Approximate Controllability of Semilinear Stochastic Generalized Systems in Hilbert Spaces

Abstract

Approximate controllability of two types of nonlinear stochastic generalized systems is investigated in the sense of mild solution in Hilbert spaces. Firstly, the approximate controllability of semilinear stochastic generalized systems with control only acting on the drift terms is discussed by GE-evolution operator and Nussbaum fixed-point theorem. Secondly, the approximate controllability of semilinear stochastic systems with control acting on both drift and diffusion terms is handled by using GE-evolution operator and Banach fixed-point theorem. At last, two illustrative examples are given.

1. Introduction

The controllability of stochastic systems has become a central problem in the study of mathematical control theory, a large number of academic papers have been published. For representative papers, see references [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87]. However, as mentioned in document [65], “ this topic is still in the early stage. Lots of fundamental problems, such as the null/approximate controllability of stochastic systems with controls only acting on the drift terms, are also open.” In terms of linear systems, this problem has been well solved for some different types of systems. For example, see [63,64,72,73,74,76,77,78,79], etc. Compared with linear stochastic systems, the research on controllability of nonlinear stochastic systems has made slow progress, but many research results have also been achieved (e.g., [80,81,82,83,84,85,86,87]). The main mathematical theory used is strongly continuous operator semigroup. Unfortunately, these studies are limited to ordinary nonlinear stochastic systems (i.e., the coefficient operator of differential term is an identity operator) and do not include nonlinear stochastic generalized systems (i.e., the coefficient operator of a differential term is a bounded linear operator). Nonlinear stochastic generalized systems have a wide range of applications. For example, the paper from [88] proposed this kind of system when studying the evolution of the free surface of seepage liquid, and the study of [89,90] studied the impulse solution, approximate and exact controllability of this kind of system under a nonrandom condition; the mathematical model of this system was established in the study of temperature distribution in document [91], the integral solution and approximate controllability of this kind of system were investigated under the nonrandom condition; references [92,93] and their references put forward this kind of system when studying models of diffusion through porous media, wave propogation, etc., and studied the well-posed problem of this kind of system. In addition, such systems can also be abstracted from the practical problems studied in document [94]. They are essentially different from ordinary nonlinear stochastic systems. A large number of controllability problems of nonlinear stochastic generalized systems are proposed in practical problems. It is very important to study these problems. In view of this, we investigate the approximate controllability of these kinds of semilinear stochastic generalized systems with time-varying states and control operators by using the GE-evolution operator and related fixed-point theorems in the sense of a mild solution in Hilbert spaces. As far as we know, no relevant research results have been seen so far.

The organization of this paper is as follows. In Section 2, the approximate controllability of semilinear stochastic generalized systems with control only acting on the drift terms and with time-varying state and control operators is analyzed by using GE-evolution operator theory and Nussbaum fixed-point theorem in the sense of mild solution in Hilbert spaces, which includes the following contents: In Section 2.1, the mild solution of the semilinear stochastic generalized system with control only acting on drift terms is considered by using GE-evolution operator theory and Banach fixed-point theorem in Hilbert spaces. Sufficient conditions for the existence and uniqueness of mild solutions are given. In Section 2.2, the approximate controllability of linear stochastic generalized system with time-varying state and control operators is investigated by using GE-evolution operator in the sense of mild solution. In Section 2.3 the approximate controllability of the semilinear stochastic generalized system with time-varying state and control operator is studied by the approximate controllability of its linear part, GE-evolution operator and Nussbaum fixed-point theorem. In Section 3, the approximate controllability of semilinear stochastic generalized system with control acting on both drift and diffusion terms and with time-varying state and control operators is handled by using GE-evolution operator theory and contract mapping principle in the sense of a mild solution in Hilbert spaces. In Section 4, two application examples are given to illustrate the effectiveness of the theoretical results obtained in this paper. The solution of approximate controllability of nonlinear stochastic generalized systems with controls acting on both drift and diffusion terms in Euclidean space is illustrated by Remark 3. At last, the conclusions are given in Section 5.

In order to investigate the approximate controllability of semilinear stochastic generalized systems, we need to make the following preparations.

Let $\mathbb{H}$ be a separable Hilbert space with an inner product ${\langle ·,·\rangle }_{\mathbb{H}}$ (or $\langle ·,·\rangle$) and norm ${\parallel ·\parallel }_{\mathbb{H}}$ (or $\parallel ·\parallel$); $L(\mathbb{U},\mathbb{H})$ denote the set of all bounded linear operators from Hilbert space $\mathbb{U}$ to Hilbert space $\mathbb{H}$. We assume that there exists a complete orthonormal set $\left\{{\varphi }_n\right\}$ in separable Hilbert space $\mathbb{V}$, a bounded sequence of nonnegative real numbers ${\mu }_n$ such that $Q{\varphi }_n={\mu }_n{\varphi }_n$ for $Q\in L(\mathbb{V},\mathbb{H})$ with finite

$$tr\left(Q\right)=\sum _{n=1}^{\infty }{\mu }_n<\infty .$$

Let $w_n\left(t\right)(n=1,2,\cdots )$ be a sequence of real valued one dimensional standard Brownian motions such that $w\left(t\right)={\sum }_{n=1}^{\infty }{\left({\mu }_n\right)}^{1/2}{w}_n\left(t\right){\varphi }_n,t\ge 0.$ Then $w\left(t\right)$ is called a $Q-$ Wiener process. Further, we assume that $\{{\mathbb{F}}_t=\sigma (w\left(s\right):0\le s\le t),0\le t\le a\},$ i.e., ${\mathbb{F}}_t$ is generated by w. For $\varphi \in L(\mathbb{V},\mathbb{H})$, define

$${\parallel \varphi \parallel }_Q^2=tr\left(\varphi Q{\varphi }^{\ast }\right)=\sum _{n=1}^{\infty }{\parallel {\left({\mu }_n\right)}^{1/2}\varphi {\varphi }_n\parallel }^2,$$

then $\varphi$ is said to be a $Q-$ Hilbert Schmidt operator. $L_Q(\mathbb{V},\mathbb{H})$ denotes the Hilbert space of all $Q-$ Hilbert Schmidt operators $\varphi$ from $\mathbb{V}$ to $\mathbb{H}$ with norm ${\parallel \varphi \parallel }_Q$. $(\Omega ,\mathbb{F},\left\{{\mathbb{F}}_t\right\},\mathbb{P})$ is a complete probability space with filtration $\left\{{\mathbb{F}}_t\right\}$, $L^2(\Omega ,\mathbb{F},\mathbb{P},\mathbb{H})$ denotes the Hilbert space of all $\mathbb{F}$-measurable $\mathbb{H}$-valued random variables $\xi$ satisfying ${\mathbb{E}\parallel \xi \parallel }_{\mathbb{H}}^2<\infty$. $C([0,a],L^2(\Omega ,\mathbb{F},\mathbb{P},\mathbb{H}))$ denotes the Banach space of all continuous maps from $[0,a]$ into $L^2(\Omega ,\mathbb{F},\mathbb{P},\mathbb{H})$ satisfying the condition

$$su{p}_{t\in [0,a]}\mathbb{E}{\parallel \xi \left(t\right)\parallel }^2<\infty .C_m=C([0,a],L^2(\Omega ,{\mathbb{F}}_t,\mathbb{P},\mathbb{H}))$$

denotes the closed subspace of $C([0,a],L^2(\Omega ,\mathbb{F},\mathbb{P},\mathbb{H}))$ consisting of all measurable and ${\mathbb{F}}_t$-adapted processes $\xi \left(t\right)$. Then $C_m$ is a Banach space with the norm ${\parallel \xi \parallel }_{C_m}=(su{p}_{t\in [0,a]}{\mathbb{E}\parallel \xi \left(t\right)\parallel }^2{)}^{1/2}$. all processes are ${\mathbb{F}}_t-$ adapted; $\mathbb{E}$ denotes the mathematical expectation; $\mathbb{R}$ is the set of all real numbers, $\mathbb{C}$ is the set of all complex numbers; $p\in [1,\infty )$, $L^p(\Omega ,{\mathbb{F}}_t,\mathbb{P},\mathbb{H})$ denotes the set of all random variables $\eta \in \mathbb{H}$ such that $\eta$ is ${\mathbb{F}}_t-$ measurable and ${\parallel \eta \parallel }_p={\left(\mathbb{E}\right(\parallel \eta \parallel }_{\mathbb{H}}^p{\left)\right)}^{1/p}<+\infty$; $L^p([0,a],{\mathbb{F}}_t,\mathbb{P},\mathbb{H})$ denotes the set of all processes $x\left(t\right)\in L^p(\Omega ,{\mathbb{F}}_t,\mathbb{P},\mathbb{H})$ such that ${\parallel x\left(t\right)\parallel }_p<+\infty ,\forall t\in [0,a]$; $L^p([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{H})$ denotes the set of all measurable processes $x\left(t\right)\in L^p([0,\tau ],{\mathbb{F}}_t,\mathbb{P},\mathbb{H})$ such that $\mathbb{E}{\int }_0^a{\parallel x\left(t\right)\parallel }_{\mathbb{H}}^{p}dt<+\infty ;$ Let A be a linear operator. $domA$ and $ranA$ denote its domain and range, respectively; $A^{\ast }$ denotes the adjoint operator of A; For a set $S,\overline{S}$ denotes the closure of S. $P([0,a],L(\mathbb{U},\mathbb{H}\left)\right)=\left\{D\right(·)\in L(\mathbb{U},\mathbb{H}):D(·)z$ is continuous for every $z\in \mathbb{U}$ and $su{p}_{0\le t\le T}{\parallel D\left(t\right)\parallel }_{L(\mathbb{U},\mathbb{H})}<+\infty \}.$

We recall here the GE-evolution operator and Nussbaum fixed-point theorem, which will be used in this paper.

Definition 1

([77,95]). Suppose $\left\{T\right(t,s):0\le s\le t\}$ is a two-parameter family of bounded linear operators in Hilbert space $\mathbb{H}$, and E is a bounded linear operator. If $T(s,s)=T_0$ is a definite operator independent of s and

$$T(t,s)=T(t,r)ET(r,s),0\le s\le r\le t,$$

then $T(t,s)$ is called a GE-evolution operator induced by E.

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

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

If there exist $b\ge 1$ and $\omega >0$ such that

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

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

Suppose $T(t,s)$ is a strongly continuous and exponentially bounded GE-evolution operator induced by E. If $M\left(t\right):domM\left(t\right)\subseteq \mathbb{H}\to \mathbb{H}$ is a linear operator and

$$M\left(t\right)\xi =\underset{h\to 0^{+}}{lim}\frac{ET(h+t,t)E-ET(t,t)E}{h}\xi ,$$

for every $\xi \in D_0\left(t\right)$, where

$$D_0\left(t\right)=\{\xi :\xi \in domM\left(t\right)\subseteq \mathbb{H},T(t,t)E\xi =\xi ,\exists \underset{h\to 0^{+}}{lim}\frac{ET(h+t,t)E-ET(t,t)E}{h}\xi \},$$

then $M\left(t\right)$ is called a generator of GE-evolution operator $T(t,s)$ induced by E.

