Controllability of Semilinear Stochastic Generalized Systems in Hilbert Spaces by GE-Evolution Operator Method

Abstract

Controllability is a basic problem in the study of stochastic generalized systems. Compared with ordinary stochastic systems, the structure of stochastic singular systems is more complex, and it is necessary to study the controllability of stochastic generalized systems in the context of different solutions. In this paper, the controllability of semilinear stochastic generalized systems was investigated by using a GE-evolution operator for integral and impulsive solutions in Hilbert spaces. Some sufficient and necessary conditions were obtained. Firstly, the existence and uniqueness of the integral solution of semilinear stochastic generalized systems were discussed using the GE-evolution operator theory and Banach fixed point theorem. The existence and uniqueness theorem of the integral solution was obtained. Secondly, the approximate controllability of semilinear stochastic generalized systems was studied in the case of the integral solution. Thirdly, the existence and uniqueness of the impulsive solution of semilinear stochastic generalized systems were considered, and some sufficient conditions were provided. Fourthly, the approximate controllability of semilinear stochastic generalized systems was studied for the impulsive solution. At last, the exact controllability of linear stochastic systems was studied in the case of the impulsive solution, with some necessary and sufficient conditions given. The obtained results have important theoretical and practical value for the study of controllability of semilinear stochastic generalized systems.

1. Introduction

In recent years, there has been an increasing interest in the controllability problem for semilinear stochastic systems (e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and their references). However, these studies are limited to ordinary semilinear stochastic systems (i.e., the coefficient operator of the differential term of the state is an identity operator), and has not yet involved semilinear stochastic generalized systems (i.e., the coefficient operator of the differential term of the state is a bounded linear operator). Semilinear stochastic generalized systems are inherent to many application fields, including input-output economics, the evolution of the free surface of seepage liquid, the stochastic generalized heat equation, stochastic models of biological, chemical and physical systems, etc. (e.g., [26,27,28,29,30,31,32,33,34,35,36]). They are essentially different from ordinary semilinear stochastic systems (e.g., [30,31]). It is necessary to study the controllability of such systems. In view of this, ref. [37] investigated the approximate controllability of two classes of semilinear stochastic generalized systems in terms of a mild solution in Hilbert space. However, the conditions for studying the approximate controllability of the semilinear stochastic generalized systems in terms of a mild solution were strong. Therefore, in this paper, we consider the approximate controllability of stochastic generalized systems in an integral solution. Compared with the mild solution, the conditions required for an integral solution are much looser. In addition, under certain conditions, stochastic generalized systems will have an impulsive solution. It is necessary to discuss the controllability of semilinear stochastic generalized systems in impulsive solutions. Therefore, in this paper, we also discuss the controllability of the stochastic generalized system in the impulsive solution. It should be emphasized that the controllability of semilinear stochastic generalized systems is studied using the GE-evolution operator method in this paper. In previous work, the controllability of semilinear stochastic systems was mainly studied by using the evolution operator method. The GE-evolution operator is a generalization of the evolution operator; generally speaking, they are completely different.

The organization of this paper is as follows:

In Section 2, we discuss the integral solution of the following semilinear stochastic generalized system, using the GE-evolution operator and Banach fixed point theorem:

$$Edx\left(t\right)=[A(t\left)x\right(t)+B(t\left)u\right(t)]dt+\alpha (t,x(t),u(t\left)\right)dt$$

$$+\beta (t,x\left(t\right),u\left(t\right))dw\left(t\right),x\left(0\right)=x_0,t\in [0,a],$$

(1)

where $x\left(t\right)$ is the state vector valued in Hilbert space H; $u\left(t\right)$ is the control vector valued in Hilbert space U; $w\left(t\right)$ is the one-dimensional standard Wiener process; $x_0$ is a given random variable; E is a bounded linear operator; $A\left(t\right):\mathrm{dom}A\left(t\right)\subseteq H\to H$ is a linear operator; and the conditions of $B\left(t\right),\alpha$ and $\beta$ are given in (H₃) and (H₄) of Section 2.1, respectively. In Section 3, we investigate the approximate controllability of semilinear stochastic generalized system (1) in an integral solution, using a GE-evolution operator. In Section 4, we discuss the impulsive solution of the following semilinear stochastic generalized system:

$$Edx\left(t\right)=[A(t\left)x\right(t)+B(t\left)u\right(t)]dt+\alpha (t,x(t),u(t\left)\right)dt$$

$$+\beta (t,x\left(t\right),u\left(t\right))dw\left(t\right),x\left(0\right)=x_0,t\in [0,a],$$

(2)

$$Ndy\left(t\right)=[y\left(t\right)+N_{1}u\left(t\right)]dt,y\left(0\right)=y_0,t\in [0,a],$$

(3)

where N is a nilpotent operator with order n; $y\left(t\right)$ is the state vector valued in Hilbert space Y; and $N_1$ is a bounded linear operator.

In Section 5, we investigate the approximate controllability of the following semilinear stochastic generalized system in an impulsive solution:

$$Edx\left(t\right)=[A(t\left)x\right(t)+B(t\left)u\right(t)]dt+\alpha (t,x(t),u(t\left)\right)dt$$

$$+\beta (t,x\left(t\right),u\left(t\right))dw\left(t\right),x\left(0\right)=x_0,t\in [0,a],$$

(4)

$$0=y\left(t\right)+N_1\left(t\right)u\left(t\right),y\left(0\right)=y_0,t\in [0,a].$$

(5)

In Section 6, we deal with the exact controllability of the following stochastic linear generalized system in an impulsive solution:

$$Edx\left(t\right)=[A\left(t\right)x\left(t\right)+B\left(t\right)u\left(t\right)]dt+A_1\left(t\right)x\left(t\right)dw\left(t\right),x\left(0\right)=x_0,t\in [0,a],$$

(6)

$$0=y\left(t\right)+N_1\left(t\right)u\left(t\right),y\left(0\right)=y_0,t\in [0,a].$$

(7)

Finally, our conclusions are given in Section 7.

Notations.$(\Omega ,F,\left\{F_t\right\},P)$ is a complete probability space, with filtration $\left\{F_t\right\}$ satisfying the usual condition (i.e., the filtration contains all $P-$null sets and is right continuous); all processes are $\left\{F_t\right\}$-adapted; $w\left(t\right)$ is a standard one-dimensional Wiener process defined on $(\Omega ,F,\left\{F_t\right\},P)$; $\mathbb{E}$ denotes the mathematical expectation; $\parallel ·\parallel$ denotes the norm; $C([0,a],\Omega ,F_t,H)$ denotes the set of all functions $f:[0,a]\to H$, which is continuous on $[0,a]$, in the context of ${\parallel f(·)\parallel }_{C([0,a],\Omega ,F_t,H)}=\mathbb{E}{\max }_{t\in [0,a]}{\parallel f\left(t\right)\parallel }_H$; $L^2(\Omega ,F_t,P,H)$ denotes the set of all random variables $\eta \in H$, such that $\eta$ is $F_t-$measurable and ${\parallel \eta \parallel }_2={\left(E\right(\parallel \eta \parallel }_H^2{\left)\right)}^{1/2}<+\infty$; $L^2([0,a],\Omega ,H)$ denotes the set of all processes $x\left(t\right)\in H$, such that ${\parallel x\left(t\right)\parallel }_2<+\infty ,\forall t\in [0,a]$; and $L^2([0,a],\Omega ,F_t,H)$ denotes the set of all processes $x\left(t\right)\in L^2([0,a],\Omega ,H)$, such that $\mathbb{E}{\int }_0^a{\parallel x\left(t\right)\parallel }_H^{2}dt<+\infty$. Let A be a linear operator; $\mathrm{dom}\left(A\right),\ker \left(A\right)$ and $\mathrm{ran}\left(A\right)$ denote its domain, kernel and range, respectively. $B(U,H)$ denotes the set of all bounded linear operators from U to H; $P([0,\tau ],B(U,H\left)\right)=\left\{C\right(·)\in B(U,H):C(·)z$ is continuous for every $z\in U$ and ${\sup }_{0\le t\le \tau }{\parallel C\left(t\right)\parallel }_{B(U,H)}<+\infty \};$ and I denotes the identical operator. $\mathbb{C}$ and $\mathbb{R}$ denote the sets of all complex and real numbers, respectively. Other mathematical symbols involved in this paper will be properly explained in the discussion.

2. Integral Solution to Semilinear Stochastic Generalized System (1)

In this section, we discuss the integral solution to semilinear stochastic generalized system (1) by using the GE-evolution operator and Banach fixed point theorem. First of all, we recall the concept of the GE-evolution operator and give some required assumptions below.

2.1. GE-Evolution Operator and Some Assumptions

Definition 1

([37,38]). Let $\left\{V\right(t,s):0\le s\le t\}$ be a two-parameter family of bounded linear operators in Hilbert space H, and let E be a bounded linear operator. If $V(s,s)=V_0$ is a definite operator independent of s and

$$V(t,s)=V(t,r)EV(r,s),0\le s\le r\le t,$$

then $V(t,s)$ is said to be a GE-evolution operator induced by E.

If the GE-evolution operator $V(t,s)$ satisfies that $V(t,·)$ is strongly continuous on $[0,t]$ and $V(·,s)$ is strongly continuous on $[s,a]$, then it is said to be strongly continuous on $[0,a]$.

If $V(t,s)$ is a compact operator for every $t>s\ge 0$, then $V(t,s)$ is said to be a compact GE-evolution operator induced by E, or a compact GE-evolution operator for short.

If there are $b\ge 1$ and $\omega >0$ so that

$$\parallel V(t,s)\parallel \le b{e}^{\omega (t-s)},t\ge s\ge 0,$$

then GE-evolution operator $V(t,s)$ is said to be exponentially bounded.

Let $V(t,s)$ be a strongly continuous and exponentially bounded GE-evolution operator, induced by E. If

$$A\left(t\right):\mathrm{dom}A\left(t\right)\subseteq H\to H$$

is a linear operator and

$$A\left(t\right)x=\underset{h\to 0^{+}}{lim}\frac{EV(h+t,t)E-EV(t,t)E}{h}x$$

for every $x\in D\left(t\right)$, where

$$D\left(t\right)=\{x:x\in H,V(t,t)Ex=x,$$

$$\exists \underset{h\to 0^{+}}{lim}\frac{EV(h+t,t)E-EV(t,t)E}{h}x\},$$

then $A\left(t\right)$ is said to be a generator of the GE-evolution operator $V(t,s)$ induced by E.

Below, it is generally assumed that

$$D\left(t\right)=D=\{x:x\in \mathrm{dom}A(t):A(t)x\in \mathrm{ran}E\}$$

is independent of t, $A\left(t\right)$ is the generator of GE-evolution operator $V(t,s)$ induced by E and $H_1=\overline{D}$, ${\left(V(t,s)E\right)|}_{H_1}$ is unique, where $\left(V(t,s)E\right){|}_{H_1}$ denotes the limitation of $V(t,s)E$ on $H_1$.

