\int _{0}^{t}exp(A(t-s))f(s,x_s)ds\\&\quad +\int _{0}^{t}exp(A(t-s))\sigma (s,x_s) d\omega (s) \end{aligned}$$

Thus, taking into account of the form of the operator $\Pi _0^T$ [12, 19] we have

$$\begin{aligned} x(t;x_0,u)&= exp(At)x_0+\Pi _0^t\bigg [exp(A^*(T-t))(\Pi _0^T)^{-1}\bigg (x_T-exp(AT)x_0 \\&\quad -\int _{0}^{T}exp(A(T-s))(f(s,x_s)ds+\sigma (s,x_s)d\omega (s))\bigg )\bigg ]\\&\quad +\int _{0}^{t}exp(A(t-s))f(s,x_s)ds+\int _{0}^{t}exp(A(t-s))\sigma (s,x_s) d\omega (s) \end{aligned}$$

Substitute $t=T$ in above equation we get

$$\begin{aligned} x(T;x_0,u)&= exp(AT)x_0+\Pi _0^T\bigg [(\Pi _0^T)^{-1}\\&\quad \times \left( x_T-exp(AT)x_0 -\displaystyle \int _{0}^{T}exp(A(T-s))(f(s,x_s)ds+\sigma (s,x_s) d\omega (s)\right) \bigg ]\\&\quad +\int _{0}^{T}exp(A(T-s))f(s,x_s)ds+\int _{0}^{T}exp(A(T-s))\sigma (s,x_s) d\omega (s)\\ x(T;x_0,u)&= x_T \end{aligned}$$

Remark 3.1

In Theorem 3.1 sufficient condition are given for the existence and uniqueness of solution of (3.1).

Lemma 2

(See [23]) For every $z\in L_2(\Omega ,\digamma _T,R^n)$, there exists a process $\varphi (.)\in L_2([0,T],R^{n\times n})$ such that

$$\begin{aligned} z&= \mathbf E z+\int _{0}^{T}\varphi (s)d\omega (s)\\ \Pi _0^Tz&= \Gamma _0^T\mathbf E z+\int _{0}^{T}\Gamma _s^T\varphi (s)d\omega (s)\\ Moreover&\\ E||\Pi _0^Tz||^2&\le ME||E\{z|\digamma _T\}||^2\\&\le ME||z||^2, \quad z\in L_2(\Omega ,\digamma _T,R^n) \end{aligned}$$

Note that if the assumption (A3) holds, then for some $\gamma >0$

$$\begin{aligned} E\langle \Pi _0^Tz,z\rangle \ge \gamma E||z||^2, \quad for\quad all\quad z\in L_2(\Omega ,\digamma _T,R^n) \end{aligned}$$

consequently

$$\begin{aligned}E||(\Pi _0^T)^{-1}||^2\le \frac{1}{\gamma }=l_4\end{aligned}$$

Now let us assume the following conditions

• (A1) $(f,\sigma )$ satisfies the Lipschitz condition with respect to $x$ i.e.,

  $||f(t,x_t)-f(t,y_t)||^2\le L_1||x_t-y_t||^2$, $||\sigma (t,x_t)-\sigma (t,y_t)||^2\le L_2||x_t-y_t||^2$    for   all   $x_t,y_t\in L^2([-h,0],R^n)$,   $0<t\le T$

• (A2) $(f,\sigma )$ is continuous on $[0,T]\times R^n$ and satisfies

  $||f(t,x_t)||^2\le L_3(||x_t||^2+1)$, $||\sigma (t,x_t)||^2\le L_4(||x_t||^2+1)$

• (A3) The linear system $(1.1)$ is completely controllable.