In the following we always assume that $D_0\left(t\right)=D_0=\{\xi :\xi \in domM\left(t\right):M\left(t\right)\xi \in ranE\}$ is independent of t, $M\left(t\right)$ is the generator of GE-evolution operator $T(t,s)$ induced by E and $\mathbb{D}=\overline{D_0}$.

Theorem 1

([96], Nussbaum). Suppose that S is a closed, bounded convex subset of a Banach space B. Suppose that $M_1,M_2$ are continuous mappings from S into B such that

 (i) 

$(M_1+M_2)B\subset B$,

 (ii) 

$\parallel M_{1}x+M_{2}y\parallel \le c\parallel x-y\parallel$ for all $x,y\in B,$ where $0\le c<1$ is a constant,

 (iii) 

$\overline{M_2[B]}$ is compact. Then the operator $M_1+M_2$ has a fixed-point in B.

2. Approximate Controllability of the Semilinear Stochastic Generalized Systems with the Control Only Acting on the Drift Terms

In this section, we consider the approximate controllability of semilinear stochastic generalized systems with the control only acting on the drift terms and with time-varying state and control operators in the sense of mild solution by using GE-evolution operator method and Nussbaum fixed-point theorem in Hilbert spaces, which is described by the following equation.

$$Ed\xi \left(t\right)=[M(t\left)\xi \right(t)+N(t\left)\eta \right(t)+\alpha (t,\xi \left(t\right)\left)\right]dt+\beta (t,\xi (t\left)\right)dw\left(t\right),$$

$$t\in [0,a],\xi \left(0\right)={\xi }_0,$$

(1)

where $\xi \left(t\right)$ is the state process valued in Hilbert space $\mathbb{H}$, the input process $\eta \left(t\right)$ belong to $L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{U})$. For the operators $E,M\left(t\right),N\left(t\right),\alpha ,\beta$ we assume the following hypotheses:

Hypothesis 1.

$M\left(t\right)$ is a generator of a compact GE-evolution operator $T(t,s)$ induced by E in $\mathbb{H}$ such that

$$ma{x}_{0\le s\le t\le a}\parallel T(t,s)\parallel \le c_T,$$

for some $c_T\ge 1$; ${\left(T(t,s)E\right)|}_{\mathbb{D}}$ is unique.

Hypothesis 1’. $M\left(t\right)$ is a generator of a compact GE-evolution operator $T(t,s)$ induced by E in $\mathbb{H}$ such that

$$ma{x}_{0\le s\le t\le a}\parallel T(t,s)\parallel \le c_T,$$

for some $c_T\ge 1$; $\left(T_{0}M\left(t\right)\right){|}_{\mathbb{D}}$ satisfies the following assumptions: $\left(P_1\right)$ For $t\in [0,T],{(\lambda I+\left(T_{0}M\left(t\right)\right){|}_{\mathbb{D}})}^{-1}$ exists for all $\lambda \in \mathbb{C}$ with $Re\lambda \le 0$ and there is a constant $M>0$ such that

$$\parallel {(\lambda I+\left(T_{0}M\left(t\right)\right){|}_{\mathbb{D}})}^{-1}{\parallel }_{L(\mathbb{H},\mathbb{H})}\le \frac{M}{\left|\lambda \right|+1}$$

for all $Re\lambda \le 0,t\in [0,T],$ where I is the identity operator on $\mathbb{H}$, $|·|$ denotes the complex modulus.

$\left(P_2\right)$ There exist constant $L>0$ and $0<\mu \le 1$ such that

$$\parallel (\left(T_{0}M\left(t\right)\right){|}_{\mathbb{D}}-\left(T_{0}M\left(s\right)\right){|}_{\mathbb{D}}){\left(\left(T_{0}M\left(\tau \right)\right){|}_{\mathbb{D}}\right)}^{-1}{\parallel }_{L(\mathbb{H},\mathbb{H})}\le L{|t-s|}^{\mu },$$

for $t,s,\tau \in [0,a].$

Hypothesis 2.

$N\left(t\right)\in P([0,a],L(\mathbb{U},\mathbb{H})),{\xi }_0\in L^2(\Omega ,{\mathbb{F}}_0,\mathbb{P},\mathbb{D}),\xi \left(t\right)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{D})$; for $\eta \in L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{U}),N\left(t\right)\eta \left(t\right)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,E\left(\mathbb{D}\right))$.

Hypothesis 3.

$\alpha :[0,a]\times \mathbb{D}\to E\left(\mathbb{D}\right),\beta :[0,a]\times \mathbb{D}\to L_Q(\mathbb{V},E\left(\mathbb{D}\right))$ are measurable and for $\forall x,y\in \mathbb{D},t\in [0,a]$ there exists a positive $l>0$ such that

$$\parallel \alpha (t,y)-\alpha (t,x)\parallel +\parallel \beta (t,y)-\beta (t,x)\parallel \le l\parallel y-x\parallel ,$$

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

Lemma 1

([77]). If $\left(P_1\right)$ and $\left(P_2\right)$ hold true in Hypothesis 1’, then ${\left(T(t,s)E\right)|}_{\mathbb{D}}$ is unique.

2.1. Mild Solution of Semilinear Stochastic Generalized System (1)

In this subsection, we investigate the mild solution of semilinear stochastic generalized system (1). Then, the theorem concerning the existence and uniqueness of the mild solution is given by GE-evolution operator in Hilbert space.

Definition 2.

For ${\xi }_0\in L^2(\Omega ,{\mathbb{F}}_0,\mathbb{P},\mathbb{D})$, a stochastic process $\xi \left(t\right),t\in [0,a]$ valued in $\mathbb{D}$, is called to be a mild solution of semilinear stochastic generalized system (1) if $\xi \left(t\right)\in L^2(\Omega ,{\mathbb{F}}_t,\mathbb{P},\mathbb{D})$,

$$\mathbb{P}[{\int }_0^a\parallel \xi \left(t\right){\parallel }^{2}dt<\infty ]=1,$$

(2)

and, for arbitrary $t\in [0,a]$, we have

$$\xi \left(t\right)=T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)N\left(s\right)\eta \left(s\right)ds+{\int }_0^{t}T(t,s)\alpha (s,\xi \left(s\right))ds$$

$$+{\int }_0^{t}T(t,s)\beta (s,\xi \left(s\right))dw\left(s\right).$$

(3)

The main theorem of this subsection is the following theorem.

Theorem 2.

Assume that Hypotheses 1–3 are satisfied. Then

 (i) 

There exists a mild solution $\xi \left(t\right)$ to semilinear stochastic generalized system (1) unique, up to equivalence, among the process satisfying (2). Moreover $\xi \left(t\right)$ possesses a continuous modification.

 (ii) 

For $\forall p\ge 2$, there exists a positive constant $b>0$ such that

$$su{p}_{t\in [0,a]}{\mathbb{E}\parallel \xi \left(t\right)\parallel }^p\le b(1+\mathbb{E}\parallel {\xi }_0{\parallel }^p),$$

(4)

where $b>0$ is a constant.

 (iii) 

For $\forall p>2$, there exists a positive constant $c>0$ such that

$$\mathbb{E}su{p}_{t\in [0,a]}{\parallel \xi \left(t\right)\parallel }^p\le c(1+\mathbb{E}\parallel {\xi }_0{\parallel }^p),$$

(5)

where $c>0$ is a constant.

Proof. 

First of all we prove the uniqueness. Suppose ${\xi }_1(·),{\xi }_2(·)$ are two processes satisfying (2) and (3). We prove that for $\forall t\in [0,a],\mathbb{P}({\xi }_1\left(t\right)={\xi }_2\left(t\right))=1.$ For a fixed positive number $r>0$, we define

${\mu }_i=inf\{t\le a:{\int }_0^t\parallel \alpha (s,{\xi }_i\left(s\right))\parallel ds\ge r$ or ${\int }_0^t{\parallel \beta (s,{\xi }_i\left(s\right))\parallel }^{2}ds\ge r$ }, $i=1,2$ and $\mu =min\{{\mu }_1,{\mu }_2\}.$ Let ${\tilde{\xi }}_i\left(t\right)=I_{[0,\mu ]}\left(t\right){\xi }_i\left(t\right),t\in [0,a],i=1,2,$ where $I_{[0,\mu ]}\left(t\right)$ denotes the characteristic function of $[0,\mu ]$. Then for $\forall t\in [0,a]$, $\mathbb{P}-a.s.$

$${\tilde{\xi }}_i\left(t\right)=I_{[0,\mu ]}\left(t\right)T(t,0)E{\xi }_0+I_{[0,\mu ]}\left(t\right){\int }_0^t{I}_{[0,\mu ]}\left(s\right)T(t,s)N\left(s\right)\eta \left(s\right)ds$$

$$+I_{[0,\mu ]}\left(t\right){\int }_0^t{I}_{[0,\mu ]}\left(s\right)T(t,s)\alpha (s,{\tilde{\xi }}_i\left(s\right))ds$$

$$+I_{[0,\mu ]}\left(t\right){\int }_0^t{I}_{[0,\mu ]}\left(s\right)T(t,s)\beta (s,{\tilde{\xi }}_i\left(s\right))dw\left(s\right).$$

Therefore, for $\forall t\in [0,a],\mathbb{P}-a.s.$

$$\mathbb{E}\parallel {\tilde{\xi }}_1\left(t\right)-{\tilde{\xi }}_2{\left(t\right)\parallel }^2\le 2c_T^2\mathbb{E}\{{\int }_0^t\parallel \alpha (s,{\tilde{\xi }}_1\left(s\right))-\alpha (s,{\tilde{\xi }}_2\left(s\right)){\parallel ds\}}^2$$

$$+2c_T^2\mathbb{E}\{{\int }_0^t\parallel \beta (s,{\tilde{\xi }}_1\left(s\right))-\beta (s,{\tilde{\xi }}_2\left(s\right)){\parallel }^{2}ds\}.$$

(6)

According to the definitions of the stopping times $\mu ,{\mu }_1,{\mu }_2$, we find that the right-hand side and therefore also the left-hand side of (6) is a bounded function on $t\in [0,a]$. Again by Hypothesis 3,

$$\mathbb{E}\parallel {\tilde{\xi }}_1\left(t\right)-{\tilde{\xi }}_2{\left(t\right)\parallel }^2\le 2c_T^2{l}^2(a+1){\int }_0^t\mathbb{E}{\parallel {\tilde{\xi }}_1\left(s\right)-{\tilde{\xi }}_2\left(s\right)\parallel }^{2}ds.$$