Now, we investigate the integral solution to semilinear generalized system (1). We need the following hypotheses:

(H₁) There are $m>0,\omega \in \mathbb{R}$, so that

$$(\omega ,+\infty )\subset \rho (E,A\left(t\right))=\{\lambda \in \mathbb{C}:\exists {(\lambda E-A\left(t\right))}^{-1}E,E{(\lambda E-A\left(t\right))}^{-1},$$

$${\parallel (\lambda E-A\left(t\right))}^{-1}{E\parallel =\parallel E(\lambda E-A\left(t\right))}^{-1}\parallel \le \frac{m}{\lambda -\omega },\lambda >\omega$$

and $E\lambda {(\lambda E-A\left(t\right))}^{-1}$ strongly converges on $\overline{\mathrm{ran}E}$ as $\lambda \to +\infty$, and $E{(\lambda E-A\left(t\right))}^{-1}$ is continuous to t in the uniform operator topology on $[0,a]$.

Obviously, $D=\mathrm{ran}\left({(\lambda E-A\left(t\right))}^{-1}E\right)$; D is independent of $\lambda$.

(H₂) $A\left(t\right)$ is a generator of a strongly continuous GE-evolution operator $V(t,s)$ induced by E, and there is $c_V>0$, so that

$$\parallel V(t,s)\parallel \le c_V,0\le s\le t\le a.$$

(H₃) $B\left(t\right)\in P([0,a],B(U,H\left)\right),$

$$\parallel B\left(t\right)\parallel \le c_B,t\in [0,a],$$

where $c_B$ is a positive constant;

$$\mathrm{ran}B\left(t\right)\subseteq \overline{\mathrm{ran}E}.$$

(H₄) The nonlinear operators $\alpha ,\beta$ are measurable, $\mathrm{ran}\alpha ,\mathrm{ran}\beta \subseteq \overline{\mathrm{ran}E}$ and satisfies that for $t\in [0,a]$, there is $a_m>0$, so that

$$\parallel \alpha (t,x_2,u_2)-\alpha (t,y_1,u_1){\parallel }^2\le c_l(\parallel x_2-x_1{\parallel }^2+\parallel u_2-u_1{\parallel }^2),$$

(8)

$${\parallel \alpha (t,x,u)\parallel }^2\le c_l{(1+\parallel x\parallel }^2+{\parallel u\parallel }^2).$$

(9)

2.2. Integral Solution

Definition 2.

A function $x:[0,a]\to H_1$ is said to be an integral solution of the semilinear stochastic generalized system (1) if the following conditions hold:

(i) $x\in C([0,a],\Omega ,F_t,H_1)$;

(ii) For $u\in L^2([0,a],\Omega ,F_t,U)$, $x\left(t\right)$ satisfies

$$x\left(t\right)=V(t,0)E{x}_0+\underset{\lambda \to +\infty }{lim}[{\int }_0^{a}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}B\left(s\right)u\left(s\right)ds$$

$$+{\int }_0^{a}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha (s,x\left(s\right),u\left(s\right))ds$$

$$+{\int }_0^{a}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\beta (s,x\left(s\right),u\left(s\right))dw\left(s\right)].$$

(10)

Theorem 1.

Under the hypotheses (H₁)–(H₄), the semilinear stochastic generalized system (1) has a unique integral solution.

Proof. 

As ${\parallel \lambda E(\lambda E-A\left(t\right))}^{-1}\parallel \le \frac{m}{\lambda -\omega }$, therefore

$$\underset{\lambda \to +\infty }{lim}\parallel E{(\lambda E-A\left(t\right))}^{-1}\parallel \le m.$$

(11)

Let O be the following operator:

$$O\left(x\right)\left(t\right)=V(t,0)E{x}_0+\underset{\lambda \to +\infty }{lim}[{\int }_0^{a}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}B\left(s\right)u\left(s\right)ds$$

$$+{\int }_0^{a}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha (s,x\left(s\right),u\left(s\right))ds$$

$$+{\int }_0^{a}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\beta (s,x\left(s\right),u\left(s\right))dw\left(s\right)]$$

$$=V(t,0)E{x}_0+O_1\left(x\right)\left(t\right)+O_2\left(x\right)\left(t\right)+O_3\left(x\right)\left(t\right),$$

$$t\in [0,a],u\in L^2([0,a],\Omega ,F_t,U).$$

We show that O maps $C([0,a],\Omega ,F_t,H_1)$ onto $C([0,a],\Omega ,F_t,H_1).$

As

$$\parallel O_1{\left(x\right)\parallel }^2\le c_V^2{m}^2{c}_B^{2}a{\int }_0^a{\parallel u\left(s\right)\parallel }^{2}dds;$$

$$\parallel O_2{\left(x\right)\parallel }^2\le c_V^2{m}^2\mathbb{E}({\int }_0^a\parallel \alpha {(s,x\left(s\right),u\left(s\right)\parallel ds)}^2$$

$$\le c_V^2{m}^{2}a\mathbb{E}{\int }_0^a{\parallel \alpha (s,x\left(s\right),u\left(s\right)\parallel }^{2}ds$$

$$\le c_V^2{m}^2{c}_{l}a\mathbb{E}{\int }_0^a{(1+\parallel x\left(s\right)\parallel }^2+{\parallel u\left(s\right)\parallel }^2)ds$$

$$\le c_V^2{m}^2{c}_l{a(a+a\parallel x\parallel }^2+\mathbb{E}{\int }_0^a{\parallel u\left(s\right)\parallel }^{2}ds),$$

$O_1$ and $O_2$ map $C([0,a],\Omega ,F_t,H_1)$ onto $C([0,a],\Omega ,F_t,H_1)$.

To show the same property for $O_3$, we remark that by [39] (Th. 4.36), we can obtain

$$\parallel O_3{\left(x\right)\parallel }^2\le c_V^2{m}^2{\int }_0^a{\mathbb{E}\parallel \beta (s,x\left(s\right),u\left(s\right)\parallel }^{2}ds$$

$$\le c_V^2{m}^2{c}_l{\int }_0^a{(1+\mathbb{E}\parallel x\left(s\right)\parallel }^2+{\mathbb{E}\parallel u\left(s\right)\parallel }^2)ds$$

$$\le c_V^2{m}^2{c}_l{(a+a\parallel x\parallel }^2+{\int }_0^a{\mathbb{E}\parallel u\left(s\right)\parallel }^{2}ds).$$

In the following, let $x_1$ and $x_2$ be arbitrary functions from $C([0,a],\Omega ,F_t,H_1)$, then

$$\parallel O\left(x_2\right)-O\left(x_1\right)\parallel =\parallel O_2\left(x_2\right)-O_2\left(x_1\right)\parallel +\parallel O_3\left(x_2\right)-O_3\left(x_1\right)\parallel$$

and

$$\parallel O_2\left(x_2\right)-O_2\left(x_1\right){\parallel }^2\le \underset{\lambda \to +\infty }{lim}\underset{t\in [0,a]}{sup}\mathbb{E}[\parallel {\int }_0^{t}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}·$$

$$(\alpha (s,x_2\left(s\right),u\left(s\right))-\alpha (s,x_1\left(s\right),u\left(s\right)))ds{\parallel }^2]$$

$$\le c_V^2{m}^2\underset{t\in [0,a]}{sup}\mathbb{E}[\parallel {\int }_0^t(\alpha (s,x_2\left(s\right),u\left(s\right))-\alpha (s,x_1\left(s\right),u\left(s\right)))ds{\parallel ]}^2$$

$$\le c_V^2{m}^{2}a\mathbb{E}{\int }_0^a{\parallel \alpha (s,x_2\left(s\right),u\left(s\right))-\alpha (s,x_1\left(s\right),u\left(s\right))\parallel }^{2}ds$$

$$\le c_V^2{m}^2{c}_{l}a\mathbb{E}{\int }_0^a{\parallel x_2\left(s\right)-x_1\left(s\right)\parallel }^{2}ds$$

$$\le c_V^2{m}^2{c}_l{a}^2{\parallel x_2-x_1\parallel }^2.$$

In a similar way, from [39] (Th. 4.36), we have

$$\parallel O_2\left(x_2\right)-O_2\left(x_1\right){\parallel }^2\le c_V^2{m}^2\underset{t\in [0,a]}{sup}\mathbb{E}{\int }_0^a{\parallel \beta (s,x_2\left(s\right),u\left(s\right))-\beta (s,x_1\left(s\right),u\left(s\right))\parallel }^{2}ds$$

$$\le c_V^2{m}^2{c}_l{\int }_0^a{\parallel x_2\left(s\right)-x_1\left(s\right)\parallel }^{2}ds$$

$$\le c_V^2{m}^2{c}_{l}a{\parallel x_2-x_1\parallel }^2$$

for all $x_1,x_2\in C([0,a],\Omega ,F_t,H_1)$. Therefore, if

$$2a{c}_V^2{m}^2{c}_l(a+1)<1,$$

(12)

then the operator O has fixed point $x\in C([0,a],\Omega ,F_t,H_1).$ This implies that $x\left(t\right)$ is an integral solution of semilinear stochastic generalized system (1). The extra condition (12) on a can be easily removed by considering the equation on $[0,a_1],[a_1,2a_1],\cdots$, with $a_1$ satisfying (12). □

3. Approximate Controllability of Semilinear Stochastic Generalized System (1)

In this section, we first consider the approximate controllability of semilinear stochastic generalized system (1) in an integral solution by using a GE-evolution operator, and then an applicable example is given to illustrate the effectiveness of the theoretical results.

3.1. Approximate Controllability of System (1)

Let $x(a,x_0,u)=x\left(a\right)$, which is given by (10). The set

$$R(a,x_0,u)=\{x(a,x_0,u):u\in L^2([0,a],\Omega ,F_t,U)\}$$

is called the reachable set of semilinear stochastic generalized system (1).

Definition 3.

Semilinear stochastic generalized system (1) is called to be approximately controllable on $[0,a]$ in an integral solution if

$$\overline{R(a,x_0,u)}=L^2([0,a],\Omega ,F_t,H_1).$$

In order to discuss the approximate controllability of semilinear stochastic generalized system (1), we need the following preparations:

The corresponding linear generalized system of a semilinear stochastic generalized system is defined as

$$Edx\left(t\right)=[A\left(t\right)x\left(t\right)+B\left(t\right)u\left(t\right)]dt,x\left(0\right)=x_0,t\in [0,a],$$

(13)

and by Theorem 1, generalized system (13) admits a unique integral solution given by

$$x\left(t\right)=V(t,0)E{x}_0+\underset{\lambda \to +\infty }{lim}[{\int }_0^{a}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}B\left(s\right)u\left(s\right)ds,t\in [0,a].$$

$\left(H_5\right)$$\mu {(\mu I+Q_t^a)}^{-1}\to 0$, as $\mu \to 0^{+}$ in the strong operator topology, where $Q_t^a={lim}_{\lambda \to +\infty }{\int }_0^{a}V(a,s)E\lambda {(\lambda E-A\left(t\right))}^{-1}B\left(s\right)B^{*}\left(s\right)\lambda {(\lambda E^{*}-A^{*}\left(s\right))}^{-1}{E}^{*}{V}^{*}(a,s)ds.$

Similar to the proof of [40] (Th. 3.2), we have Proposition 1.