Let $\mathbf S$ be an operator defined as

$$\begin{aligned}&\mathbf {S}(x)(t)\\&\quad =\left\{ \begin{array}{l} \textstyle \psi (t)\quad for\quad t\in [-h,0]\\ \textstyle exp(At)x_0+\Pi _0^t\bigg [exp(A^*(T-t))\bigg ((\Pi _0^T)^{-1}\times \bigg (x_T-exp(AT)x_0\\ \textstyle \quad -\int _{0}^{T}exp(A(T-r))f(r,x_r)dr-\int _{0}^{T}exp(A(T-r))\sigma (r,x_r)d\omega (r)\bigg )\bigg )\bigg ]\\ \textstyle \quad +\int _{0}^{t}exp(A(t-s))f(s,x_s)ds +\int _{0}^{t}exp(A(t-s))\sigma (s,x_s)d\omega (s)\quad for \quad t\in [0,T] \end{array}\right. \end{aligned}$$

From Lemma 1, the control $u(t)$ transfer the system (3.1) from the initial state $x_0$ to the final state $x_T$ provided that the operator $\mathbf S$ has a fixed point. So, if the operator $\mathbf S$ has a fixed point then the system (1.2) has a unique mild solution and completely controllable.

Now for convenience, let us introduce the notation

$$\begin{aligned}&l_1=max{||exp(At)||^2:t\in [0,T]},\quad l_2=max(||B_0||^2,||B_1||^2)\\&l_3=E||x_T||^2, \quad M=max{||\Gamma _{s}^{T}||^2: s\in [0,T]} \end{aligned}$$

Theorem 3.1

Assume that the conditions (A1), (A2) and (A3) hold. In addition if the inequality

$$\begin{aligned} \bigg (4l_1L(Ml_1l_4+1)(T+1)T^2\bigg )^{\frac{1}{2}}<1 \end{aligned}$$

(3.2)

holds, then the system (1.2) is completely controllable.

Proof As mentioned above, to prove the complete controllability it is enough to show that $\mathbf S$ has a fixed point in $X_2$. To do this, we use the contraction mapping principle. To apply the contraction mapping principle, first we show that $\mathbf S$ maps $X_2$ into itself.

Now by Lemma 1 we have

$$\begin{aligned}&E\bigg |\bigg |(\mathbf S x)(t)\bigg |\bigg |^2\\&\quad =E\bigg |\bigg |\psi (t)+exp(At)x_0+\Pi _0^t\left[ exp(A^*(T-t))\times (\Pi _0^T)^{-1}(x_T-exp(AT)x_0\right. \\&\qquad \left. -\int _{0}^{T}exp(A(T-r))f(r,x_r)dr-\int _{0}^{T}exp(A(T-r))\sigma (r,x_r)d\omega (r))\right] \\&\qquad +\int _{0}^{t}exp(A(t-s))f(s,x_s)ds+\int _{0}^{t}exp(A(t-s))\sigma (s,x_s)d\omega (s)\bigg |\bigg |^2\\&\quad \le 5||\psi (t)||^2 +5||exp(At)x_0||^2 +5E\bigg |\bigg |\Pi _0^t[exp(A^*(T-t))\times (\Pi _0^T)^{-1} (x_T-exp(AT)x_0\\&\qquad -\int _{0}^{T}exp(A(T-r))f(r,x_r)dr-\int _{0}^{T}exp(A(T-r))\sigma (r,x_r)d\omega (r))]\bigg |\bigg |^2\\&\qquad +5t\int _{0}^{t}||exp(A(t-r))||^2E||f(r,x_r||^2dr+5\int _{0}^{t}||exp(A(t-r))||^2E||\sigma (r,x_r)||^2dr\\&\quad \le 5||\psi (t)||^2+5l_1||x_0||_{L^2[-h,0]}^2+20Ml_1l_4\bigg (l_3+l_1||x_0||^2+Tl_1\int _{0}^{T}E||f(r,x_r)||^2dr\\&\qquad +l_1\int _{0}^{T}E||\sigma (r,x_r)||^2dr\bigg )+5l_1\int _{0}^{t}\left( TE||f(r,x_r)||^2+E||\sigma (r,x_r)||^2\right) dr\\&\quad \le B_1+B_2\left( \int _{0}^{T}\left( TE||f(r,x_r)||^2+E||\sigma (r,x_r)||^2\right) dr\right) \end{aligned}$$