The boundedness of $\mathbb{E}\parallel {\tilde{\xi }}_1\left(t\right)-{\tilde{\xi }}_2{\left(t\right)\parallel }^2,t\in [0,a]$, and the Gronwall Lemma imply $\mathbb{E}\parallel {\tilde{\xi }}_1\left(t\right)-{\tilde{\xi }}_2\left(t\right)\parallel =0.$ Therefore, for $\forall t\in [0,a],$ we have $\mathbb{P}({\tilde{\xi }}_1\left(t\right)={\tilde{\xi }}_2\left(t\right))=1$. So, the process ${\tilde{\xi }}_1\left(t\right),{\tilde{\xi }}_2\left(t\right)$ are ${\mathbb{P}}_a-a.s.$ identical. Since this is true for $\forall r>0,$ therefore ${\tilde{\xi }}_1\left(t\right),{\tilde{\xi }}_2\left(t\right)$ are $\mathbb{P}-a.s.$ identical. Taking into account that ${\tilde{\xi }}_1\left(t\right),{\tilde{\xi }}_2\left(t\right)$ are a mild solution of Equation (3). We can easily deduce that for $\forall t\in [0,a],{\tilde{\xi }}_1\left(t\right)={\tilde{\xi }}_2\left(t\right),\mathbb{P}-a.s.$

Now, we prove the existence, which is based on the classical fixed-point theorem for contractions. Let ${\mathbb{D}}_p,p\ge 2,$ be the Banach space of all $\mathbb{D}$-valued processes x defined on the $[0,a]$ such that

$${\parallel x\parallel }_{{\mathbb{D}}_p}=(su{p}_{t\in [0,a]}{\mathbb{E}\parallel x\left(t\right)\parallel }^p{)}^{1/p}<\infty .$$

If one identifies process which are identical $\mathbb{P}-a.s.$ then ${\mathbb{D}}_p$, with the norm ${\parallel ·\parallel }_{{\mathbb{D}}_p}$, becomes a Banach space. Let O be the following transformation:

$$O\left(x\right)\left(t\right)=T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)N\left(s\right)\eta \left(s\right)ds+{\int }_0^{t}T(t,s)\alpha (s,x\left(s\right))ds$$

$$+{\int }_0^{t}T(t,s)\beta (s,x\left(s\right))dw\left(s\right)$$

$$=T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)N\left(s\right)\eta \left(s\right)ds+O_1\left(x\right)\left(t\right)+O_2\left(x\right)\left(t\right),t\in [0,a],x\in {\mathbb{D}}_p.$$

We suppose that ${\mathbb{E}\parallel x\left(t\right)\parallel }^p<\infty$ and show that O maps ${\mathbb{D}}_p$ into ${\mathbb{D}}_p$. As the composition of measurable mappings is measurable therefore, taking into account Hypotheses 1–3, we can obtain that $O_1,O_2$ are well defined. Moreover

$$\mathbb{E}\parallel O_1{\left(x\right)\parallel }_{{\mathbb{D}}_p}^p\le c_T^p\mathbb{E}({\int }_0^a{\parallel \alpha (s,x\left(s\right))\parallel ds)}^p\le a^{p-1}{c}_T^p\mathbb{E}{\int }_0^a{\parallel \alpha (s,x\left(s\right))\parallel }^{p}ds$$

$$\le 2^{(p-2)/2}{a}^{p-1}{\left(l{c}_T\right)}^p\mathbb{E}{\int }_0^a(1+\parallel x\left(s\right){\parallel }^p)ds$$

$$\le 2^{(p-2)/2}{\left(al{c}_T\right)}^p{(1+\parallel x\parallel }_{{\mathbb{D}}_p}^p).$$

Consequently $O_1$ maps ${\mathbb{D}}_p$ into ${\mathbb{D}}_p$. In order to prove the same property for $O_2$ we remark that, similar to the proof of ([97], Th.4.36) we find that there exists constant $c_p>0$ such that

$$\mathbb{E}\parallel O_2{\left(x\right)\parallel }_{{\mathbb{D}}_p}^p\le su{p}_{t\in [0,a]}\mathbb{E}{\parallel {\int }_0^{a}T(t,s)\beta (s,x\left(s\right))dw\left(s\right)\parallel }^p$$

$$\le c_T^p{c}_p\mathbb{E}({\int }_0^a{\parallel \beta (s,x\left(s\right))\parallel }^2{ds)}^{p/2}$$

$$\le c_T^p{c}_p{l}^p\mathbb{E}({\int }_0^a{(1+\parallel x\left(s\right)\parallel }^2{\left)ds\right)}^{p/2}$$

$$\le c_T^p{c}_{p}l{a}^{(p-2)/2}\mathbb{E}{\int }_0^a{(1+\parallel x\left(s\right)\parallel }^2{)}^{p/2}ds$$

$$\le c_T^p{c}_{p}l{a}^{(p-2)/2}{2}^{(p-2)/2}\mathbb{E}({\int }_0^a{(1+\parallel x\left(s\right)\parallel }^p)ds$$

$$\le c_T^p{c}_{p}l{\left(2a\right)}^{(p-2)/2}{a(1+\parallel x\parallel }_{{\mathbb{D}}_p}^p).$$

Now let $x_1,x_2$ be arbitrary processes from ${\mathbb{D}}_p$ then

$$\parallel O\left(x_1\right)-O\left(x_2\right){\parallel }_{{\mathbb{D}}_p}\le \parallel O_1\left(x_1\right)-O_1\left(x_2\right){\parallel }_{{\mathbb{D}}_p}+{\parallel O_2\left(x_1\right)-O_2\left(x_2\right)\parallel }_{{\mathbb{D}}_p}=i_1+i_2$$

and

$$i_1^p\le su{p}_{t\in [0,a]}\mathbb{E}{\parallel {\int }_0^t[T(t,s)(\alpha (s,x_1\left(s\right))-\alpha (s,x_2\left(s\right)))]ds\parallel }^p$$

$$\le c_T^{p}su{p}_{t\in [0,a]}\mathbb{E}[{\int }_0^a\parallel \alpha (s,x_1\left(s\right))-\alpha (s,x_2\left(s\right)){\parallel ds]}^p$$

$$\le {\left(c_{T}l\right)}^p{a}^{p-1}[{\int }_0^a\mathbb{E}\parallel x_1\left(s\right)-x_2\left(s\right){\parallel }^{p}ds]$$

$$\le {\left(c_{T}l\right)}^p{a}^{p}su{p}_{t\in [0,a]}\mathbb{E}{\parallel x_1\left(t\right)-x_2\left(t\right)\parallel }^p$$

$$\le {\left(c_{T}l\right)}^p{a}^p{\parallel x_1-x_2\parallel }_{{\mathbb{D}}_p}^p.$$

In a similar way, by ([97], Th.4.36) we can obtain

$$i_2^p\le c_p{\left(c_{T}l\right)}^p{a}^{p/2}{\parallel x_1-x_2\parallel }_{{\mathbb{D}}_p}^p.$$

Summing up the obtained estimates, we obtain

$$\parallel O\left(x_1\right)-O\left(x_2\right){\parallel }_{{\mathbb{D}}_p}\le c_{T}l{(a^p+c_p{a}^{p/2})}^{1/p}{\parallel x_1-x_2\parallel }_{{\mathbb{D}}_p}^p$$

(7)

for all $x_1,x_2\in {\mathbb{D}}_p$. Consequently, if

$$c_{T}l{(a^p+c_p{a}^{p/2})}^{1/p}<1,$$

(8)

then the operator O has unique fixed point $\xi$ in ${\mathbb{D}}_p$ which, as it is easy to see, is a solution of the semilinear stochastic generalized system (1). The extra condition (8) on a can be easily removed by consider the equation on interval $[0,a_1],[a_1,2a_1],\cdots$, with $a_1$ satisfying (8). Thus, we have proved assertion (ii) of the theorem since (4) follows easily by using Gronwall’s lemma.

If $\mathbb{E}\parallel {\xi }_0{\parallel }^p=\infty$, to construct a solution, we show first that if ${\xi }_0,x_0$ are two initial condition satisfying $\parallel {\xi }_0\parallel <\infty ,\parallel x_0\parallel <\infty$, and if $\xi (·),x(·)\in {\mathbb{D}}_p$ are the corresponding solution of semilinear stochastic generalized system (1), then

$$I_d\xi (·)=I_{d}x(·),\mathbb{P}-a.s.$$

(9)

where $d=\{\omega \in \Omega :\xi (\omega )=x(\omega \left)\right\}$.

To see this, define

$${\xi }^0=T(·,0)E{\xi }_0,{\xi }^{k+1}=O\left({\xi }^k\right),t\in [0,a],k=0,1,2,\cdots .$$

Then, for $t\in [0,a],\mathbb{P}-a.s.$

$${\xi }^{k+1}\left(t\right)=T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)N\left(s\right)\eta \left(s\right)ds+{\int }_0^{t}T(t,s)\alpha (s,{\xi }^k\left(s\right))ds$$

$$+{\int }_0^{t}T(t,s)\beta (s,{\xi }^k\left(s\right))dw\left(s\right).$$

Since $I_d$ is an ${\mathbb{F}}_0$-measurable random variable, therefore $I_d\beta (·,{\xi }^k(·))$ is an $L_Q(\mathbb{V},\mathbb{D})$-process and for $t\in [0,a],$

$${\int }_0^{t}T(t,s)I_d\beta (s,{\xi }^k\left(s\right))dw\left(s\right)=I_d{\int }_0^{t}T(t,s)\beta (s,{\xi }^k\left(s\right))dw\left(s\right).$$

Therefore, for $t\in [0,a]$,

$$I_d{\xi }^{k+1}\left(t\right)=T(t,0)E{I}_d{\xi }_0+{\int }_0^{t}T(t,s)I_{d}N\left(s\right)\eta \left(s\right)ds+{\int }_0^{t}T(t,s)I_d\alpha (s,{\xi }^k\left(s\right))ds$$

$$+{\int }_0^{t}T(t,s)I_d\beta (s,{\xi }^k\left(s\right))dw\left(s\right).$$

(10)

If, for a similar-defined sequence

$$x^0=T(·,0)E{x}_0,x^{k+1}=O\left(x^k\right),t\in [0,a],k=0,1,2,\cdots ,$$

and some k we have

$$I_d{\xi }^k(·)=I_d{x}^k(·),{\mathbb{P}}_a-a.s.$$

where ${\mathbb{P}}_a$ is defined to be the product of the Lebesgue measure in $[0,a]$ and the measure $\mathbb{P}$, then also

$$I_d\alpha (·,{\xi }^k(·))=I_d\alpha (·,x^k(·)),I_d\beta (·,{\xi }^k(·))=I_d\beta (·,x^k(·)),{\mathbb{P}}_a-a.s.$$

Since the process $\xi ,x$ are limits in the ${\parallel ·\parallel }_{{\mathbb{D}}_p}$ norms of the sequence $\left\{{\xi }^k(·)\right\}$ and $\left\{x^k(·)\right\}$, respectively; therefore, (9) must be true. Moreover, the process $I_d\xi (·)$ satisfies the semilinear stochastic generalized system (1) with the initial condition $I_d\xi \left(0\right)=I_d{\xi }_0$.

We now prove existence. Let us define, for $n\in \mathbb{N}$,

$${\xi }_n={\xi }_0,\parallel {\xi }_0\parallel \le n;{\xi }_n=0,\parallel {\xi }_0\parallel >n,$$

and denote by ${\xi }_n$ the corresponding solution of (8). By the previous argument we have ${\xi }_n\left(t\right)={\xi }_{n+1}\left(t\right)$ on $\{\omega \in \Omega ,\parallel {\xi }_0\parallel \le n\}$. It is now easy to see that the process

$$\xi \left(t\right)=li{m}_{n\to \infty }{\xi }_n\left(t\right),t\in [0,a],$$

is $\mathbb{P}-a.s.$ well defined and satisfies the semilinear stochastic generalized system (1).