Proposition 1.

Under the assumptions (H₁)–(H₃), linear stochastic generalized system (13) is approximately controllable on $[0,a]$ in an integral solution, if and only if (H₅) holds.

Proposition 2

([3]). For any $x_a\in L^2(\Omega ,F_a,P,H_1)$, there exists $\psi \in L^2([0,a],\Omega ,F_t,H_1)$, such that

$$x_a=\mathbb{E}{x}_a+{\int }_0^a\psi \left(s\right)dw\left(s\right).$$

According to Proposition 2, for any $\mu >0,x_a\in L^2(\Omega ,F_a,P,H_1)$, we define the input function as

$$u_{\mu }(t,x)=[\underset{\lambda \to +\infty }{lim}{B}^{*}\left(t\right)\lambda {(\lambda E^{*}-A^{*}\left(t\right))}^{-1}{E}^{*}{V}^{*}(a,t)]{(\mu I+Q_0^a)}^{-1}·$$

$$(\mathbb{E}{x}_a-V(a,0)E{x}_0)$$

$$+\underset{\lambda \to +\infty }{lim}{B}^{*}\left(t\right)\lambda {(\lambda E^{*}-A^{*}\left(t\right))}^{-1}{E}^{*}{V}^{*}(a,t){\int }_0^t{(\mu I+Q_s^a)}^{-1}\psi \left(s\right)dw\left(s\right)$$

$$-\underset{\lambda \to +\infty }{lim}[B^{*}\left(t\right)\lambda {(\lambda E^{*}-A^{*}\left(t\right))}^{-1}{E}^{*}{V}^{*}(a,t){\int }_0^t{(\mu I+Q_s^a)}^{-1}V(a,s)·$$

$$E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha (s,x\left(s\right),u\left(s\right))ds]$$

$$-\underset{\lambda \to +\infty }{lim}[B^{*}\left(t\right)\lambda {(\lambda E^{*}-A^{*}\left(t\right))}^{-1}{E}^{*}{V}^{*}(a,t){\int }_0^t{(\mu I+Q_s^a)}^{-1}V(a,s)·$$

$$E\lambda {(\lambda E-A\left(s\right))}^{-1}\beta (s,x\left(s\right),u\left(s\right))dw\left(s\right)].$$

(14)

Proposition 3.

There exist $p_1,p_2$ and $p_3$, such that for all $x_1,x_2\in C([0,a],\Omega ,F_t,H_1)$, we have

$$\parallel u_{\mu }(t,x_2)-u_{\mu }(t,x_1){\parallel }^2\le {\mu }^{-2}{p}_1{\int }_0^t\mathbb{E}{\parallel x_2\left(s\right)-x_1\left(s\right)\parallel }^{2}ds,$$

(15)

$$\parallel u_{\mu }(t,x_1){\parallel }^2\le {\mu }^{-2}{p}_2+{\mu }^{-2}{p}_3{\int }_0^t(1+\mathbb{E}\parallel x_1\left(s\right){\parallel }^2+\mathbb{E}\parallel u\left(s\right){\parallel }^2)ds.$$

(16)

Proof. 

Let $x_1,x_2\in C([0,a],\Omega ,F_t,H_1)$. According to (H₁)–(H₅), we have

$$\parallel u_{\mu }(t,x_2)-u_{\mu }(t,x_1){\parallel }^2\le \mathbb{E}[\parallel \underset{\lambda \to +\infty }{lim}{B}^{*}\left(t\right)\lambda {(\lambda E^{*}-A^{*}\left(t\right))}^{-1}{E}^{*}{V}^{*}(a,t)·$$

$${\int }_0^t{(\mu I+Q_s^a)}^{-1}V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}·$$

$$(\alpha (s,x_2\left(s\right),u\left(s\right))-\alpha (s,x_1\left(s\right),u\left(s\right)))ds\parallel$$

$$+\parallel \underset{\lambda \to +\infty }{lim}{B}^{*}\left(t\right)\lambda {(\lambda E^{*}-A^{*}\left(t\right))}^{-1}{E}^{*}{V}^{*}(a,t)·$$

$${\int }_0^t{(\mu I+Q_s^a)}^{-1}V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}·$$

$$(\beta (s,x_2\left(s\right),u\left(s\right))-\beta (s,x_1\left(s\right),u\left(s\right)))dw\left(s\right){\parallel ]}^2$$

$$\le \mathbb{E}[c_B{m}^2{c}_V^2{\mu }^{-1}{\int }_0^t\parallel \alpha (s,x_2\left(s\right),u\left(s\right))-\alpha (s,x_1\left(s\right),u\left(s\right))\parallel ds$$

$$+c_{B}m{c}_V\underset{\lambda \to +\infty }{lim}\parallel {\int }_0^t{(\mu I+Q_s^a)}^{-1}V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}·$$

$${(\beta (s,x_2\left(s\right),u\left(s\right))-\beta (s,x_1\left(s\right),u\left(s\right))dw\left(s\right)\parallel ]}^2$$

$$\le 2c_B^2{m}^4{c}_V^4{\mu }^{-2}a\mathbb{E}{\int }_0^t{\parallel \alpha (s,x_2\left(s\right),u\left(s\right))-\alpha (s,x_1\left(s\right),u\left(s\right))\parallel }^{2}ds$$

$$+2c_B^2{m}^4{c}_V^4{\mu }^{-2}\mathbb{E}{\int }_0^t{\parallel \beta (s,x_2\left(s\right),u\left(s\right))-\beta (s,x_1\left(s\right),u\left(s\right))\parallel }^{2}ds$$

$$\le 2c_B^2{m}^4{c}_V^4{\mu }^{-2}a{c}_l\mathbb{E}{\int }_0^t{\parallel x_2\left(s\right)-x_1\left(s\right)\parallel }^{2}ds$$

$$+2c_B^2{m}^4{c}_V^4{\mu }^{-2}{c}_l\mathbb{E}\parallel {\int }_0^t{\parallel x_2\left(s\right)-x_1\left(s\right)\parallel }^{2}ds$$

$$=2{\mu }^{-2}{c}_B^2{m}^4{c}_V^4{c}_l(a+1){\int }_0^t\mathbb{E}{\parallel x_2\left(s\right)-x_1\left(s\right)\parallel }^{2}ds,$$

where $\parallel \mu {(\mu I+Q_s^a)}^{-1}\parallel <1$ is used. Let $p_1=2c_B^2{m}^4{c}_V^4{c}_l(a+1)$. Then (15) holds. Since

$$\parallel u_{\mu }(t,x_1){\parallel }^2\le 4c_B^2{m}^2{c}_V^2{\mu }^{-2}(\parallel \mathbb{E}{x}_a\parallel +c_V\parallel E\parallel \parallel x_0{\parallel )}^2$$

$$+4c_B^2{m}^2{c}_V^2{\mu }^{-2}{\int }_0^t\mathbb{E}{\parallel \psi \left(s\right)\parallel }^{2}ds$$

$$+4c_B^2{m}^4{c}_V^{4}a{\mu }^{-2}{\int }_0^t\mathbb{E}{\parallel \alpha (s,x_1\left(s\right),u\left(s\right))\parallel }^{2}ds$$

$$+4c_B^2{m}^4{c}_V^4{\mu }^{-2}{\int }_0^t\mathbb{E}{\parallel \beta (s,x_1\left(s\right),u\left(s\right))\parallel }^{2}ds$$

$$\le {\mu }^{-2}{p}_2+{\mu }^{-2}{p}_3{\int }_0^t(1+\mathbb{E}\parallel x_1{\left(s\right)\parallel }^2+{\mathbb{E}\parallel u\left(s\right)\parallel }^2)ds,$$

where

$$p_2=4c_B^2{m}^2{c}_V^2[\mathbb{E}(\parallel x_a{\parallel }^2+c_V\parallel E\parallel \parallel x_0{\parallel )}^2+{\int }_0^a{\mathbb{E}\parallel \psi \left(s\right)\parallel }^{2}ds],$$

$$p_3=4c_B^2{m}^4{c}_V^4(a+1)c_l,$$

we have that (16) is true. □

For every $\mu >0,$ we define the operator $P:C([0,a],\Omega ,F_t,H_1)\to C([0,a],\Omega ,F_t,H_1)$ by

$$P\left(x\right)\left(t\right)=V(t,0)E{x}_0+\underset{\lambda \to +\infty }{lim}[{\int }_0^{a}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}B\left(s\right)u_{\mu }(s,x\left(s\right))ds$$

$$+{\int }_0^{a}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha (s,x\left(s\right),u_{\mu }(s,x\left(s\right))ds$$

$$+{\int }_0^{a}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\beta (s,x\left(s\right),u_{\mu }(s,x\left(s\right)))dw\left(s\right)].$$

(17)

From Theorem 1 and Proposition 3, we can obtain the following propositions.

Proposition 4.

Assume that $\left(H_1\right)-\left(H_5\right)$ hold. If

$$3a{c}_V^2{m}^2[c_B^2{p}_1{\mu }^{-2}{a}^2+c_l(a+{\mu }^{-2}{p}_1{a}^2)+c_l(1+{\mu }^{-2}{p}_{1}a)]<1,$$

(18)

then the operator P has a fixed point $x_{\mu }\left(t\right)\in C([0,a],\Omega ,F_t,H_1)$.

Proposition 5.

For any $x_a\in L^2(\Omega ,F_a,P,H_1)$, the control $u_{\mu }(t,x)$ in (14) transfers semilinear stochastic generalized system (1) to some neighborhood of $x_a$ at time a, and

$$x_{\mu }\left(a\right)=x_a-\mu {(\mu I+Q_0^a)}^{-1}(\mathbb{E}{x}_a-V(a,0)E{x}_0)$$

$$+\underset{\lambda \to +\infty }{lim}{\int }_0^a\mu {(\mu I+Q_s^a)}^{-1}V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))ds$$

$$+\underset{\lambda \to +\infty }{lim}{\int }_0^a\mu {(\mu I+Q_s^a)}^{-1}(V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}·$$

$$\beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))-\psi \left(s\right))dw\left(s\right).$$

(19)

Proof. 