where $B_1>0$ and $B_2>0$ are suitable constants. It follows from the above and the condition (A2) that there exists $C_1>0$ such that

$$\begin{aligned} E\bigg |\bigg |(Sx)(t)\bigg |\bigg |^2&\le C_1\left( 1+\int _{0}^{T}E||x_r||^2dr\right) \\&= C_1\left( 1+\left( \int _{0}^{T}E\int _{-h}^{0}||x(r+s)||^2dsdr\right) \right) \\&= C_1\left( 1+\left( \int _{0}^{T}E\int _{r-h}^{r}||x(v)||^2dvdr\right) \right) \\&\le C_1\left( 1+\left( \int _{0}^{T}E\int _{-h}^{T}||x(v)||^2dvdr\right) \right) \\&\le C_1\left( 1+T\left( \int _{-h}^{T}E||x(v)||^2dv\right) \right) \end{aligned}$$

taking $L_2$ norm both side

$$\begin{aligned} \left( \int _{-h}^{T}E||(Sx)(t)||^2dt\right) ^{1/2}&\le \left( \int _{-h}^{T}C_1\left( 1+T\left( \int _{-h}^{T}E||x(v)||^2dv\right) \right) dt\right) ^{1/2}\\&\le \sqrt{C_1}\sqrt{T+h}\left( 1+T\left( \int _{-h}^{T}E||x(t)||^2dt\right) \right) ^{1/2} \end{aligned}$$

for all $t\in [-h,T]$. Therefore $\mathbf S$ maps $X_2$ into itself. Secondly, we show that $\mathbf S$ is a contraction mapping. Indeed

$$\begin{aligned}&E\bigg |\bigg |(\mathbf S x)(t)-(\mathbf S y)(t)\bigg |\bigg |^2\\&\quad =E\bigg |\bigg |\Pi _0^t[exp(A^*(T-t))(\Pi _0^T)^{-1}\times \left( \int _{0}^{T}exp(A(T-s))(f(s,y_s)-f(s,x_s))\right) ds\\&\qquad +\int _{0}^{T}exp(A(T-s))(\sigma (s,y_s)-\sigma (s,x_s)))d\omega (s))]\\&\qquad +\int _{0}^{t}exp(A(t-s))(f(s,x_s)-f(s,y_s))ds\\&\qquad +\int _{0}^{t}exp(A(t-s))(\sigma (s,x_s)-\sigma (s,y_s))d\omega (s)\bigg |\bigg |^2\\&\quad \le 4Ml_1^2l_4(T\int _{0}^{T}E||f(s,x_s)-f(s,y_s)||^2ds+\int _{0}^{T}E||\sigma (s,x_s)-\sigma (s,y_s)||^2ds)\\&\qquad +4l_1(T\int _{0}^{t}E||f(s,x_s)-f(s,y_s)||^2ds+\int _{0}^{t}E||\sigma (s,x_s)-\sigma (s,y_s)||^2ds)\\&\quad =4Ml_1^2l_4L(T+1)\int _{0}^{T}E||x_s-y_s)||^2ds+4l_1L(T+1)\int _{0}^{t}E||x_s-y_s||^2ds\\&\quad \le 4l_1L(Ml_1l_4+1)(T+1)\int _{0}^{T}E||x_s-y_s||^2ds\\&\quad = 4l_1L(Ml_1l_4+1)(T+1)\left( \int _{0}^{T}E\int _{-h}^{0}||x(t+s)-y(t+s)||^2dtds\right) \\&\quad = 4l_1L(Ml_1l_4+1)(T+1)\left( \int _{0}^{T}E\int _{s-h}^{s}||x(v)-y(v)||^2dvds\right) \\&\quad \le 4l_1L(Ml_1l_4+1)(T+1)\left( \int _{0}^{T}E\int _{-h}^{T}||x(v)-y(v)||^2dvds\right) \\&\quad \le 4l_1L(Ml_1l_4+1)(T+1)T\left( \int _{-h}^{T}E||x(v)-y(v)||^2dv\right) \end{aligned}$$