For proof of existence of continuous modification of the mild solution assume first that $\mathbb{E}\parallel {\xi }_0{\parallel }^{2r}<\infty$ for some $r>1.$ From the first part of the theorem one knows that

$$su{p}_{t\in [0,a]}\mathbb{E}{\parallel \xi \left(t\right)\parallel }^{2r}<\infty .$$

(11)

Define $\varphi \left(t\right)=\beta (t,\xi (t\left)\right),t\in [0,a]$, and

$$i=\mathbb{E}{\int }_0^a{\parallel \varphi \left(t\right)\parallel }^{2r}dt=\mathbb{E}{\int }_0^a{\parallel \beta (t,\xi \left(t\right))\parallel }^{2r}dt.$$

By Hypothesis 3 we have

$$i\le l^{2r}\mathbb{E}({\int }_0^a{(1+\parallel \xi \left(t\right)\parallel }^2{)}^r<\infty .$$

Consequently, ([97], Proposition 7.3) implies that the process

$${\int }_0^{t}T(t,s)\beta (s,\xi \left(s\right))dw\left(s\right),t\in [0,a]$$

and therefore also $\xi \left(t\right),t\in [0,a]$, has a continuous modification.

The case of initial conditions satisfying $\mathbb{E}\parallel {\xi }_0{\parallel }^{2r}<\infty$ can be reduced to the case just considered by regarding initial condition ${\xi }_n,$

$${\xi }_n={\xi }_0,\parallel {\xi }_0\parallel \le n;{\xi }_n=0,\parallel {\xi }_0\parallel >n,$$

as in the proof of existence. Finally, (5) follows again from Gronwell’s Lemma. □

We assume the following Hypothesis:

Hypothesis 4.

$\alpha :[0,a]\times \mathbb{D}\to E\left(\mathbb{D}\right),\beta :[0,a]\times \mathbb{D}\to L_Q(\mathbb{V},E\left(\mathbb{D}\right))$ are measurable and for $\forall x,y\in \mathbb{D},t\in [0,a]$ there exists a positive $l>0$ such that

$${\parallel \alpha (t,y)-\alpha (t,x)\parallel }^2{+\parallel \beta (t,y)-\beta (t,x)\parallel }^2\le l{\parallel y-x\parallel }^2,$$

$${\parallel \alpha (t,x)\parallel }^2+{\parallel \beta (t,x)\parallel }^2\le l,$$

According to the above, if Hypotheses 1 and 2 and Hypothesis 4 hold, then we have that the time varying semilinear stochastic generalized system (1) admits a mild solution $\xi (·)\in C_m=C([0,a],L^2(\Omega ,{\mathbb{F}}_t,\mathbb{P},\mathbb{D}))$ for $\eta (·)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{U})$. This mild solution satisfies the semilinear stochastic generalized Equation (3).

In order to investigate the approximate controllability of semilinear stochastic generalized system (1), we consider its corresponding linear part

$$Ed\xi \left(t\right)=[M\left(t\right)\xi \left(t\right)+N\left(t\right)\eta \left(t\right)]dt+Ldw\left(t\right),t\in [0,a],\xi \left(0\right)={\xi }_0,$$

(12)

where $L\in L_Q(\mathbb{V},\mathbb{D})$ and assume the approximate controllability of linear stochastic generalized system (12).

2.2. Approximate Controllability of Linear Stochastic Generalized System and Stochastic Generalized Linear Regulator Problem

In this subsection, the necessary and sufficient condition concerning the approximate controllability of linear stochastic generalized system (12) is given and the stochastic generalized linear regulator problem is solved. We find an optimal control in terms of stochastic control operators which drives a point ${\xi }_0\in \mathbb{D}$ to a small neighborhood of an arbitrary point ${\xi }_a\in L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{D})$. Further, we discuss relation between controllability operator $Q_0^a$ and stochastic analog $G_0^a$.

Define the stochastic generalized linear regulator problem: to minimize

$$\Sigma \left(\eta \right)=\mathbb{E}\parallel {\xi }_{\lambda }\left(a\right)-{\xi }_a{\parallel }^2+\lambda \mathbb{E}{\int }_0^a{\parallel \eta \left(t\right)\parallel }^{2}dt$$

(13)

over all $\eta (·)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{U})$, where ${\xi }_{\lambda }(·)$ is a solution of the following equation

$$\xi \left(t\right)=T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)N\left(s\right)\eta \left(s\right)ds+{\int }_0^{t}T(t,s)\alpha \left(s\right)ds$$

$$+{\int }_0^{t}T(t,s)\beta \left(s\right)dw\left(s\right),$$

(14)

when $\eta (·)={\eta }_{\lambda }(·);{\xi }_a\in L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{D})$ and $\lambda >0$ is a parameter;

$$\alpha (·)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,E\left(\mathbb{D}\right)),\beta (·)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,L_Q(\mathbb{V},E\left(\mathbb{D}\right))),$$

$$\beta (·)dw(·)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,E\left(\mathbb{D}\right)).$$

It is convenient at this point to introduce relevant operators the basic controllability condition

$$C_t^a\eta ={\int }_t^{a}T(a,s)N\left(s\right)\eta \left(s\right)ds,$$

$$Q_t^a={\int }_t^{a}T(a,s)N\left(s\right)N^{\ast }\left(s\right)T^{\ast }(a,s)ds,$$

$$G_t^a(·)={\int }_t^{a}T(a,s)N\left(s\right)N^{\ast }\left(s\right)T^{\ast }(a,s)\mathbb{E}(·|{\mathbb{F}}_s)ds.$$

Hypothesis 5.

$\lambda {(\lambda I+G_0^a)}^{-1}\to 0$, as $\lambda \to 0^{+}$ in the strong topology.

Similar to the proof of ([77], Th.3.2) it is known that Hypothesis 5 holds if and only if the linear stochastic generalized system (12) is approximate controllable on $[0,a]$; one can prove that

$$\lambda {(\lambda I+Q_0^a)}^{-1}\to 0,$$

as $\lambda \to 0^{+}$ in the strong topology.

Proposition 1.

 (i) 

For $\forall {\xi }_a\in L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{D})$ there exists $k_{{\xi }_a}(·)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,L_Q(\mathbb{V},\mathbb{D}))$ such that

$$\mathbb{E}\left({\xi }_a\right|{\mathbb{F}}_t)=\mathbb{E}{\xi }_a+{\int }_0^t{k}_{{\xi }_a}\left(s\right)dw\left(s\right),$$

(15)

$$G_0^a{\xi }_a=Q_0^a\mathbb{E}{\xi }_a+{\int }_0^t{Q}_s^a{k}_{{\xi }_a}\left(s\right)dw\left(s\right),$$

(16)

$${(\lambda I+G_0^a)}^{-1}{\xi }_a={(\lambda I+Q_0^a)}^{-1}\mathbb{E}{\xi }_a+{\int }_0^t{(\lambda I+Q_s^a)}^{-1}{k}_{{\xi }_a}\left(s\right)dw\left(s\right),$$

(17)

$$G_0^t{E}^{\ast }{T}^{\ast }(a,t){(\lambda I+G_0^a)}^{-1}{\xi }_a=Q_0^t{E}^{\ast }{T}^{\ast }(a,t){(\lambda I+Q_0^a)}^{-1}\mathbb{E}{\xi }_a$$

$$+{\int }_0^t{Q}_r^t{E}^{\ast }{T}^{\ast }(a,t){(\lambda I+Q_r^a)}^{-1}{k}_{{\xi }_a}\left(r\right)dw\left(r\right),$$

(18)

 (ii) 

If $\alpha :[0,a]\times \mathbb{D}\to E\left(\mathbb{D}\right)$ satisfies Hypothesis 4 and $\xi (·)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{D}),$ then there exists $k_{\alpha }(·,\xi )\in L^2([0,a],\Omega ,{\mathbb{F}}_t,L_Q(\mathbb{V},\mathbb{D}))$ such that

$${\int }_0^{a}T(a,s)\alpha (s,\xi \left(s\right))ds=\mathbb{E}{\int }_0^{a}T(a,s)\alpha (s,\xi \left(s\right))ds+{\int }_0^a{k}_{\alpha }(s,\xi \left(s\right))dw\left(s\right),$$

(19)

and for all $\xi (·),x(·)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{D})$

$$\mathbb{E}{\int }_0^a\parallel k_{\alpha }(s,\xi )-k_{\alpha }{(s,x)\parallel }^{2}ds\le c_T^{2}la\mathbb{E}{\int }_0^a{\parallel \xi \left(s\right)-x\left(s\right)\parallel }^{2}ds,$$

(20)

$$\mathbb{E}{\int }_0^a{\parallel k_{\alpha }(s,\xi )\parallel }^2\le c_T^{2}l{a}^2.$$

(21)

Proof. (i) Similar to the proof of [27], we can obtain Formulas (15)–(17). Now we prove (18). In fact,

$$G_0^t{E}^{\ast }{T}^{\ast }(a,t){(\lambda I+G_0^a)}^{-1}{\xi }_a$$

$$={\int }_0^{t}T(t,s)N\left(s\right)N^{\ast }\left(s\right)T^{\ast }(t,s)E^{\ast }{T}^{\ast }(a,t)\mathbb{E}\left({(\lambda I+G_0^a)}^{-1}{\xi }_a\right|{\mathbb{F}}_s)ds$$

$$={\int }_0^{t}T(t,s)N\left(s\right)N^{\ast }\left(s\right)T^{\ast }(t,s)E^{\ast }{T}^{\ast }(a,t){(\lambda I+Q_0^a)}^{-1}\mathbb{E}{\xi }_{a}ds$$

$$+{\int }_0^{t}T(t,s)N\left(s\right)N^{\ast }\left(s\right)T^{\ast }(t,s)E^{\ast }{T}^{\ast }(a,t){\int }_0^s{(\lambda I+Q_r^a)}^{-1}{k}_{{\xi }_a}\left(r\right)dw\left(r\right)$$

$$=Q_0^t{E}^{\ast }{T}^{\ast }(a,t){(\lambda I+Q_0^a)}^{-1}\mathbb{E}{\xi }_a$$

$$+{\int }_0^t{\int }_r^{t}T(t,s)N\left(s\right)N^{\ast }\left(s\right)T^{\ast }(t,s)E^{\ast }{T}^{\ast }(a,t){(\lambda I+Q_r^a)}^{-1}{k}_{{\xi }_a}\left(r\right)dsdw\left(r\right)$$

$$=Q_0^t{E}^{\ast }{T}^{\ast }(a,t){(\lambda I+Q_0^a)}^{-1}\mathbb{E}{\xi }_a$$

$$+{\int }_0^t{Q}_r^t{E}^{\ast }{T}^{\ast }(a,t){(\lambda I+Q_r^a)}^{-1}{k}_{{\xi }_a}\left(r\right)dw\left(r\right).$$

Here, we used (17) and the stochastic Fubini theorem.