By substituting (14) in (17), we obtain

$$x_{\mu }\left(t\right)=V(t,0)E{x}_0+\underset{\lambda \to +\infty }{lim}[{\int }_0^{t}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}B\left(s\right)·$$

$$(B^{*}\left(s\right)\lambda {(\lambda E^{*}-A^{*}\left(s\right))}^{-1}{E}^{*}{V}^{*}(a,s){(\mu I+Q_0^a)}^{-1}(\mathbb{E}{x}_a-V(a,0)E{x}_0)$$

$$+B^{*}\left(s\right)\lambda {(\lambda E^{*}-A^{*}\left(s\right))}^{-1}{E}^{*}{V}^{*}(a,s){\int }_0^s\mu {(\mu I+Q_r^a)}^{-1}\psi \left(r\right)dw\left(r\right)$$

$$-B^{*}\left(s\right)\lambda {(\lambda E^{*}-A^{*}\left(s\right))}^{-1}{E}^{*}{V}^{*}(a,s){\int }_0^s\mu {(\mu I+Q_r^a)}^{-1}V(a,r)·$$

$$E\lambda {(\lambda E-A\left(r\right))}^{-1}\alpha (r,x_{\mu }\left(r\right),u_{\mu }(r,x_{\mu }\left(r\right)))dr$$

$$-B^{*}\left(s\right)\lambda {(\lambda E^{*}-A^{*}\left(s\right))}^{-1}{E}^{*}{V}^{*}(a,s){\int }_0^s\mu {(\mu I+Q_r^a)}^{-1}V(a,r)·$$

$$E\lambda {(\lambda E-A\left(r\right))}^{-1}\beta (r,x_{\mu }\left(r\right),u_{\mu }(r,x_{\mu }\left(r\right)))dw\left(r\right))ds$$

$$+{\int }_0^{t}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))ds$$

$$+{\int }_0^{t}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))dw\left(s\right)]$$

$$=V(t,0)E{x}_0+\underset{\lambda \to +\infty }{lim}{\int }_0^{t}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))ds$$

$$+\underset{\lambda \to +\infty }{lim}{\int }_0^{t}V(t,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))dw\left(s\right)$$

$$+Q_0^t{E}^{*}{V}^{*}(a,t){(\mu I+Q_0^a)}^{-1}(\mathbb{E}{x}_a-V(a,0)E{x}_0)$$

$$-\underset{\lambda \to +\infty }{lim}{\int }_0^t{Q}_s^t{E}^{*}{V}^{*}(a,s){(\mu I+Q_s^a)}^{-1}V(a,s)·$$

$$E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))ds$$

$$-\underset{\lambda \to +\infty }{lim}{\int }_0^t{Q}_s^t{E}^{*}{V}^{*}(a,t){(\mu I+Q_s^a)}^{-1}·$$

$$(V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))-\psi \left(s\right))dw\left(s\right).$$

Let $t=a.$ Then

$$x_{\mu }\left(a\right)=V(a,0)E{x}_0+\underset{\lambda \to +\infty }{lim}{\int }_0^{a}V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))ds$$

$$+\underset{\lambda \to +\infty }{lim}{\int }_0^{a}V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))dw\left(s\right)$$

$$+(-\mu I+\mu I+Q_0^a){(\mu I+Q_0^a)}^{-1}(\mathbb{E}{x}_a-V(a,0)E{x}_0)$$

$$-\underset{\lambda \to +\infty }{lim}{\int }_0^t(-\mu I+\mu I+Q_s^a){(\mu I+Q_s^a)}^{-1}V(a,s)·$$

$$E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))ds$$

$$-\underset{\lambda \to +\infty }{lim}{\int }_0^t(-\mu I+\mu I+Q_s^a){(\mu I+Q_s^a)}^{-1}·$$

$$(V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))-\psi \left(s\right))dw\left(s\right)$$

$$=x_a-\mu {(\mu I+Q_0^a)}^{-1}(\mathbb{E}{x}_a-V(a,0)E{x}_0)$$

$$+\underset{\lambda \to +\infty }{lim}{\int }_0^a\mu {(\mu I+Q_s^a)}^{-1}V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))ds$$

$$+\underset{\lambda \to +\infty }{lim}{\int }_0^a\mu {(\mu I+Q_s^a)}^{-1}(V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}·$$

$$\beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))-\psi \left(s\right))dw\left(s\right).$$

This implies that (19) holds. □

Theorem 2.

Assume that (H₁)–(H₅) hold. If $\alpha ,\beta$ are uniformly bounded and $V(t,s)$ is compact, then semilinear stochastic generalized system (1) is approximately controllable on $[0,a]$.

Proof. 

According to the property of $\alpha$ and $\beta$, we have

$$\parallel \alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right))){\parallel }^2+{\parallel \beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))\parallel }^2<c_q.$$

Therefore, there exists a subsequence, still denoted by

$$\{\alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right))),\beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))\},$$

which converges weakly to $\left\{\alpha \right(s),\beta (s\left)\right\},$ in $\overline{\mathrm{ran}E}\times \overline{\mathrm{ran}E}$. On the other hand, $\mu {(\mu I+Q_s^a)}^{-1}\to 0$ strongly as $\mu \to 0$ and $\parallel \mu {(\mu I+Q_s^a)}^{-1}\parallel \le 1$, together with the Lebesgue-dominated convergence theorem. By (19), we have

$$\mathbb{E}\parallel x_{\mu }\left(a\right)-x_a{\parallel }^2\le \mathbb{E}[\parallel \mu {(\mu I+Q_0^a)}^{-1}(\mathbb{E}{x}_a-V(a,0)E{x}_0)\parallel$$

$$+\parallel \underset{\lambda \to +\infty }{lim}{\int }_0^a\mu {(\mu I+Q_s^a)}^{-1}V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}·$$

$$(\alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))-\alpha \left(s\right))ds\parallel$$

$$+\parallel \underset{\lambda \to +\infty }{lim}{\int }_0^a\mu {(\mu I+Q_s^a)}^{-1}V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha \left(s\right))ds\parallel$$

$$+\parallel \underset{\lambda \to +\infty }{lim}{\int }_0^a\mu {(\mu I+Q_s^a)}^{-1}(V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}·$$

$$(\beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))-\beta \left(s\right))dw\left(s\right)\parallel$$

$$+\parallel \underset{\lambda \to +\infty }{lim}{\int }_0^a\mu {(\mu I+Q_s^a)}^{-1}(V(a,s)E\lambda {(\lambda E-A\left(s\right))}^{-1}\beta \left(s\right)dw\left(s\right)\parallel$$

$$+\parallel {\int }_0^a\mu {(\mu I+Q_s^a)}^{-1}\psi \left(s\right))dw\left(s\right){\parallel ]}^2$$

$$\le 6\mathbb{E}\parallel \mu {(\mu I+Q_0^a)}^{-1}(\mathbb{E}{x}_a-V(a,0)E{x}_0){\parallel }^2$$

$$+6a\mathbb{E}{\int }_0^a\parallel \mu {(\mu I+Q_s^a)}^{-1}V(a,s)\underset{\lambda \to +\infty }{lim}E\lambda {(\lambda E-A\left(s\right))}^{-1}·$$

$$(\alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))-\alpha \left(s\right)){\parallel }^{2}ds$$

$$+6a\mathbb{E}{\int }_0^a\parallel \mu {(\mu I+Q_s^a)}^{-1}V(a,s)\underset{\lambda \to +\infty }{lim}E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha \left(s\right){)\parallel }^{2}ds$$

$$+6\mathbb{E}{\int }_0^a\parallel \mu {(\mu I+Q_s^a)}^{-1}(V(a,s)\underset{\lambda \to +\infty }{lim}E\lambda {(\lambda E-A\left(s\right))}^{-1}·$$

$$(\beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))-\beta \left(s\right)){\parallel }^{2}ds$$

$$+6\mathbb{E}{\int }_0^a\parallel \mu {(\mu I+Q_s^a)}^{-1}{(V(a,s)\underset{\lambda \to +\infty }{lim}E\lambda {(\lambda E-A\left(s\right))}^{-1}\beta \left(s\right)\parallel }^{2}ds$$

$$+6\mathbb{E}{\int }_0^a{\parallel \mu {(\mu I+Q_s^a)}^{-1}\psi \left(s\right)\parallel }^{2}ds$$

$$\le 6\mathbb{E}[\parallel \mu {(\mu I+Q_0^a)}^{-1}(\mathbb{E}{x}_a-V(a,0)E{x}_0){\parallel }^2$$

$$+6a\mathbb{E}{\int }_0^a\parallel V(a,s)\underset{\lambda \to +\infty }{lim}E\lambda {(\lambda E-A\left(s\right))}^{-1}·$$

$$(\alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))-\alpha \left(s\right)){\parallel }^{2}ds$$

$$+6a\mathbb{E}{\int }_0^a\parallel \mu {(\mu I+Q_s^a)}^{-1}V(a,s)\underset{\lambda \to +\infty }{lim}E\lambda {(\lambda E-A\left(s\right))}^{-1}\alpha \left(s\right){)\parallel }^{2}ds$$

$$+6\mathbb{E}{\int }_0^a\parallel (V(a,s)\underset{\lambda \to +\infty }{lim}E\lambda {(\lambda E-A\left(s\right))}^{-1}·$$

$$(\beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))-\beta \left(s\right)){\parallel }^{2}ds$$

$$+6\mathbb{E}{\int }_0^a\parallel \mu {(\mu I+Q_s^a)}^{-1}{(V(a,s)\underset{\lambda \to +\infty }{lim}E\lambda {(\lambda E-A\left(s\right))}^{-1}\beta \left(s\right)\parallel }^{2}ds$$

$$+6\mathbb{E}{\int }_0^a{\parallel \mu {(\mu I+Q_s^a)}^{-1}\psi \left(s\right)\parallel }^{2}ds\to 0$$

as $\mu \to 0^{+}$, which implies that semilinear stochastic generalized system (1) is approximately controllable on $[0,a]$. □

3.2. Application Example

Here, we give an example to illustrate the validity of Theorem 2.

Example 1.

Consider the semilinear stochastic generalized heat equation:

$$\left[\begin{array}{cc}{I}_1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}d{x}_1(t,\xi )\\ d{x}_2(t,\xi )\end{array}\right]=\left[\begin{array}{cc}\frac{{\partial }^2}{\partial {\xi }^2}& 0\\ 0& (1+t)I_2\end{array}\right]\left[\begin{array}{c}{x}_1(t,\xi )\\ x_2(t,\xi )\end{array}\right]dt+$$

$$\left[\begin{array}{c}(1+t^2)I_1\\ 0\end{array}\right]u(t,\xi )dt$$

$$+\left[\begin{array}{c}{\alpha }_1(t,x(t,\xi ),u(t,\xi ))\\ 0\end{array}\right]dt+\left[\begin{array}{c}{\beta }_1(t,x(t,\xi ),u(t,\xi ))\\ 0\end{array}\right]dw\left(t\right),$$

$$x_1(t,0)=x_1(t,\pi )=0,x_2(t,0)=x_2(t,\pi )=0,0\le t\le a,0<\xi <\pi ;$$

$$x_1(0,\xi )=x_{10}\left(\xi \right),x_2(0,\xi )=x_{20}\left(\xi \right),$$

(20)

where $x(t,\xi )=\left[\begin{array}{c}{x}_1(t,\xi )\\ x_2(t,\xi )\end{array}\right].$

Let $H_2$ be a Hilbert space, $H_1=L^2(0,\pi ),H=H_1\oplus H_2,E=\left[\begin{array}{cc}{I}_1& 0\\ 0& 0\end{array}\right]$

$$A\left(t\right)=\left[\begin{array}{cc}{A}_1& 0\\ 0& (1+t)I_2\end{array}\right],A_1=\frac{d^2}{d{\xi }^2}$$

with domain $\mathrm{dom}{A}_1=\{x_1\in H_1:x_1,\frac{d{x}_1}{d\xi }$ are absolutely continuous, $\frac{d^2{x}_1}{d{\xi }^2}\in H_1,x_1\left(0\right)=x_1\left(\pi \right)=0\},B\left(t\right)=\left[\begin{array}{c}(1+t^2)I_1\\ 0\end{array}\right],$

$$\alpha (t,x,u)=\left[\begin{array}{c}{\alpha }_1(t,x\left(t\right),u\left(t\right))\\ 0\end{array}\right],\beta (t,x,u)=\left[\begin{array}{c}{\beta }_1(t,x\left(t\right),u\left(t\right))\\ 0\end{array}\right].$$