taking $L_2$ norm both side we get

$$\begin{aligned}&\left( \int _{-h}^{T}E\bigg |\bigg |(\mathbf S x)(t)-(\mathbf S y)(t)\bigg |\bigg |^2dt\right) ^{1/2}=\left( \int _{0}^{T}E\bigg |\bigg |(\mathbf S x)(t)-(\mathbf S y)(t)\bigg |\bigg |^2dt\right) ^{1/2}\\&\quad \le \left( \int _{0}^{T}4l_1L(Ml_1l_4+1)(T+1)T\left( \int _{-h}^{T}E||x(v)-y(v)||^2dv\right) dt\right) ^{1/2}\\&\quad \le \bigg (4l_1L(Ml_1l_4+1)(T+1)T^2\bigg )^{\frac{1}{2}}\left( \int _{-h}^{T}E||x(t)-y(t)||^2dt\right) ^{1/2} \end{aligned}$$

Therefore $\mathbf S$ is a contraction mapping if the inequality (3.2) holds. Then the mapping $\mathbf S$ has a unique fixed point $x(.)$ in $X_2$ which is the solution of the Eq. (1.2). Thus the system (1.2) is completely controllable. The theorem is proved.

Remark 3.2

In this paper the sufficient conditions for complete controllability are obtained in Theorem (3.1) for the system (1.2)–(1.4) using $L_2$ norm. In [21] Mahmudov et al. have considered a particular case of system (1.2)–(1.4) by taking $B_1=0,\; f(t,x_t)=f(t,x(t)),\; \sigma (t,x_t)=\sigma (t,x(t))$ and obtained the results using $L_{\infty }$ (supremum) norm.

Remark 3.3

In [12] Balachandran et al. have considered a particular case of system (1.2)–(1.4) for deterministic system by taking $B_1=0,\; f(t,x_t)=f(t,x(t)),\; \sigma (t,x_t)=0$ and obtained the results using $L_{\infty }$ (supremum) norm.

Remark 3.4

Inequality (3.2) is fulfilled if $L$ is sufficiently small.

4 Example

Example 1

Consider a two-dimensional semi-linear stochastic system with delay in state and control terms

$$\begin{aligned} dx(t)&= [A_0x(t)+B_0u(t)+B_1u(t-h)+f(t,x_t)]dt+\sigma (t,x_t)d\omega (t)\nonumber \\&{\text {for}}\, t\in [0,T] \end{aligned}$$

(4.1)

with initial condition (1.3)

where $\omega (t)$ is a one dimensional Wiener process and

$$\begin{aligned} A_0=\left[ \begin{array}{r@{\quad }r} -1 &{} 1 \\ -1 &{} -1 \end{array}\right] , \;\; B_0=\left[ \begin{array}{r@{\quad }r} 1 &{} 0 \\ 0 &{} 1 \end{array}\right] ,\;\; B_1=\left[ \begin{array}{r@{\quad }r} 0 &{} -1 \\ 1 &{} 0 \end{array}\right] \\ f(t,x_t)=\frac{1}{a}\left[ \begin{array}{r} \sin x_t \\ x_t \end{array}\right] , \;\;\sigma (t,x_t)=\frac{1}{b} \left[ \begin{array}{r@{\quad }r} x_t&{} 0 \\ 0 &{} \cos x_t \end{array}\right] \end{aligned}$$

If we take Euclidean norm then

$$\begin{aligned}&||f(t,x_t)-f(t,y_t)||^2\le \frac{2}{a^2}||x_t-y_t||^2 \quad and \nonumber \\&||\sigma (t,x_t)-\sigma (t,y_t)||^2\le \frac{2}{b^2}||x_t-y_t||^2\quad so, \nonumber \\&||f(t,x_t)-f(t,y_t)||^2+||\sigma (t,x_t)-\sigma (t,y_t)||^2\le L||x_t-y_t||^2\end{aligned}$$