(ii) It is obvious that

$${\int }_0^{a}T(a,s)\alpha (s,\xi \left(s\right))ds\in L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{D}).$$

Therefore, (19) follows from (15). Formulas (20) and (21) follow from (19) and Hypothesis 4. □

The next proposition gives a formula for a control which steers the system from ${\xi }_0\in L^2(\Omega ,{\mathbb{F}}_0,\mathbb{P},\mathbb{D})$ to a neighborhood of ${\xi }_a\in L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{D}).$

Proposition 2.

There exists a unique optimal control ${\eta }_{\lambda }(·)$ such that

$${\eta }_{\lambda }\left(t\right)=N^{\ast }\left(t\right)T^{\ast }(a,t)\mathbb{E}[{(\lambda I+G_0^a)}^{-1}({\xi }_a-T(a,0)E{\xi }_0-{\int }_0^{a}T(a,s)\alpha \left(s\right)ds$$

$$-{\int }_0^{a}T(a,s)\beta \left(s\right)dw\left(s\right)\left)\right|{\mathbb{F}}_t]$$

(22)

and

$${\xi }_{\lambda }\left(a\right)-{\xi }_a=-\lambda {(\lambda I+Q_0^a)}^{-1}[\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0-\mathbb{E}{\int }_0^{a}T(a,s)\alpha \left(s\right)ds]$$

$$-\lambda {\int }_0^a{(\lambda I+Q_s^a)}^{-1}(k_{{\xi }_a}\left(s\right)-T(a,s)\beta \left(s\right)-k_{\alpha }\left(s\right))dw\left(s\right),$$

(23)

where

$${\int }_0^{a}T(a,s)\alpha \left(s\right)ds=\mathbb{E}{\int }_0^{a}T(a,s)\alpha \left(s\right)ds+{\int }_0^a{k}_{\alpha }\left(s\right)dw\left(s\right)$$

Proof. 

The problem of minimizing the function (13) has a unique solution ${\eta }_{\lambda }(·)\in L^2([0,a]$, $\Omega ,{\mathbb{F}}_t,\mathbb{U})$, which is completely characterized by the stochastic maximum principle (see [98,99] for details) and has the following form:

$${\eta }_{\lambda }\left(t\right)=-{\lambda }^{-1}{N}^{\ast }\left(t\right)T^{\ast }(a,t)\mathbb{E}({\xi }_{\lambda }\left(a\right)-{\xi }_a|{\mathbb{F}}_t).$$

(24)

Using this in (14), we have

$${\xi }_{\lambda }\left(a\right)=T(a,0)E{\xi }_0+{\int }_0^{a}T(a,s)N\left(s\right)[-\lambda N^{\ast }\left(s\right)T^{\ast }(a,s)\mathbb{E}({\xi }_{\lambda }\left(a\right)-{\xi }_a|{\mathbb{F}}_s)]ds$$

$$+{\int }_0^{a}T(a,s)\alpha \left(s\right)ds+{\int }_0^{a}T(a,s)\beta \left(s\right)dw\left(s\right)$$

$$=T(t,0)E{\xi }_0-{\lambda }^{-1}{G}_0^a({\xi }_{\lambda }\left(a\right)-{\xi }_a)$$

$$+{\int }_0^{a}T(a,s)\alpha \left(s\right)ds+{\int }_0^{a}T(a,s)\beta \left(s\right)dw\left(s\right).$$

This implies that

$$(\lambda I+G_0^a)({\xi }_{\lambda }\left(a\right)-{\xi }_a)=-\lambda ({\xi }_a-T(a,0)E{\xi }_0$$

$$-{\int }_0^{a}T(a,s)\alpha \left(s\right)ds-{\int }_0^{a}T(a,s)\beta \left(s\right)dw\left(s\right)).$$

Therefore,

$${\xi }_{\lambda }\left(a\right)-{\xi }_a=-\lambda {(\lambda I+G_0^a)}^{-1}({\xi }_a-T(a,0)E{\xi }_0$$

$$-{\int }_0^{a}T(a,s)\alpha \left(s\right)ds-{\int }_0^{a}T(a,s)\beta \left(s\right)dw\left(s\right)).$$

(25)

According to Proposition 1 and (25) we have

$${\xi }_{\lambda }\left(a\right)-{\xi }_a=-\lambda {(\lambda I+Q_0^a)}^{-1}(\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0-\mathbb{E}{\int }_0^{a}T(a,s)\alpha \left(s\right)ds)$$

$$-\lambda {\int }_0^a{(\lambda I+Q_s^a)}^{-1}(k_{{\xi }_a}\left(s\right)-k_{\alpha }\left(s\right)-T(a,s)\beta \left(s\right))dw\left(s\right).$$

Thus, (23) holds. Substituting (25) into (24), we have (22). □

Formulas (23) and (25) imply that the linear stochastic generalized system (14) is approximately controllable on $[0,a]$ if and only if $\lambda {(\lambda I+G_0^a)}^{-1}$ converges to zero operator as $\lambda \to 0^{+}$ in the strong operator topology.

2.3. The Approximate Controllability of Semilinear Stochastic Generalized System (1)

In this subsection, conditions are formulated under which the approximate controllability of the semilinear stochastic generalized system (1) is implied by the approximate controllability of its linear part. The investigation is carried through by the Nussbaum fixed-point theorem and GE-evolution operator.

Definition 3.

The semilinear stochastic generalized System (1) is called to be approximate controllable on the interval $[0,a]$ if

$$\overline{R(a,{\xi }_0,\eta )}=L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{D}),$$

where

$$R(a,{\xi }_0,\eta )=\{\xi (a,{\xi }_0,\eta ):\eta \in L^p([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{U})\},$$

$\xi (a,{\xi }_0,\eta )=\xi \left(a\right)$ is given by (3).

Define the control

$${\eta }_{\lambda }\left(t\right)=N^{\ast }\left(t\right)T^{\ast }(a,t)\mathbb{E}[{(\lambda I+G_0^a)}^{-1}({\xi }_a-T(a,0)E{\xi }_0-{\int }_0^{a}T(a,s)\alpha (s,\xi \left(s\right))ds$$

$$-{\int }_0^{a}T(a,s)\beta (s,\xi \left(s\right))dw\left(s\right)\left)\right|{\mathbb{F}}_t].$$

(26)

In order to formulate the controllability problem in the form suitable for application of the Nussbaum fixed-point theorem, we put the control ${\eta }_{\lambda }(·)$ into the semilinear stochastic generalized system (3) and obtain a nonlinear operator $f_{\lambda }:C_m\to C_m$:

$$\left(f_{\lambda }\xi \right)\left(t\right)=T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)\alpha (s,\xi \left(s\right))ds+{\int }_0^{t}T(t,s)\beta (s,\xi \left(s\right))dw\left(s\right)$$

$$+G_0^t{E}^{\ast }{T}^{\ast }(a,t){(\lambda I+G_0^a)}^{-1}({\xi }_a-T(a,0)E{\xi }_0-{\int }_0^{a}T(a,s)\alpha (s,\xi \left(s\right))ds$$

$$-{\int }_0^{a}T(a,s)\beta (s,\xi \left(s\right))dw\left(s\right)).$$

We shall prove that the semilinear stochastic generalized system (3) is approximate controllable if for all $\lambda >0$ there exists a fixed point of the operator $f_{\lambda }$. To show that $f_{\lambda }$ has a fixed point we employ the Nussbaum fixed-point theorem in $C_m.$

We define the operator $f_{\lambda }^1:C_m\to C_m;f_{\lambda }^2:C_m\to C([0,a],\mathbb{D})$ as follows

$$\left(f_{\lambda }^1\xi \right)\left(t\right)=T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)\alpha (s,\xi \left(s\right))ds+{\int }_0^{t}T(t,s)\beta (s,\xi \left(s\right))dw\left(s\right)$$

$$+{\int }_0^t{Q}_s^t{E}^{\ast }{T}^{\ast }(a,s){(\lambda I+Q_s^a)}^{-1}\times$$

$$(k_{{\xi }_a}\left(s\right)-T(a,s)\beta (s,\xi \left(s\right))-k_{\alpha }(s,\xi ))dw\left(s\right),$$

(27)

and

$$\left(f_{\lambda }^2\xi \right)\left(t\right)=Q_0^t{E}^{\ast }{T}^{\ast }(a,t){(\lambda I+Q_0^a)}^{-1}(\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0$$

$$-\mathbb{E}{\int }_0^{a}T(a,s)\alpha (s,\xi \left(s\right))ds),$$

(28)

where $k_{{\xi }_a}\left(s\right),k_{\alpha }(s,\xi )$ come from (15) and (19), respectively. By (18) from Proposition 1 (i) with $y_a$ as ${\xi }_a,$ where

$$y_a={\xi }_a-T(a,0)E{\xi }_0-{\int }_0^{a}T(t,s)\alpha (s,\xi \left(s\right))ds-{\int }_0^{t}T(t,s)\beta (s,\xi \left(s\right))dw\left(s\right),$$

we can see that

$$f_{\lambda }\xi =(f_{\lambda }^1+f_{\lambda }^2)\xi .$$

We are now ready to employ Nussbaum fixed-point theorem with

$${\mathbb{Y}}_c=\{\xi (·)\in C_m{:\mathbb{E}\parallel \xi \left(t\right)\parallel }^2\le c\},$$

where c is a positive constant. Next, for convenience, let us introduce the following notation

$$c_N=\parallel N\parallel ,c_{TN}=amax\{\parallel T(t,s)N\left(s\right)N^{\ast }\left(s\right)T^{\ast }(t,s)\parallel :0\le s\le t\le a\}.$$

Then, we have the following theorem.

Theorem 3.

Assume Hypotheses 1, 2, 4, and 5 are satisfied. Then, the semilinear stochastic generalized system (1) is approximate controllable on $[0,a].$

Proof. 

The proof of the theorem is divided into six steps.

Step 1. For arbitrary $\lambda >0$ there is a positive constant $c_0=c_0\left(\lambda \right)$ such that $f:{\mathbb{Y}}_{c_0}\to {\mathbb{Y}}_{c_0}$.