Then, (20) can be written in the form of (1). According to [41], $A_1$ is a generator of compact $C_0-$semigroup $V_1\left(t\right)$. Thus, $A_1\left(t\right)$ is a generator of the compact GE-evolution operator $V(t,s)$, which is given by $V(t,s)=\left[\begin{array}{cc}{V}_1(t-s)& 0\\ 0& 0\end{array}\right].$ It is obvious that (H₁)–(H₃) hold. We take $\alpha ,\beta$ to satisfy (H₄). The linear generalized system corresponding to (20) is

$$\left[\begin{array}{cc}{I}_1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}d{x}_1(t,\xi )\\ d{x}_2(t,\xi )\end{array}\right]=\left[\begin{array}{cc}\frac{{\partial }^2}{\partial {\xi }^2}& 0\\ 0& (1+t)I_2\end{array}\right]\left[\begin{array}{c}{x}_1(t,\xi )\\ x_2(t,\xi )\end{array}\right]dt$$

$$+\left[\begin{array}{c}(1+t^2)I_1\\ 0\end{array}\right]u(t,\xi )dt,$$

$$x_1(t,0)=x_1(t,\pi )=0,x_2(t,0)=x_2(t,\pi )=0,0\le t\le a,0<\xi <\pi ;$$

$$x_1(0,\xi )=x_{10}\left(\xi \right),x_2(0,\xi )=x_{20}\left(\xi \right).$$

(21)

It is clear that linear generalized system (21) is approximately controllable on $[0,a]$. Therefore, (H₅) holds. Hence, semilinear stochastic generalized system (20) is approximately controllable on $[0,a]$ by Theorem 2.

4. Impulsive Solution of Semilinear Stochastic Generalized Systems (2) and (3)

In general, if there is a term containing $dw\left(t\right)$ in system (3), then the impulsive solution of system (3) does not exist. In view of this, in this section, we consider the impulsive solution of semilinear stochastic generalized systems (2) and (3) using a GE-evolution operator, where $dw\left(t\right)$ is not included in generalized system (3).

In order to discuss the impulsive solution of systems (2) and (3), we need the following hypotheses: Assume that (H₂)–(H₄) are true: $\overline{D}=H$.

(H₆): assume that $u\in C^{n-1}([0,+\infty ),U)$, and there exist constants $c_u>0$ and $\omega >0$, such that

$$\parallel u^{\left(k\right)}{\parallel }_U\le c_U{e}^{\omega t},k=0,1,\cdots ,n-1,$$

where $u^{\left(k\right)}$ denotes the kth derivative of u.

Definition 4.

A stochastic process $x\in C([0,a],\Omega ,F_t,H)$ is a mild solution of semilinear stochastic generalized system (2) if, for each $u\in L^2([0,a],\Omega ,F_t,U)$, it satisfies the following equation:

$$x\left(t\right)=V(t,0)E{x}_0+{\int }_0^{a}V(t,s)B\left(s\right)u\left(s\right)ds$$

$$+{\int }_0^{a}V(t,s)\alpha (s,x\left(s\right),u\left(s\right))ds$$

$$+{\int }_0^{a}V(t,s)\beta (s,x\left(s\right),u\left(s\right))dw\left(s\right).$$

(22)

Remark 1.

If $\overline{D}=H$, then we can obtain that the integral solution to semilinear stochastic generalized system (2) is the mild solution.

According to the GE-evolution operator theory and Banach fixed point theorem, similar to the proof of Theorem 1, we obtain Theorem 3.

Theorem 3.

Under the hypotheses (H₂)–(H₄), semilinear stochastic generalized system (2) has a unique mild solution.

According to [30], under (H₆), system (3) has a unique impulsive solution:

$$y\left(t\right)=-\sum _{k=1}^{n-1}{N}^k{\delta }^{(k-1)}\left(t\right)[y_0+\sum _{k=0}^{n-1}{N}^k{N}_1{u}^{\left(k\right)}\left(0\right)]-\sum _{k=0}^{n-1}{N}^k{N}_1{u}^{\left(k\right)}\left(t\right),$$

(23)

where $\delta \left(t\right)$ is the Dirac function and ${\delta }^{\left(k\right)}\left(t\right)$ is the kth derivative of $\delta \left(t\right)$; see [30] for the details.

Definition 5.

If $x\left(t\right)$ is the mild solution of system (2) and $y\left(t\right)$ is the impulsive solution of system (3), then $\left[\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right]$ is called the impulsive solution of systems (2) and (3).

From Theorem 3, formula (23) and Definition 5, we obtain Theorem 4.

Theorem 4.

Under the hypotheses (H₂)–(H₄) and (H₆), semilinear stochastic generalized systems (2) and (3) have a unique impulsive solution.

For semilinear stochastic generalized systems (4) and (5), we still call

$$\left[\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right]=\left[\begin{array}{c}x\left(t\right)\\ -N_1\left(t\right)u\left(t\right)\end{array}\right]$$

its impulsive solution, where $x\left(t\right)$ is the mild solution to semilinear stochastic generalized system (4), $N_1\left(t\right)\in P([0,\tau ],B(U,Y))$.

5. Approximate Controllability of Semilinear Stochastic Generalized Systems (4) and (5)

In general, if $N\ne 0$ in system (3), semilinear stochastic generalized systems (2) and (3) are not approximately controllable. Therefore, we investigate the approximate controllability of semilinear stochastic generalized systems (4) and (5) in terms of their impulsive solution, and give two examples to illustrate the validity of the theoretical results.

5.1. Approximate Controllability

Definition 6.

Semilinear stochastic generalized systems (4) and (5) are deemed approximately controllable on $[0,a]$ if, for any $\left[\begin{array}{c}{x}_a\\ y_a\end{array}\right]\in L^2(\Omega ,F_a,P,H\oplus Y)$, any initial $\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]\in L^2(\Omega ,F_0,P,H\oplus Y)$ and any $ϵ>0,$ there exists a $u\in L^2([0,a],\Omega ,P,U)$, such that the impulsive solution $\left[\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right]$ to systems (4) and (5) satisfies

$$(\mathbb{E}\parallel \left[\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right]-\left[\begin{array}{c}{x}_a\\ y_a\end{array}\right]{{\parallel }^2)}^{1/2}<ϵ.$$

In order to discuss the approximate controllability of semilinear stochastic generalized systems (4) and (5), we need the following preparations:

Let

$$Q_t^a={\int }_t^a{s}^2{(s-a)}^{2}V(a,s)B\left(s\right)B^{*}\left(s\right)V^{*}(a,s)ds.$$

(H₇): $\mu {(\mu I+Q_t^a)}^{-1}\to 0$, as $\mu \to 0^{+}$ in the strong operator topology.

The corresponding linear generalized system of semilinear stochastic generalized system (4) is defined as

$$Edx\left(t\right)=[A\left(t\right)x\left(t\right)+B\left(t\right)u\left(t\right)]dt,x\left(0\right)=x_0,t\in [0,a],$$

(24)

and admits a unique mild solution given by

$$x\left(t\right)=V(t,0)E{x}_0+{\int }_0^{a}V(t,s)B\left(s\right)u\left(s\right)ds,t\in [0,a].$$

Similar to the proof of [40] (Th. 3.2), we have the following proposition:

Proposition 6.

Under the assumptions (H₂)–(H₄), linear generalized system (24) is approximately controllable on $[0,a]$ in the mild solution, if and only if (H₇) holds.

Similar to the proof of [30] (Th. 3), we can obtain the following theorem:

Theorem 5.

System (5) is approximately controllable on $[0,a]$, up to $L^2(\Omega ,F_a,P,Y)$, if and only if

$$N_1\left(a\right):L^2(\Omega ,F_a,P,U)\to L^2(\Omega ,F_a,P,Y)$$

satisfies

$$\overline{\mathrm{ran}{N}_1\left(a\right)}=L^2(\Omega ,F_a,P,Y).$$

(25)

Proposition 7

([1]). For any $x_a\in L^2(\Omega ,F_a,P,H)$, there exists $\psi \in L^2([0,a],\Omega ,F_t,H)$, such that

$$\mathbb{E}\left(x_a\right|F_t)=\mathbb{E}{x}_a+{\int }_0^a\psi \left(s\right)dw\left(s\right).$$

From Proposition 7, for any $\mu >0,x_a\in L^2(\Omega ,F_a,P,H)$, we define the input function

$$u_{\mu }(t,x)=t^2{(t-a)}^2{B}^{*}\left(t\right)V^{*}(a,t)]{(\mu I+Q_0^a)}^{-1}(\mathbb{E}{x}_a^{\prime }-V(a,0)E{x}_0)$$

$$+t^2{(t-a)}^2{B}^{*}\left(t\right)V^{*}(a,t){\int }_0^t{(\mu I+Q_s^a)}^{-1}\psi \left(s\right)dw\left(s\right)$$

$$-t^2{(t-a)}^2{B}^{*}\left(t\right)V^{*}(a,t){\int }_0^t{(\mu I+Q_s^a)}^{-1}V(a,s)\alpha (s,x\left(s\right),u\left(s\right))ds$$

$$-t^2{(t-a)}^2{B}^{*}\left(t\right)V^{*}(a,t){\int }_0^t{(\mu I+Q_s^a)}^{-1}V(a,s)\beta (s,x\left(s\right),u\left(s\right))dw\left(s\right)$$

$$+\mathbb{E}\left({\beta }_a\right|F_t),$$

(26)

where ${\beta }_a\in L^2(\Omega ,F_a,P,U)$,

$$x_a^{\prime }=x_a-{\int }_0^{a}V(a,s)N_1\left(s\right)\mathbb{E}\left({\beta }_a\right|F_s)ds.$$

Similar to the proofs of Propositions 3–5, we can obtain the following results:

Proposition 8.

There exist $p_1,p_2,p_3$ and $p_4$, such that, for all $x_1,x_2\in C([0,a],\Omega ,F_t,H)$, we have

$$\parallel u_{\mu }(t,x_2)-u_{\mu }(t,x_1){\parallel }^2\le {\mu }^{-2}{p}_1{\int }_0^t\mathbb{E}{\parallel x_2\left(s\right)-x_1\left(s\right)\parallel }^{2}ds,$$

(27)

$$\parallel u_{\mu }(t,x_1){\parallel }^2\le {\mu }^{-2}{p}_2+{\mu }^{-2}{p}_3{\int }_0^t(1+\mathbb{E}\parallel x_1\left(s\right){\parallel }^2+\mathbb{E}{\parallel u\left(s\right)\parallel }^{2}ds+c_4.$$

(28)

Proposition 9.