(4.2)

$$\begin{aligned}&where \quad L=\left( \frac{2}{a^2}+\frac{2}{b^2}\right) \end{aligned}$$

(4.3)

$$\begin{aligned} ||A_0||=2,||B_0||=\sqrt{2},||B_1||=\sqrt{2} \end{aligned}$$

We can see that conditions of Theorem 3.1 with the help definition of $M$ and Lemma 2 for sufficiently small $L$ [using Eq. (4.3)] for any time $T$ are satisfied. So system (4.1) is completely controllable.

Example 2

In particular let us consider the system (1.2) without any delay means $B_1=0$, $f(t,x_t)=f(t,x(t))$ and $\sigma (t,x_t)=\sigma (t,x(t))$

$$\begin{aligned} dx(t)&= [Ax(t)+Bu(t)+f(t,x(t))]dt+\sigma (t,x(t))d\omega (t)\nonumber \\ x(0)&= x_0 \in R^n \end{aligned}$$

(4.4)

$$\begin{aligned} A=\left[ \begin{array}{r@{\quad }r} -1 &{} 1 \\ -1 &{} -1 \end{array}\right] , \; B=\left[ \begin{array}{r@{\quad }r} 1 &{} 0 \\ 0 &{} 1 \end{array}\right] ,\; f(t,x(t))=\left[ \begin{array}{r} \sin x(t) \\ x(t) \end{array}\right] , \; \sigma (t,x(t))=\left[ \begin{array}{r@{\quad }r} x(t)&{} 0 \\ 0 &{} \cos x(t) \end{array}\right] \end{aligned}$$

Here $f(t,x(t))$ and $\sigma (t,x(t))$ are satisfying conditions (A1) and (A2). For $x=(x_1,x_2)$ with the initial value $x_0$ and final point $x_T\in R^2$. For this system the controllability matrix is

$$\begin{aligned} \Gamma _s^t=\frac{1}{2}\left[ \begin{array}{r@{\quad }r} 1-\exp ^{-2(t-s)} &{} 0 \\ 0 &{} 1-\exp ^{-2(t-s)} \end{array}\right] \end{aligned}$$

If we take Euclidean norm then

$$\begin{aligned} ||A||=2,\;||B||=\sqrt{2},\;||\Gamma _s^t||=\frac{1-\exp ^{-2(t-s)}}{\sqrt{2}}>0\;\;\;\;\forall \;\; 0\le s<t; \end{aligned}$$

In [16] obtained sufficient condition for controllability of system $(4.4)$ in $L_\infty$ (supremum) norm as below

$$\begin{aligned} 4l_1L(Ml_1l_4+1)(T+1)T<1 \end{aligned}$$

In this paper sufficient condition for controllability of system $(4.4)$ is obtained in $L_2$ norm as below

$$\begin{aligned} \bigg (4l_1L(Ml_1l_4+1)(T+1)T^2\bigg )^{1/2}<1 \end{aligned}$$

Suppose $C=(4l_1L(Ml_1l_4+1))$ then sufficient condition for controllability for [16] will be $C(T+1)T<1$. Substitute $T\;(time)=5 \;unit$ then $C \in (0, 1/30)$. For all these values of $C$ system will be controllable. Now in this paper Left hand side for sufficient condition is $\sqrt{C(T+1)}\;T=C_1\;(let)$ substitute values of $C$ and $T$ we get $C_1\in (0,\sqrt{5})$. It means $C_1$ can be greater than $1$ for some cases. For these cases sufficient condition of Theorem $(3.1)$ is not satisfied but the system $(4.4)$ is completely controllable form [16] in $L_\infty$ norm.

But if we proceed in similar manner as above and suppose the sufficient conditions of this paper are satisfied for some values of $T<1$ then sufficient conditions of [16] paper may not be satisfied.