According to the definition of $f_{\lambda }^1$ and $f_{\lambda }^2$, for ∀$\xi (·)\in {\mathbb{Y}}_{c_0}$, we have

$$\parallel f_{\lambda }^1\xi \parallel \le c_T\parallel E\parallel \parallel {\xi }_0\parallel +a{c}_T{l}^{1/2}+c_T{\left(al\right)}^{1/2}+{\lambda }^{-1}{c}_{TN}{c}_T\parallel E\parallel ({\int }_0^a\mathbb{E}{\parallel k_{{\xi }_a}\left(s\right)\parallel }^{2}ds$$

$$+c_T^{2}la+{\int }_0^a\mathbb{E}\parallel k_{\alpha }(s,\xi ){{\parallel }^{2}ds)}^{1/2}$$

$$\le c_T\parallel E\parallel \parallel {\xi }_0\parallel +a{c}_T{l}^{1/2}+c_T{\left(al\right)}^{1/2}$$

$$+3{\lambda }^{-1}\parallel E\parallel c_{TN}{c}_T({\int }_0^a\mathbb{E}\parallel k_{{\xi }_a}\left(s\right){{\parallel }^{2}ds+a{c}_T^{2}l+c_T^2{a}^{2}l)}^{1/2};$$

$$\parallel f_{\lambda }^2\xi \parallel \le {\lambda }^{-1}\parallel E\parallel c_{TN}{c}_T(\parallel \mathbb{E}{\xi }_a\parallel +c_T\parallel E\parallel \parallel {\xi }_0\parallel +c_{T}a{l}^{1/2}),$$

where $\parallel \lambda {(\lambda I+Q_s^a)}^{-1}\parallel <1$ is used. This implies for sufficient large $c_0,$

$$\parallel f_{\lambda }\xi \parallel \le \parallel f_{\lambda }^1\xi \parallel +\parallel f_{\lambda }^2\xi \parallel \le c_0.$$

Hence, $f_{\lambda }$ maps ${\mathbb{Y}}_{c_0}$ into itself for some $c_0$.

Step 2. For $\forall \lambda >0$ the operator $f_{\lambda }^2$ maps ${\mathbb{Y}}_{c_0}$ into a relatively compact subset of ${\mathbb{Y}}_{c_0}$.

From the infinite-dimensional version of the Ascoli–Arzela theorem we have to show that

(i) for $\forall t\in [0,a]$ the set

$$v\left(t\right)=\{\left(f_{\lambda }^2\xi \right)\left(t\right):\xi \in {\mathbb{Y}}_{c_0}\}\subset \mathbb{D}$$

is relative compact.

(ii) for $\forall ϵ>0$ there exists $\delta >0$ such that

$$\parallel \left(f_{\lambda }^2\xi \right)(t+\tau )-\left(f_{\lambda }^2\xi \right)\left(t\right)\parallel <ϵ,$$

if $\parallel \xi \parallel \le c,\left|\tau \right|<\delta ,$ and $t,t+\tau \in [0,a]$. Notice that the uniform boundedness is proved in step 1.

Let us prove (i). In fact, if $t=0$, the proof is trivial, since $v\left(0\right)=\left\{{\xi }_0\right\}.$ So let $t,0<t\le a$, be fixed and let $0<\gamma <t$. Define

$$f_{\lambda ,\gamma }^2\xi \left(t\right)={\int }_0^{t-\gamma }T(t,s)N\left(s\right)N^{\ast }\left(s\right)T^{\ast }(a,s)ds{(\lambda I+Q_0^a)}^{-1}\times$$

$$(\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0-\mathbb{E}{\int }_0^{a}T(a,s)\alpha (s,\xi \left(s\right))ds)$$

$$=T(\gamma ,0)E\left(f_{\lambda }^2\left(\xi \right)\right)(t-\gamma ).$$

Since $T(\gamma ,0)$ is compact and $\left(f_{\lambda }^2\left(\xi \right)\right)(t-\gamma )$ is bounded on ${\mathbb{Y}}_{c_0}$ the set

$$v_{\gamma }\left(t\right)=\{\left(f_{\lambda ,\gamma }^2\xi \right)\left(t\right):\xi \in {\mathbb{Y}}_c\}\subset \mathbb{D}$$

is relative compact set in $\mathbb{D}$, that is, we can find a finite set $\{{\gamma }_i:1\le i\le n\}$ in $\mathbb{D}$ such that

$$v_{\gamma }\left(t\right)\subset {\cup }_{i=1}^{n}o({\gamma }_i,ϵ/2).$$

On the other hand, there exists $\gamma >0$ such that

$$\parallel \left(f_{\lambda }^2\left(\xi \right)\right)\left(t\right)-\left(f_{\lambda ,\gamma }^2\xi \right)\left(t\right)\parallel =\parallel {\int }_{t-\gamma }^{t}T(t,s)N\left(s\right)N^{\ast }\left(s\right)T^{\ast }(a,s)ds{(\lambda I+Q_0^a)}^{-1}\times$$

$$(\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0-\mathbb{E}{\int }_0^{a}T(a,s)\alpha (s,\xi \left(s\right))ds)\parallel$$

$$\le {\lambda }^{-1}{c}_T^2{c}_N^2(\parallel \mathbb{E}{\xi }_a\parallel +c_T\parallel E\parallel \parallel {\xi }_0\parallel +c_T{l}^{1/2}a)\gamma \le \frac{1}{2}ϵ,$$

where $\parallel \lambda {(\lambda I+Q_s^a)}^{-1}\parallel <1$ is used. Consequently

$$v\left(t\right)\subset {\cup }_{i=1}^{n}o({\gamma }_i,ϵ).$$

Therefore, for each $t\in [0,a]$, $v\left(t\right)$ is relatively compact in $\mathbb{D}$.

Next, we prove (ii), we have to prove that

$$v(·)=\{\left(f_{\lambda }^2\xi \right)(·):\xi \in {\mathbb{Y}}_{c_0}\}$$

is equicontinuous on $[0,a]$. In fact, for $0<t<t+\tau \le a$ and $0<\gamma \le t$

$$\parallel \left(f_{\lambda }^2\left(\xi \right)\right)(t+\tau )-\left(f_{\lambda }^2\xi \right)\left(t\right)\parallel \le \parallel Q_0^{t+\tau }{E}^{\ast }{T}^{\ast }(a-t,\tau )-Q_0^t{E}^{\ast }{T}^{\ast }(a-t,0)\parallel \times$$

$$\parallel {(\lambda I+Q_0^a)}^{-1}(\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0-\mathbb{E}{\int }_0^{a}T(a,s)\alpha (s,\xi \left(s\right))ds)\parallel$$

$$\le \parallel {\int }_t^{t+\tau }T(t+\tau ,s)N\left(s\right)N^{\ast }\left(s\right)T^{\ast }(a,s)ds$$

$$-(T(\tau ,0)E-I){\int }_0^{t}T(t,s)N\left(s\right)N^{\ast }\left(s\right)T^{\ast }(a,s)ds\parallel \times$$

$${\lambda }^{-1}(\parallel \mathbb{E}{\xi }_a\parallel +c_T\parallel E\parallel \parallel {\xi }_0\parallel +c_{T}a{l}^{1/2})$$

$$\le {\lambda }^{-1}(\tau +a\parallel T(\tau ,0)E-I\parallel )c_T^2{c}_N^2(\parallel \mathbb{E}{\xi }_a|+c_T\parallel E\parallel \parallel {\xi }_0\parallel +c_{T}a{l}^{1/2}),$$

(29)

where $\parallel \lambda {(\lambda I+Q_s^a)}^{-1}\parallel <1$ is used. The right-hand side of (29) does not depend on particular choice of $\xi$ and approaches zero as $\tau \to 0^{+}.$ The case $0<t+\tau <t\le a$ can be considered in a similar manner; therefore, we obtain the equicontinuity of $v(·)$. Thus, $f_{\lambda }^2$ maps ${\mathbb{Y}}_{c_0}$ into an equicontinuous family of deterministic functions, which are also bounded. According to the Ascoli–Arzela theorem $f_{\lambda }^2\left({\mathbb{Y}}_{c_0}\right)$ is relatively compact in $C_m$.

Step 3. $f_{\lambda }^1$ is a contraction mapping. In fact

$$\parallel f_{\lambda }^1\xi -f_{\lambda }^{1}x\parallel \le \parallel {\int }_0^{t}T(t,s)[\alpha (s,\xi \left(s\right))-\alpha (s,x\left(s\right))]ds\parallel$$

$$+\parallel {\int }_0^{t}T(t,s)[\beta (s,\xi \left(s\right))-\beta (s,x\left(s\right))]dw\left(s\right)\parallel$$

$$+c_T{c}_{TN}\parallel E\parallel \parallel {\int }_0^t{(\lambda I+Q_0^a)}^{-1}T(a,s)[\beta (s,\xi \left(s\right))-\beta (s,x\left(s\right))]dw\left(s\right)\parallel$$

$$+c_T{c}_{TN}\parallel E\parallel \parallel {\int }_0^t{(\lambda I+Q_0^a)}^{-1}T(a,s)[k_{\alpha }(s,\xi )-k_{\alpha }(s,x)]dw\left(s\right)\parallel$$

$$\le (c_{T}a{l}^{1/2}+c_T{\left(al\right)}^{1/2}+{\lambda }^{-1}{c}_{TN}{c}_T^2{\left(al\right)}^{1/2}\parallel E\parallel +{\lambda }^{-1}{c}_{TN}{c}_T^3{\left(al\right)}^{1/2}\parallel E\parallel )\parallel \xi -x\parallel .$$

Here, we used the inequality (20) and $\parallel \lambda {(\lambda I+Q_s^a)}^{-1}\parallel <1$. Therefore, if

$$c_{T}a{l}^{1/2}+c_T{\left(al\right)}^{1/2}+{\lambda }^{-1}{c}_{TN}{c}_T^2{\left(al\right)}^{1/2}\parallel E\parallel +{\lambda }^{-1}{c}_{TN}{c}_T^3{\left(al\right)}^{1/2}\parallel E\parallel <1,$$

(30)

$f_{\lambda }^1$ is a contraction mapping.

Step 4. $f_{\lambda }^2$ is continuous on $C([0,a],\mathbb{D})$. Here $C([0,a];\mathbb{D})$ denotes the set of all functions $f:[0,a]\to \mathbb{D}$, which is continuous on $[0,a]$ in the sense of ${\parallel f(·)\parallel }_{C([0,a];\mathbb{D})}=ma{x}_{t\in [0,a]}{\parallel f\left(t\right)\parallel }_{\mathbb{D}}$. To apply the Nussbaum fixed-point theorem, it remains to show that $f_{\lambda }^2$ is continuous on $C_m$. Let $\left\{{\xi }_n(·)\right\}\subset C_m$ with ${\xi }_n(·)\to \xi (·)$ in $C_m$. Then the Lebesgue-domain convergence theorem implies

$$\parallel f_{\lambda }^2{\xi }_n\left(t\right)-f_{\lambda }^2\xi \left(t\right)\parallel \le {\lambda }^{-1}{c}_{TN}{c}_T\parallel E\parallel \mathbb{E}{\int }_0^a\parallel T(a,s)[\alpha (s,{\xi }_n\left(s\right))-\alpha (s,\xi \left(s\right))]\parallel ds$$

$$\le {\lambda }^{-1}{c}_{TN}{c}_T^2{a}^{1/2}\parallel E\parallel ({\int }_0^a\mathbb{E}\parallel \alpha (s,{\xi }_n\left(s\right))-\alpha (s,\xi \left(s\right)){{\parallel }^{2}ds)}^{1/2}$$

$$\le {\lambda }^{-1}{c}_{TN}{c}_T^2{\left(la\right)}^{1/2}\parallel E\parallel ({\int }_0^a\mathbb{E}\parallel {\xi }_n\left(s\right)-\xi \left(s\right){{\parallel }^{2}ds)}^{1/2}$$

$$\le {\lambda }^{-1}{c}_{TN}{c}_T^2{\left(la\right)}^{1/2}\parallel E\parallel \parallel {\xi }_n-\xi \parallel \to 0,n\to \infty ,$$

where $\parallel \lambda {(\lambda I+Q_s^a)}^{-1}\parallel <1$ is used. Thus, $f_{\lambda }^2$ is continuous on $C_m$.