Assume that (H₂)–(H₄) and (H₇) hold. For every $\mu >0,$ let

$$P:C([0,a],\Omega ,F_t,H)\to C([0,a],\Omega ,F_t,H)$$

be defined by

$$P\left(x\right)\left(t\right)=V(t,0)E{x}_0+{\int }_0^{a}V(t,s)B\left(s\right)u_{\mu }(s,x\left(s\right))ds$$

$$+{\int }_0^{t}V(t,s)\alpha (s,x\left(s\right),u_{\mu }(s,x\left(s\right)))ds$$

$$+{\int }_0^{a}V(t,s)\beta (s,x\left(s\right),u_{\mu }(s,x\left(s\right)))dw\left(s\right).$$

(29)

If

$$3a{c}_V^2[c_l(a+1)+p_1{\mu }^{-2}(a{c}_B^2+c_l(a+1)]<1,$$

(30)

then the operator P has a fixed point $x_{\mu }\left(t\right)\in C([0,a],\Omega ,F_t,H)$.

Proposition 10.

For any $x_a\in L^2(\Omega ,F_a,P,H)$, the control $u_{\mu }(t,x)$ in (26) transfers the semilinear stochastic generalized system (29) to some neighborhood of $x_a$ at time a and

$$x_{\mu }\left(a\right)=x_a-\mu {(\mu I+Q_0^a)}^{-1}(\mathbb{E}{x}_a^{\prime }-V(a,0)E{x}_0)$$

$$+{\int }_0^a\mu {(\mu I+Q_s^a)}^{-1}V(a,s)\alpha (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))ds$$

$$+{\int }_0^a\mu {(\mu I+Q_s^a)}^{-1}(V(a,s)·$$

$$\beta (s,x_{\mu }\left(s\right),u_{\mu }(s,x_{\mu }\left(s\right)))-\psi \left(s\right))dw\left(s\right).$$

(31)

Similar to the proof of Theorem 2, we have Theorem 6, as follows:

Theorem 6.

Assume that (H₂)–(H₄) and (H₇) hold. If $\alpha ,\beta$ are uniformly bounded, $V(t,s)$ is compact and system (5) is approximately controllable, then semilinear stochastic generalized systems (4)–(5) are approximately controllable on $[0,a]$.

Theorem 7.

If semilinear stochastic generalized systems (4) and (5) are approximately controllable, then systems (4) and (5) are approximately controllable on $[0,a]$, respectively.

Proof. 

Assuming $x_0\in L^2(\Omega ,F_0,P,H),Y_0\in L^2(\Omega ,F_0,P,Y),$

$$x_a\in L^2(\Omega ,F_a,P,H),Y_a\in L^2(\Omega ,F_a,P,Y).$$

For any $ϵ>0$, we have to find

$$u_{\mu }(·)\in L^2([0,a],\Omega ,F_t,U),$$

such that

$$\parallel x\left(a\right)-x_a\parallel <ϵ,$$

$$\parallel y\left(a\right)-y_a\parallel <ϵ.$$

By Theorem 5, there exists

$${\beta }_a\in L^2(\Omega ,F_a,P,U),$$

such that

$$\parallel y\left(a\right)-y_a\parallel =\parallel -N_1\left(a\right){\beta }_a-y_a\parallel <ϵ.$$

Let $u_{\mu }(·)$ be described by (26). Then, $u_{\mu }(·)$ can establish

$$\parallel y\left(a\right)-y_a\parallel <ϵ.$$

Similar to the proof of Theorem 2, $u_{\mu }(·)$ can establish

$$\parallel x\left(a\right)-x_a\parallel <ϵ.$$

This implies that semilinear stochastic generalized systems (4) and (5) are approximately controllable on $[0,a]$. □

Remark 2.

In this section, if $x\left(t\right)$ is the integral solution to semilinear stochastic generalized system (4) and H is changed to H₁, the results remain valid.

5.2. Example

Here, we give two examples to illustrate the validity of Theorem 6.

Example 2.

Consider the semilinear stochastic generalized heat equation:

$$\left[\begin{array}{cc}{I}_1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}dx(t,\xi )\\ dy(t,\xi )\end{array}\right]=\left[\begin{array}{cc}\frac{{\partial }^2}{\partial {\xi }^2}& 0\\ 0& I_2\end{array}\right]\left[\begin{array}{c}x(t,\xi )\\ y(t,\xi )\end{array}\right]dt$$

$$+\left[\begin{array}{c}(1+2t)I_1\\ (1+t^2)I_2\end{array}\right]u(t,\xi )dt$$

$$+\left[\begin{array}{c}\alpha (t,x(t,\xi ),u(t,\xi \left)\right)\\ 0\end{array}\right]dt+\left[\begin{array}{c}\beta (t,x(t,\xi ),u(t,\xi \left)\right)\\ 0\end{array}\right]dw\left(t\right),$$

$$x(t,0)=x(t,\pi )=0,y(t,0)=y(t,\pi )=0,0\le t\le a,0<\xi <\pi ;$$

$$x(0,\xi )=x_0\left(\xi \right),y(0,\xi )=y_0\left(\xi \right).$$

(32)

Let Y be a Hilbert space, $H=L^2(0,\pi )$,

$$E=I_1,B\left(t\right)=(1+2t)I_1,N_1\left(t\right)=(1+t^2)I_2,A=\frac{{\partial }^2}{\partial {\xi }^2}$$

with domain $\mathrm{dom}A=\{x\in H:x,\frac{dx}{d\xi }$ are absolutely continuous, $\frac{d^{2}x}{d{\xi }^2}\in H,x\left(0\right)=x\left(\pi \right)=0\},$ where $I_1$ and $I_2$ are identical operators on H and Y, respectively. Then, (32) can be written in the form of (4) and (5), i.e.,

$$Edx\left(t\right)=[A(t\left)x\right(t)+B(t\left)u\right(t)]dt+\alpha (t,x(t),u(t\left)\right)dt$$

$$+\beta (t,x\left(t\right),u\left(t\right))dw\left(t\right),x\left(0\right)=x_0,t\in [0,a],$$

(33)

$$0=y\left(t\right)+N_1\left(t\right)u\left(t\right),y\left(0\right)=y_0,t\in [0,a].$$

(34)

According to [41], A is a generator of compact $C_0-$semigroup $V_1\left(t\right)$. Thus, A is a generator of compact GE-evolution operator $V(t,s)$, which is given by $V(t,s)=V_1(t-s).$ It is obvious that (H₂) and (H₃) hold. We take $\alpha ,\beta$ to satisfy (H₄). The linear generalized system corresponding to (33) is

$$dx(t,\xi )=\frac{{\partial }^2}{\partial {\xi }^2}x(t,\xi )dt+(1+t^2)u(t,\xi )dt$$

$$x(t,0)=x(t,\pi )=0,0\le t\le a;$$

$$x(0,\xi )=x_0\left(\xi \right),0<\xi <\pi .$$

(35)

It is clear that system (34) and linear system (35) are approximately controllable on $[0,a]$. Therefore, (H₇) holds. Hence, semilinear stochastic generalized system (32) is approximately controllable on $[0,a]$ by Theorem 6.

Example 3.

According to [27], the stochastic version of the economics Leontief dynamic input-output model in Euclid space can be written as

$$Edx\left(t\right)=[A(t\left)x\right(t)+B(t\left)u\right(t)]dt+\alpha (t,x(t),u(t\left)\right)dt$$

$$+\beta (t,x\left(t\right),u\left(t\right))dw\left(t\right),x\left(0\right)=x_0,t\in [0,a],$$

(36)

$$0=y\left(t\right)+N_1\left(t\right)u\left(t\right),y\left(0\right)=y_0,t\in [0,a],$$

(37)

which is the same with (4) and (5), where $H=Y=U$, and H is a Euclid space. The linear system corresponding to semilinear stochastic generalized system (36) is

$$Edx\left(t\right)=[A\left(t\right)x\left(t\right)+B\left(t\right)u\left(t\right)]dt,x\left(0\right)=x_0,t\in [0,a].$$

(38)

Next, we take the data as follows:

$$E=I,A\left(t\right)=-(3t^2+1)I,B\left(t\right)=(1+t)I,N_1\left(t\right)=(t^2+2)I,$$

where I is the identical operator in Euclid space H. Systems (36) and (37) are the form

$$dx\left(t\right)=[-(3t^2+1)x\left(t\right)+(1+t)u\left(t\right)]dt+\alpha (t,x\left(t\right),u\left(t\right))dt$$

$$+\beta (t,x\left(t\right),u\left(t\right))dw\left(t\right),x\left(0\right)=x_0,t\in [0,a],$$

(39)

$$0=y\left(t\right)+(t^2+2)u\left(t\right),y\left(0\right)=y_0,t\in [0,a].$$

(40)

We can obtain that

$$V(t,s)=e^{-t^3-t+s^3+s}I$$

is a compact GE-evolution operator, induced by E with generator $A\left(t\right)$. It is obvious that (H₂) and (H₃) hold. We take $\alpha ,\beta$ to satisfy (H₄). It is clear that system (40) and the linear system corresponding to semilinear stochastic generalized system (39) are approximately controllable on $[0,a]$. Therefore, (H₇) holds. Hence, semilinear stochastic generalized systems (36) and (37) are approximately controllable on $[0,a]$ by Theorem 6.

Remark 3.

In this section, if $x\left(t\right)$ is the integral solution to semilinear stochastic generalized system (4), and H is changed to be (H₁), the results are still valid.

6. Exact Controllability of Linear Stochastic Generalized Systems (6) and (7)

For the exact controllability of ordinary linear stochastic systems, refer to [26,42,43]. In this section, we discuss the exact controllability of linear stochastic generalized systems (6) and (7) in the impulsive solution. Firstly, the existence and uniqueness of the impulsive solution to linear stochastic generalized systems (6) and (7) is considered by the GE-evolution operator and stochastic GE-evolution operator. Secondly, the exact controllability of linear stochastic generalized systems (6) and (7) is considered in the impulsive solution. Thirdly, three examples are given to illustrate the effectiveness of the theoretical results obtained in this section.

In the following, we assume that

$$A_1\left(t\right)\in P([0,\tau ],B(H,H)),\parallel A_1\left(t\right)\parallel <c_{A_1},t\in [0,a],$$

where $c_{A_1}$ is a constant.

6.1. Impulsive Solution to Linear Stochastic Generalized Systems (6) and (7)

Definition 7.

A stochastic process $x\in C([0,a],\Omega ,F_t,H)$ is a mild solution of linear stochastic generalized system (6), if for each $u\in L^2([0,a],\Omega ,F_t,U)$, it satisfies the following equation:

$$x\left(t\right)=V(t,0)E{x}_0+{\int }_0^{a}V(t,s)B\left(s\right)u\left(s\right)ds$$

$$+{\int }_0^{a}V(t,s)A_1\left(s\right)x\left(s\right)dw\left(s\right).$$

(41)

Similar to the proof of Theorem 1, we have Theorem 8, as follows:

Theorem 8.

Under the hypotheses (H₂) and (H₃), linear stochastic generalized system (6) has a unique mild solution.

Definition 8.