Step 5. $f_{\lambda }$ has a fixed point. From the Nussbaum fixed-point theorem $f_{\lambda }$ has a fixed point provided that the inequality (30) is satisfied. We can see that this fixed point is a mild solution of the semilinear stochastic generalized system (1). The extra condition (30) can easily be removed by considering semilinear stochastic generalized system (1) on interval $[0,b],[b,2b],\cdots ,$ with b satisfies (30).

Step 6. The semilinear stochastic generalized system (1) is approximately controllable. Let ${\xi }_{\lambda }^{\ast }(·)$ be a fixed point of the operator $f_{\lambda }$ in ${\mathbb{Y}}_{c_0}$. Any fixed point of $f_{\lambda }$ is a mild solution of semilinear stochastic generalized system (1) on $[0,a]$ under the control ${\eta }_{\lambda }\left(t\right)$ defined by (26), where $\xi$ is replaced by ${\xi }_{\lambda }^{\ast }$ and, by Proposition 2, satisfies

$$\left(f_{\lambda }{\xi }_{\lambda }^{\ast }\right)\left(a\right)={\xi }_{\lambda }^{\ast }\left(a\right)={\xi }_a+\lambda {(\lambda I+G_0^a)}^{-1}\times (T(a,0)E{\xi }_0+{\int }_0^{a}T(a,s)\alpha (s,{\xi }_{\lambda }^{\ast }\left(s\right))ds$$

$$+{\int }_0^{a}T(a,s)\beta (s,{\xi }_{\lambda }^{\ast }\left(s\right))dw\left(s\right)-{\xi }_a).$$

(31)

Let

$${\xi }_{\lambda }=T(a,0)E{\xi }_0+{\int }_0^{a}T(a,s)\alpha (s,{\xi }_{\lambda }^{\ast }\left(s\right))ds+{\int }_0^{a}T(a,s)\beta (s,{\xi }_{\lambda }^{\ast }\left(s\right))dw\left(s\right)-{\xi }_a.$$

By Hypothesis 4

$$\parallel \alpha (s,{\xi }_{\lambda }^{\ast }\left(s\right)){\parallel }^2+{\parallel \beta (s,{\xi }_{\lambda }^{\ast }\left(s\right))\parallel }^2\le l$$

in $[0,a]\times \Omega$. Then, there is a subsequence, still denoted by $\left\{(\alpha (s,{\xi }_{\lambda }^{\ast }\left(s\right)),\beta (s,{\xi }_{\lambda }^{\ast }\left(s\right)))\right\}$, weakly convergence to, say $\left(\alpha \right(s,\omega ),\beta (s,\omega \left)\right)$ in $E\left(\mathbb{D}\right)\times L_Q(\mathbb{V},E\left(\mathbb{D}\right))$. From the compactness of $T(t,s),t>s\ge 0$, we find that

$$T(a,s)\alpha (s,{\xi }_{\lambda }^{\ast }\left(s\right))\to T(a,s)\alpha \left(s\right),$$

$$T(a,s)\beta (s,{\xi }_{\lambda }^{\ast }\left(s\right))\to T(a,s)\beta \left(s\right),$$

a.e. in $[0,a]\times \Omega$. On the other hand

$$\parallel T(a,s)\alpha (s,{\xi }_{\lambda }^{\ast }\left(s\right)){\parallel }^2+{\parallel T(a,s)\beta (s,{\xi }_{\lambda }^{\ast }\left(s\right))\parallel }^2\le c_T^{2}l,$$

a.e. in $[0,a]\times \Omega$; therefore, by the Lebesgue-dominated convergence theorem

$$\mathbb{E}\parallel {\xi }_{\lambda }{-\xi \parallel }^2\to 0,\lambda \to 0^{+},$$

where

$$\xi =T(a,0)E{\xi }_0+{\int }_0^{a}T(a,s)\alpha \left(s\right)ds+{\int }_0^{a}T(a,s)\beta \left(s\right)dw\left(s\right)-{\xi }_a.$$

Since

$$\mathbb{E}\parallel \lambda {(\lambda I+G_0^a)}^{-1}{\parallel }^2\le 1$$

and

$$\lambda {(\lambda I+G_0^a)}^{-1}\to 0$$

strongly by Hypothesis 5, according to (31) we have

$$(\mathbb{E}\parallel {\xi }_{\lambda }^{\ast }-{\xi }_a{\parallel }^2{)}^{1/2}\le (\mathbb{E}\parallel \lambda {(\lambda I+G_0^a)}^{-1}({\xi }_{\lambda }-\xi ){\parallel }^2{)}^{1/2}+(\mathbb{E}\parallel \lambda {(\lambda I+G_0^a)}^{-1}\xi {{\parallel }^2)}^{1/2}$$

$$\le (\mathbb{E}\parallel {\xi }_{\lambda }{-\xi \parallel }^2{)}^{1/2}+(\mathbb{E}\parallel \lambda {(\lambda I+G_0^a)}^{-1}\xi {{\parallel }^2)}^{1/2}\to 0$$

as $\lambda \to 0^{+}$. This implies that semilinear stochastic generalized system (1) is approximate controllability. □

3. Approximate Controllability of the Semilinear Stochastic Generalized Systems with Control Acting on Both Drift and Diffusion Terms

In this section, we investigate the approximate controllability of the following semilinear stochastic generalized system with control acting on both drift and diffusion terms and with time-varying state and control operators by using compact GE-evolution operator theory in the sense of mild solution in Hilbert spaces:

$$Ed\xi \left(t\right)=[M(t\left)\xi \right(t)+N(t\left)\eta \right(t)+\alpha (t,\xi \left(t\right),\eta \left(t\right)\left)\right]dt+\beta (t,\xi (t),\eta (t\left)\right)dw\left(t\right),$$

$$t\in [0,a],\xi \left(0\right)={\xi }_0,$$

(32)

where $\xi \left(t\right)$ is the state process valued in Hilbert space $\mathbb{H}$, the input process $\eta \left(t\right)$ belong to $L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{U})$; the nonlinear operator $\alpha :[0,a]\times \mathbb{H}\times \mathbb{U}\to \mathbb{H}$ satisfies

$$su{p}_{(t,\xi ,\eta )\in [0,a]\times \mathbb{H}\times \mathbb{U}}\mathbb{E}\parallel \alpha (t,\xi ,\eta )\parallel <\infty ;$$

$\beta :[0,a]\times \mathbb{H}\times \mathbb{U}\to L_Q(\mathbb{V},\mathbb{H})$ satisfies ${\mathbb{E}\parallel \beta (t,\xi ,\eta )\parallel }^2<\infty .$

3.1. Preliminaries

For the operators $E,M\left(t\right),N\left(t\right),\alpha ,\beta$ we assume the following hypotheses:

Hypothesis 6.

$M\left(t\right):domM\left(t\right)\subseteq \mathbb{H}\to \mathbb{H}$ satisfies Hypothesis 1.

Hypothesis 7.

$N\left(t\right)\in P([0,a],L(\mathbb{U},\mathbb{H})),\parallel N\left(t\right)\parallel \le c_N,$

$${\xi }_0\in L^2(\Omega ,{\mathbb{F}}_0,\mathbb{P},\mathbb{D}),\xi \left(t\right)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{D});$$

for $\eta \in L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{U}),N\left(t\right)\eta \left(t\right)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,E\left(\mathbb{D}\right))$.

Hypothesis 8.

The nonlinear operators $\alpha :[0,a]\times \mathbb{D}\times \mathbb{U}\to E\left(\mathbb{D}\right),\beta :[0,a]\times \mathbb{D}\times \mathbb{U}\to L_Q(\mathbb{V},E\left(\mathbb{D}\right))$ are measurable and satisfy the Lipschitz condition and linear growth condition, i.e.,

$$\parallel \alpha (t,{\xi }_1,{\eta }_1)-\alpha (t,{\xi }_2,{\eta }_2){\parallel }^2+{\parallel \beta (t,{\xi }_1,{\eta }_1)-\beta (t,{\xi }_2,{\eta }_2)\parallel }^2$$

$$\le l_1(\parallel {\xi }_1-{\xi }_2{\parallel }^2+\parallel {\eta }_1-{\eta }_2{\parallel }^2),$$

$${\parallel \alpha (t,\xi ,\eta )\parallel }^2+{\parallel \beta (t,\xi ,\eta )\parallel }^2\le l_2{(1+\parallel \xi \parallel }^2+{\parallel \eta \parallel }^2).$$

The corresponding deterministic linear generalized system of semilinear stochastic generalized system (32) is defined as

$$Ed\xi \left(t\right)=[M\left(t\right)\xi \left(t\right)+N\left(t\right)\eta \left(t\right)]dt,t\in [0,a],\xi \left(0\right)={\xi }_0,$$

(33)

admits a unique mild solution given by

$$\xi \left(t\right)=T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)N\left(s\right)\eta \left(s\right)ds,t\in [0,a].$$

Definition 4.

The deterministic linear generalized system (33) is said to be approximate controllable on $[0,a]$ if for any ${\xi }_0,{\xi }_a\in \mathbb{D},ϵ>0$, there exists $\eta \in L^2([0,a],\mathbb{U})$ such that the mild solution $\xi \left(t\right)$ of deterministic time varying linear generalized system (33) corresponding η satisfy $\xi \left(0\right)={\xi }_0,\parallel \xi \left(a\right)-{\xi }_a\parallel <ϵ.$

Here $L^2([0,a],\mathbb{U})$ denotes the Hilbert space of all functions $f:[0,a]\to \mathbb{U}$ satisfying

$${\parallel f(·)\parallel }_{L^2([0,a],\mathbb{U})}=({\int }_0^a{\parallel f(·)\parallel }_{\mathbb{U}}{)}^{1/2}<+\infty .$$

In order to obtain the main results, we need the following hypothesis.

Hypothesis 9.

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

Similar to the proof of [77], we can obtain that the deterministic linear generalized system (33) corresponding to semilinear stochastic generalized system (32) is approximate controllable on $[0,a]$ if and only if Hypothesis 9 holds true.

Definition 5.

A stochastic process $\xi \in C_m=C_m([0,a],L^2(\Omega ,{\mathbb{F}}_t,\mathbb{P},\mathbb{D}))$ is a mild solution of semilinear stochastic generalized system (32) if for each $\eta \in L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{U})$, it satisfies the following integral equation:

$$\xi \left(t\right)=T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)N\left(s\right)\eta \left(s\right)ds+{\int }_0^{t}T(t,s)\alpha (s,\xi \left(s\right),\eta \left(s\right))ds$$

$$+{\int }_0^{t}T(t,s)\beta (s,\xi \left(s\right),\eta \left(s\right))dw\left(s\right).$$

(34)

Let $\xi (a,{\xi }_0,\eta )=\xi \left(a\right)$ is given by (34). The set

$$R(a,{\xi }_0,\eta )=\{\xi (a,{\xi }_0,\eta ):\eta \in L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{U})\}$$

is called the reachable set of semilinear stochastic generalized system (32).