Let ${\Delta }_a=\{(t,s):0\le s\le t\le a\}$. A family of stochastic operators $\{S(t,s):(t,s)\in {\Delta }_a\}$ on H is said to be a stochastic generalized operator (i.e., a stochastic GE-evolution operator) induced by E on $[0,a]$ if it has the following properties:

(i) $S:{\Delta }_a\times \Omega \to B(H,H)$ is strongly measurable;

(ii) $S(t,s)$ is strongly $F_t$-measurable for $t\ge s$;

(iii) $S(s,s)=S_0,0\le s\le a$ and $S(t,r)ES(r,s)=S(t,s)$ for any $0\le s\le r\le t\le a$, where $S_0\in B(H,H)$ is a steady operator;

(iv) For any $x\in H,(t,s)\to S(t,s)x$ is mean square continuous from ${\Delta }_a$ into H.

Definition 9.

We say that the stochastic GE-evolution operator $S(t,s)$ induced by E is related to the linear stochastic generalized homogeneous equation

$$Edx\left(t\right)=A\left(t\right)x\left(t\right)dt+A_1\left(t\right)x\left(t\right)dw\left(t\right),x\left(s\right)=x_0,0\le s\le t\le a,$$

(42)

if $x\left(t\right)=S(t,s)E{x}_0$ is the mild solution to linear stochastic generalized system (42), with $x\left(s\right)=S(s,s)E{x}_0$ for arbitrary $x_0\in L^2(\Omega ,F_s,P,H)$.

In the following, we assume that the following hypothesis (H₈) holds.

(H₈) For any $0\le s\le t\le a$ and $x\in L^2(\Omega ,F_s,P,H)$, we have that

$$\mathbb{E}{\int }_s^t{\parallel S(r,s)x\parallel }^{2}dr\le c_s{\parallel x\parallel }^2,$$

$$\underset{r\in [s,t]}{sup}{\mathbb{E}\parallel S(r,s)x\parallel }^2\le c_s{\parallel x\parallel }^2.$$

Similar to the proof of [44] (Th. 50), we have the following theorem:

Theorem 9.

Under (H₂), (H₃) and (H₈), the mild solution $x\left(t\right)$ to linear stochastic generalized system (6) can be written in the form

$$x\left(t\right)=S(t,0)E{x}_0+{\int }_0^{t}S(t,s)B\left(s\right)u\left(s\right)ds.$$

(43)

For linear stochastic generalized systems (6) and (7), we still call

$$\left[\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right]=\left[\begin{array}{c}x\left(t\right)\\ -N_1\left(t\right)u\left(t\right)\end{array}\right]$$

its impulse solution, where $x\left(t\right)$ is the mild solution to linear stochastic generalized system (6).

6.2. Exact Controllability of Systems (6) and (7)

In general, if $N\ne 0$ in linear generalized system (3), then linear stochastic generalized systems (6) and (3) are not exactly controllable. Therefore, we discuss the exact controllability of linear stochastic generalized systems (6) and (7).

Definition 10.

Linear stochastic generalized systems (6) and (7) are called to be exactly controllable on $[0,a]$, if, for any state $\left[\begin{array}{c}{x}_a\\ y_a\end{array}\right]\in L^2(\Omega ,F_a,P,H\oplus Y)$ and any initial state $\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]\in L^2(\Omega ,F_0,P,H\oplus Y)$, there exists a control $u\in L^2(\Omega ,F_0,P,U)$, such that the impulsive solution $\left[\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right]$ of the linear stochastic generalized system (6)–(7) satisfies that

$$(\mathbb{E}\parallel \left[\begin{array}{c}x\left(a\right)\\ y\left(a\right)\end{array}\right]-\left[\begin{array}{c}{x}_a\\ y_a\end{array}\right]{{\parallel }^2)}^{1/2}=0.$$

In order to obtain the main theorem of exact controllability, the following concepts are introduced. Controllability operator

$$C_0^{a}u={\int }_0^{a}S(a,r)B\left(r\right)u\left(r\right)dr.$$

(44)

It is obvious that $C_0^a$ is a bounded linear operator, and its dual

$$C_0^{a*}:L^2(\Omega ,F_a,P,H)\to L^2([0,a],\Omega ,F_t,U)$$

is defined by

$$C_0^{a*}x=B^{*}\left(r\right)S^{*}(a,r)\mathbb{E}\left(x\right|F_r),$$

where $x\in L^2(\Omega ,F_a,P,H).$ Similar to the proof of [44] (Th. 52), we have the following theorem:

Theorem 10.

Linear stochastic generalized system (6) is exactly controllable on $[0,a]$ if and only if the following condition holds for some $c_a>0$ and for all $x\in L^2(\Omega ,F_a,P,H)$:

$$\mathbb{E}{\int }_0^a\parallel B^{*}\left(r\right)S^{*}(a,r)\mathbb{E}\left(x\right|F_r){\parallel }^{2}dr\ge c_a\mathbb{E}{\parallel x\parallel }^2.$$

(45)

According to Theorem 10, we obtain the following theorem:

Theorem 11.

Linear stochastic generalized system (6) is exactly controllable on $[0,a]$ if and only if the following condition holds for some $c_a>0$ and for all $x\in L^2(\Omega ,F_a,P,H)$:

$$\mathbb{E}{\int }_0^{a}f\left(t\right)\parallel B^{*}\left(r\right)S^{*}(a,r)\mathbb{E}\left(x\right|F_r){\parallel }^{2}dr\ge c_a\mathbb{E}{\parallel x\parallel }^2,$$

(46)

where $f\left(t\right)=t^2{(t-a)}^2$. In this case,

$$C_0^a{C}_0^{a*}(·)={\int }_0^{a}S(a,r)B\left(r\right)B^{*}\left(r\right)S^{*}(a,r)\mathbb{E}(·|F_r)dr$$

has bounded inverse.

Proof. 

Sufficiency. As

$$\underset{t\in [0,a]}{max}f\left(t\right)={(a/2)}^4,$$

from (46), we have that

$${(a/2)}^4\mathbb{E}{\int }_0^a{\parallel B^{*}\left(r\right)S^{*}(a,r)\mathbb{E}\left(x\right|F_r)\parallel }^{2}dr$$

$$\ge \mathbb{E}{\int }_0^a{\parallel f\left(t\right)B^{*}\left(r\right)S^{*}(a,r)\mathbb{E}\left(x\right|F_r)\parallel }^{2}dr$$

$$\ge c_a\mathbb{E}{\parallel x\parallel }^2.$$

Therefore, (45) is true. By Theorem 10, linear stochastic generalized system (6) is exactly controllable on $[0,a]$.

Necessity. Assume (45) is true. If (46) is false, then for any positive integer k, there exists

$$x_k\in L^2(\Omega ,F_a,P,H)$$

and

$$\parallel x_k{\parallel }_{L^2(\Omega ,F_a,P,H)}=1,$$

such that

$$\mathbb{E}{\int }_0^a{\parallel f\left(t\right)B^{*}\left(t\right)S(a,t)\mathbb{E}\left(x_k\right|F_t)\parallel }^{2}dt<\frac{1}{k}.$$

(47)

As $f\left(t\right)$ is an increasing function when $t\in [0,a/2]$ and $f\left(t\right)$ is a decreasing function when $t\in [a/2,a]$, we have that

$$f\left(t\right)\ge {(1/k)}^{1/2}{[{(1/k)}^{1/4}-a]}^2,$$

$$t\in [{(1/k)}^{1/4},a-{(1/k)}^{1/4}].$$

According to (47), we can obtain that

$$\begin{array}{ccc}\hfill \frac{1}{k}& >& {\left(\frac{1}{k}\right)}^{1/2}{[{(1/k)}^{1/4}-a]}^2\hfill \\ & & \times \mathbb{E}{\int }_{{(1/k)}^{1/4}}^{a-{(1/k)}^{1/4}}{\parallel B^{*}\left(t\right)S^{*}(a,t)\mathbb{E}\left(x_k\right|F_t)\parallel }^{2}dt\hfill \\ & =& {\left(\frac{1}{k}\right)}^{1/2}{[{(1/k)}^{1/4}-a]}^2[\mathbb{E}{\int }_0^a{\parallel B^{*}\left(t\right)S^{*}(a,t)\mathbb{E}\left(x_k\right|F_t)\parallel }^{2}dt\hfill \\ & & -\mathbb{E}{\int }_0^{{(1/k)}^{1/4}}{\parallel B^{*}\left(t\right)S^{*}(a,t)\mathbb{E}\left(x_k\right|F_t)\parallel }^{2}dt\hfill \\ & & -\mathbb{E}{\int }_{a-{(1/k)}^{1/4}}^a\parallel B^{*}\left(t\right)S^{*}(a,t)\mathbb{E}\left(x_k\right|F_t){\parallel }^{2}dt].\hfill \end{array}$$

From (45), we can obtain that

$$\begin{array}{ccc}\hfill {\left(\frac{1}{k}\right)}^{1/2}& >& {[{(1/k)}^{1/4}-a]}^2[c_a\hfill \\ & & -\mathbb{E}{\int }_0^{{(1/k)}^{1/4}}{\parallel B^{*}\left(t\right)S^{*}(a,t)\mathbb{E}\left(x_k\right|F_t)\parallel }^{2}dt\hfill \\ & & -\mathbb{E}{\int }_{\tau -{(1/m)}^{1/4}}^a\parallel B^{*}\left(t\right)S^{*}(a,t)\mathbb{E}\left(x_k\right|F_t){\parallel }^{2}dt].\hfill \end{array}$$

As $k\to +\infty$, we have that $0\ge a^2{c}_a>0$. This contradiction indicates that (46) is true. By [41] (Example A.4.2), the inverse of $C_0^a{C}_0^{a*}(·)$ is bounded. □

Similar to the proof of [31] (Th. 4), we have the following theorem:

Theorem 12.

System (7) is exactly controllable on $[0,a]$ up to $L^2(\Omega ,F_a,P,Y)$, if and only if

$$N_1\left(a\right):L^2(\Omega ,F_a,P,U)\to L^2(\Omega ,F_a,P,Y)$$

satisfies

$$\mathrm{ran}{N}_1\left(a\right)=L^2(\Omega ,F_a,P,Y).$$

Now, we investigate the exact controllability of linear stochastic generalized systems (6) and (7).

Theorem 13.

Under hypotheses (H₂), (H₃) and (H₈), linear stochastic generalized systems (6) and (7) are exactly controllable on $[0,a]$, if and only if both systems (6) and (7) are exactly controllable on $[0,a]$.

Proof. 

The necessity is obvious, so we only need to prove sufficiency. Assume that

$$\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]\in L^2(\Omega ,F_0,P,H\oplus Y);$$

$$\left[\begin{array}{c}{x}_a\\ y_a\end{array}\right]\in L^2(\Omega ,F_a,P,H\oplus Y).$$

We have to find

$$u(·)\in L^2([0,a],\Omega ,F_t,U),$$

such that

$$x\left(t\right)=S(t,0)E{x}_0+{\int }_0^{t}S(t,r)B\left(r\right)u\left(r\right)dr,$$

$$y\left(t\right)=-N_1\left(t\right)u\left(t\right)$$

and

$$x\left(a\right)=x_a,y\left(a\right)=y_a.$$

(48)

We choose

$$u\left(t\right)=u_1\left(t\right)+u_2\left(t\right).$$

Then,

$$x\left(t\right)=S(t,0)E{x}_0+{\int }_0^{t}S(t,r)B\left(r\right)u_1\left(r\right)dr$$

$$+{\int }_0^{t}S(t,r)B\left(r\right)u_2\left(r\right)dr$$

and

$$y\left(t\right)=-N_1\left(t\right)u_1\left(t\right)-N_1\left(t\right)u_2\left(t\right).$$

We select

$$u_1\left(t\right)=t(a-t)v\left(t\right)$$

(49)

for some

$$v(·)\in L^2([0,a],\Omega ,F_t,U).$$

Then, $u_1\left(a\right)=0$. By Theorem 12, there exists

$$u_2(·)\in L^2([0,a],\Omega ,F_t,U),$$

such that

$$y\left(a\right)=-N_1\left(a\right)u_2\left(a\right)=y_a.$$

(50)

According to Theorem 11, there exists

$$x_a\in L^2(\Omega ,F_t,P,H)$$

such that

$${\int }_0^{a}f\left(t\right)S(a,t)B\left(t\right)B^{*}\left(t\right)S^{*}(a,t)\mathbb{E}\left(x_a\right|F_t)dt$$

$$-[x_a-S(a,0)E{x}_0$$

$$-{\int }_0^{a}S(a,t)B\left(t\right)u_2\left(t\right)dt]=0,$$

(51)

where

$$f\left(t\right)=t^2{(a-t)}^2$$

and

$$u_2\left(t\right)=\mathbb{E}\left(u_2\left(a\right)\right|F_t).$$

Let

$$v\left(t\right)=t(a-t)B^{*}\left(t\right)S^{*}(a,t)\mathbb{E}\left(x_a\right|F_t).$$

Then, by (49) and (51), we obtain

$$u_1\left(t\right)=f\left(t\right)B^{*}\left(t\right)S^{*}(a,t)\mathbb{E}\left(x_a\right|F_t)$$

and

$${\int }_0^{a}S(a,t)B\left(t\right)u_1\left(t\right)dt$$

$$-[x_a-S(a,0)E{x}_0$$

$$-{\int }_0^{a}S(a,t)B\left(t\right)u_2\left(t\right)dt]=0,$$

i.e.,

$$x_a=S(a,0)E{x}_0+{\int }_0^{a}S(a,t)B\left(t\right)u\left(t\right)dt.$$

(52)

From (50) and (52), we have that (48) is true. Therefore, linear stochastic generalized systems (6) and (7) are exactly controllable on $[0,a]$. □

6.3. Applicable Examples

Here, three examples are provided to illustrate the effectiveness of Theorem 13

Example 4.

Consider the linear stochastic generalized equation:

$$\left[\begin{array}{cc}{I}_1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}dx(t,\xi )\\ dy(t,\xi )\end{array}\right]=\left[\begin{array}{cc}(1+t)I_1& 0\\ 0& I_2\end{array}\right]\left[\begin{array}{c}x(t,\xi )\\ y(t,\xi )\end{array}\right]dt$$

$$+\left[\begin{array}{c}(1+2t)I_1\\ (1+t^2)I_2\end{array}\right]u(t,\xi )dt$$

$$+\left[\begin{array}{cc}{I}_1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}x(t,\xi )\\ y(t,\xi )\end{array}\right]dw\left(t\right),$$

$$x(t,0)=x(t,\pi )=0,y(t,0)=y(t,\pi )=0,0\le t\le a,0<\xi <\pi ;$$

$$x(0,\xi )=x_0\left(\xi \right),y(0,\xi )=y_0\left(\xi \right).$$

(53)

Let Y be a Hilbert space, $H=L^2(0,\pi )$,

$$E=I_1,B\left(t\right)=(1+2t)I_1,A\left(t\right)=(1+t)I_1;$$

$$A_1\left(t\right)=I_1,N_1\left(t\right)=(1+t^2)I_2,$$

where $I_1$ and $I_2$ are identical operators on H and Y, respectively. Then, (53) can be written in the form of (6) and (7), i.e.,

$$Edx\left(t\right)=[A\left(t\right)x\left(t\right)+B\left(t\right)u\left(t\right)]dt+A_1\left(t\right)x\left(t\right)dw\left(t\right)$$

$$x\left(0\right)=x_0,t\in [0,a],$$

(54)

$$0=y\left(t\right)+N_1\left(t\right)u\left(t\right),y\left(0\right)=y_0,t\in [0,a].$$

(55)

According to [45], $A\left(t\right)$ is a generator of the GE-evolution operator $V(t,s)$ which is given by

$$V(t,s)=e^{\frac{1}{2}{t}^2+t-\frac{1}{2}{s}^2-s}{I}_1.$$

It is obvious that (H₂) and (H₃) hold. The linear stochastic generalized homogeneous system corresponding to (54) is

$$Edx\left(t\right)=A\left(t\right)x\left(t\right)dt+A_1\left(t\right)x\left(t\right)dw\left(t\right)$$

$$x\left(0\right)=x_0,t\in [0,a].$$

(56)

The stochastic GE-evolution operator $S(t,s)$ induced by E related to the linear stochastic generalized homogeneous Equation (56) is

$$S(t,s)=e^{\frac{1}{2}(t+t^2-s-s^2)+w\left(t\right)-w\left(s\right)}{I}_1.$$

It is clear that (H₈) holds. Hence, linear stochastic generalized systems (54) and (55) are exactly controllable on $[0,a]$, respectively. Therefore, linear stochastic generalized system (53) is exactly controllable on $[0,a]$ by Theorem 13.

Example 5.

According to [27], in input-output economics, many models were established to describe real economics. The Leontief dynamic input-output model can be extended as a linear generalized differential equation of the form:

$$E_{1}dz\left(t\right)=M\left(t\right)z\left(t\right)+L\left(t\right)u\left(t\right)$$

(57)

in Hilbert spaces. However, in reality, there are many unpredicted parameters and different types of uncertainties that have not been implemented in system (57). Nonetheless, from [46,47,48], we can consider a stochastic version of system (57) with one-dimensional standard Wiener process $w\left(t\right)$ used to model the uncertainties of the form:

$$E_{1}dz=M\left(t\right)z\left(t\right)dt+L\left(t\right)u\left(t\right)dt+C\left(t\right)z\left(t\right)dw\left(t\right).$$

(58)

If, for some concrete engineering practices, the following data are taken in (58):

$$dx\left(t\right)=-(1+t)x\left(t\right)+(1+t^2)u\left(t\right)dt+2x\left(t\right)dw\left(t\right),$$

$$x\left(0\right)=x_0,t\in [0,a],$$

(59)

$$0=y\left(t\right)+(1+2t^2)u\left(t\right),y\left(0\right)=y_0,t\in [0,a],$$

(60)

which is the same with (6) and (7), where $H=Y=U$ and H is a Hilbert space. It is obvious that if

$$V(t,s)=e^{-(t+\frac{1}{2}{t}^2-s-\frac{1}{2}{s}^2)}I,$$

$$S(t,s)=e^{-[t-s+\frac{1}{2}(t^2-s^2)]+2w\left(t\right)-2w\left(s\right)}I$$

and hypotheses (H₂), (H₃) and (H₈) are true, linear stochastic generalized systems (59) and (60) are exactly controllable on $[0,a]$, respectively. Therefore, linear stochastic generalized systems (59) and (60) are exactly controllable on $[0,a]$ by Theorem 13.

Example 6.

In this example, we consider the linear stochastic generalized equation:

$$\left[\begin{array}{cc}I& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}dx(t,\xi )\\ dy(t,\xi )\end{array}\right]=\left[\begin{array}{cc}\frac{{\partial }^2}{\partial {\xi }^2}& 0\\ 0& I\end{array}\right]\left[\begin{array}{c}x(t,\xi )\\ y(t,\xi )\end{array}\right]dt$$

$$+\left[\begin{array}{c}(1+2t^2)I\\ (1+t^2)I\end{array}\right]u(t,\xi )dt$$

$$+\frac{1}{2}\left[\begin{array}{cc}I& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}x(t,\xi )\\ y(t,\xi )\end{array}\right]dw\left(t\right),$$

$$x(0,\xi )=x_0\left(\xi \right),y(0,\xi )=y_0\left(\xi \right),t\in [0,a].$$

(61)

Let $H=Y=U$, H be a Hilbert space, $H=L^2\left(\mathbb{R}\right)$,

$$Ax=\frac{{\partial }^{2}x(t,\xi )}{\partial {\xi }^2},\forall x\left(\xi \right)\in H^2\left(\mathbb{R}\right),A_1=\frac{1}{2}I$$

and

$$B\left(t\right)=(1+2t^2)I,N_1\left(t\right)=(1+t^2)I.$$

Then, linear stochastic generalized system (61) can be written in the following linear stochastic generalized system in abstract form:

$$Edx\left(t\right)=[Ax\left(t\right)+B\left(t\right)u\left(t\right)]dt+A_{1}x\left(t\right)dw\left(t\right),$$

$$x\left(0\right)=x_0,t\in [0,a],$$

(62)

$$0=y\left(t\right)+N_1\left(t\right)u\left(t\right),y\left(0\right)=y_0,t\in [0,a].$$

(63)

The linear stochastic generalized homogeneous system corresponding to (62) is

$$Edx\left(t\right)=A\left(t\right)x\left(t\right)dt+A_1\left(t\right)x\left(t\right)dw\left(t\right),$$

$$x\left(0\right)=x_0,t\in [0,a].$$

(64)

Similar to the proof of Proposition 5.1 and Theorem 5.2 of [49], we have that A is a generator of a $C_0-$semigroup $V_1\left(t\right)$, and there exists a stochastic GE-evolution operator $S(t,s)$, induced by E related to the linear stochastic generalized homogeneous Equation (64), such that $\left(H_8\right)$ holds. Thus, A is a generator of GE-evolution operator $V(t,s)$, which is given by $V(t,s)=V_1(t-s).$ It is obvious that (H₂) and (H₃) hold and (63) is exactly controllable on $[0,a]$. Hence, linear stochastic generalized system (61) is exactly controllable on $[0,a]$ if and only if linear stochastic generalized system (62) is exactly controllable on $[0,a]$ by Theorem 13.

7. Conclusions

We have discussed the existence and uniqueness of the integral and impulsive solutions to semilinear stochastic generalized systems by using the GE-evolution operator theory and Banach fixed point theorem under different conditions. The existence and uniqueness theorems of solutions were obtained for these two cases. The approximate controllability of semilinear stochastic generalized systems was studied in integral and impulsive solutions, and the exact controllability of linear stochastic generalized systems was studied in impulsive solution. The criteria of approximate and exact controllability of correlated stochastic generalized systems were given, and some examples were provided to illustrate the validity of the theoretical results. The results obtained are very convenient and important for judging the controllability of these stochastic generalized systems. Future research should investigate how to solve the approximate and exact controllability of semilinear stochastic generalized systems with delay.