Definition 6.

The semilinear stochastic generalized system (32) is called to be approximate controllable on the interval $[0,a]$ if

$$\overline{R(a,{\xi }_0,\eta )}=L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{D}).$$

Proposition 3.([32]) For any ${\xi }_a$ in $L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{D})$ there exists

$$\psi (·)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,L_Q(\mathbb{V},\mathbb{D}))$$

such that

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

According to Proposition 3, for any $\lambda >0,{\xi }_a\in L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{D}),$ we define the input function

$${\eta }_{\lambda }(t,\xi )=N^{\ast }\left(t\right)T^{\ast }(a,t){(\lambda I+Q_0^a)}^{-1}(\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0)$$

$$+N^{\ast }\left(t\right)T^{\ast }(a,t){\int }_0^t{(\lambda I+Q_s^a)}^{-1}\psi \left(s\right)dw\left(s\right)$$

$$-N^{\ast }\left(t\right)T^{\ast }(a,t){\int }_0^t{(\lambda I+Q_s^a)}^{-1}T(a,s)\alpha (s,\xi \left(s\right),\eta \left(s\right))ds$$

$$-N^{\ast }\left(t\right)T^{\ast }(a,t){\int }_0^t{(\lambda I+Q_s^a)}^{-1}T(a,s)\beta (s,\xi \left(s\right),\eta \left(s\right))dw\left(s\right).$$

(35)

Proposition 4.

There exist positive constants $p_1,p_2,p_3$ such that for all ${\xi }_1,{\xi }_2\in C_m$, we have

$$\mathbb{E}\parallel {\eta }_{\lambda }(t,{\xi }_1)-{\eta }_{\lambda }(t,{\xi }_2){\parallel }^2\le {\lambda }^{-2}{p}_1{\int }_0^t\mathbb{E}{\parallel {\xi }_1\left(s\right)-{\xi }_2\left(s\right)\parallel }^{2}ds,$$

$$\mathbb{E}\parallel {\eta }_{\lambda }(t,{\xi }_1){\parallel }^2\le {\lambda }^{-2}{p}_2+{\lambda }^{-2}{p}_3{\int }_0^t(1+\mathbb{E}\parallel {\xi }_1\left(s\right){\parallel }^2+\mathbb{E}\parallel \eta \left(s\right){\parallel }^2)ds$$

Proof. 

Let ${\xi }_1,{\xi }_2\in C_m$. According to Proposition 3 and Hypothesis 8, we have

$$\mathbb{E}\parallel {\eta }_{\lambda }(t,{\xi }_1)-{\eta }_{\lambda }(t,{\xi }_2){\parallel }^2=\mathbb{E}\parallel N^{\ast }\left(t\right)T^{\ast }(a,t)[{\int }_0^t{(\lambda I+Q_s^a)}^{-1}T(a,s)(\alpha (s,{\xi }_1\left(s\right),\eta \left(s\right))$$

$$-\alpha (s,{\xi }_2\left(s\right),\eta \left(s\right)))ds+{\int }_0^t{(\lambda I+Q_s^a)}^{-1}T(a,s)(\beta (s,{\xi }_1\left(s\right),\eta \left(s\right))$$

$$-\beta (s,{\xi }_2\left(s\right),\eta \left(s\right)){\left)dw\left(s\right)\right]\parallel }^2$$

$$\le 2{\lambda }^{-2}{c}_T^2{c}_N^2[c_T^{2}a{\int }_0^t\mathbb{E}{\parallel \alpha (s,{\xi }_1\left(s\right),\eta \left(s\right))-\alpha (s,{\xi }_2\left(s\right),\eta \left(s\right))\parallel }^{2}ds$$

$$+c_T^{2}tr\left(Q\right){\int }_0^t\mathbb{E}\parallel \beta (s,{\xi }_1\left(s\right),\eta \left(s\right))-\beta (s,{\xi }_2\left(s\right),\eta \left(s\right)){)\parallel }^{2}ds]$$

$$\le 2{\lambda }^{-2}{c}_T^4{c}_N^2(a+tr\left(Q\right))l_1{\int }_0^t\mathbb{E}{\parallel {\xi }_1\left(s\right)-{\xi }_2\left(s\right)\parallel }^{2}ds$$

$$\le {\lambda }^{-2}{p}_1{\int }_0^t\mathbb{E}{\parallel {\xi }_1\left(s\right)-{\xi }_2\left(s\right)\parallel }^{2}ds,$$

where $p_1=2c_T^4{c}_N^2(a+tr\left(Q\right))l_1,$ and $\parallel \lambda {(\lambda I+Q_s^a)}^{-1}\parallel <1$ is used. Similarly

$$\mathbb{E}\parallel {\eta }_{\lambda }(t,{\xi }_1){\parallel }^2\le {\lambda }^{-2}{p}_2+{\lambda }^{-2}{p}_3{\int }_0^t(1+\mathbb{E}\parallel {\xi }_1\left(s\right){\parallel }^2+\mathbb{E}\parallel \eta \left(s\right){\parallel }^2)ds,$$

where $p_2=4c_T^2{c}_N^2{(\parallel \xi \parallel }^2+c_T^2{\parallel E\parallel }^2\parallel {\xi }_0{\parallel }^2),p_3=4c_T^4{c}_N^2(a+tr\left(Q\right))l_2$. □

3.2. Approximate Controllability of Semilinear Stochastic Generalized System (32)

For every $\lambda >0$, we define the operator $O_{C_m}:C_m\to C_m$ by

$$\left(O_{C_m}\xi \right)\left(t\right)=T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)N\left(s\right){\eta }_{\lambda }(s,\xi \left(s\right))ds+{\int }_0^{t}T(t,s)\alpha (s,\xi \left(s\right),\eta \left(s\right))ds$$

$$+{\int }_0^{t}T(t,s)\beta (s,\xi \left(s\right),\eta \left(s\right))dw\left(s\right).$$

(36)

In order to prove the approximate controllability of semilinear stochastic generalized system (32), first of all we prove existence of a fixed point of the operator $O_{C_m}$ as above by using the contraction mapping principle. The existence of mild solution to semilinear stochastic generalized system (32) is a natural premise to carry out the study of approximate controllability.

Theorem 4.

Assume that Hypotheses 6–9 hold true. If

$$[3a^2{c}_T^2{c}_N^2{\lambda }^{-2}{p}_1+3(a+tr\left(Q\right))c_T^2{l}_1]a<1,$$

(37)

then the operator $O_{C_m}$ has a fixed point in $C_m$.

Proof. 

First of all we prove that operator $O_{C_m}$ maps $C_m$ to itself. Let $\xi \in C_m$, from the Hypotheses 6–8, Holder and Minkowski inequalities, we have

$$\mathbb{E}\parallel \left(O_{C_m}\xi \right)\left(t\right){\parallel }^2=\mathbb{E}\parallel T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)N\left(s\right){\eta }_{\lambda }(s,\xi \left(s\right))ds$$

$$+{\int }_0^{t}T(t,s)\alpha (s,\xi \left(s\right),\eta \left(s\right))ds$$

$$+{\int }_0^t{T(t,s)\beta (s,\xi \left(s\right),\eta \left(s\right))dw\left(s\right)\parallel }^2$$

$$\le 4c_T^2{\parallel E\parallel }^2\parallel \parallel {\xi }_0{\parallel }^2+4a{c}_T^2{c}_N^2{\int }_0^t\mathbb{E}{\parallel {\eta }_{\lambda }(s,\xi \left(s\right))\parallel }^{2}ds$$

$$+4c_T^{2}a{\int }_0^t\mathbb{E}{\parallel \alpha (s,\xi \left(s\right),\eta \left(s\right))\parallel }^{2}ds$$

$$+4c_T^{2}tr\left(Q\right){\int }_0^t\mathbb{E}{\parallel \beta (s,\xi \left(s\right),\eta \left(s\right))\parallel }^{2}ds$$

$$\le 4c_T^2{\parallel E\parallel }^2\parallel \parallel {\xi }_0{\parallel }^2+4a{c}_T^2{c}_N^{2}a(p_2{\lambda }^{-2}+p_3{\lambda }^{-2}{a(1+\mathbb{E}\parallel \xi \parallel }^2+{\mathbb{E}\parallel \eta \parallel }^2\left)\right)$$

$$+4c_T^2(a+tr\left(Q\right))l_2{a(1+\mathbb{E}\parallel \xi \parallel }^2+{\mathbb{E}\parallel \eta \parallel }^2)<\infty .$$

We can obtain that

$$\mathbb{E}\parallel \left(O_{C_m}\xi \right)\left(t\right){\parallel }^2<\infty ,$$

i.e., $\left(O_{C_m}\xi \right)\left(t\right)\in C_m$, for each $\xi \left(t\right)\in C_m$; therefore, $O_{C_m}$ is self map. Next, we prove that $O_{C_m}$ is a contract mapping on $C_m$. For any ${\xi }_1,{\xi }_2\in C_m$,

$$\mathbb{E}\parallel \left(O_{C_m}{\xi }_1\right)\left(t\right)-\left(O_{C_m}{\xi }_2\right)\left(t\right){\parallel }^2=\mathbb{E}\parallel {\int }_0^{t}T(t,s)N\left(s\right)({\eta }_{\lambda }(s,{\xi }_1\left(s\right))-{\eta }_{\lambda }(s,{\xi }_2\left(s\right)))ds$$

$$+{\int }_0^{t}T(t,s)(\alpha (s,{\xi }_1\left(s\right),\eta \left(s\right))-\alpha (s,{\xi }_2\left(s\right),\eta \left(s\right)))ds$$

$$+{\int }_0^{t}T(t,s)(\beta (s,{\xi }_1\left(s\right),\eta \left(s\right))-\beta (s,{\xi }_2\left(s\right),\eta \left(s\right))){dw\left(s\right)\parallel }^2$$

$$\le 3a{c}_T^2{c}_N^2{\int }_0^t\mathbb{E}{\parallel {\eta }_{\lambda }(s,{\xi }_1\left(s\right))-{\eta }_{\lambda }(s,{\xi }_2\left(s\right))\parallel }^{2}ds$$

$$+3a{c}_T^2{\int }_0^t\mathbb{E}{\parallel \alpha (s,{\xi }_1\left(s\right),\eta \left(s\right))-\alpha (s,{\xi }_2\left(s\right),\eta \left(s\right))\parallel }^{2}ds$$

$$+3c_T^{2}tr\left(Q\right){\int }_0^t\mathbb{E}{\parallel \beta (s,{\xi }_1\left(s\right),\eta \left(s\right))-\beta (s,{\xi }_2\left(s\right),\eta \left(s\right))\parallel }^{2}ds$$

$$\le 3a^2{c}_T^2{c}_N^2{p}_1{\lambda }^{-2}{\int }_0^t\mathbb{E}{\parallel {\xi }_1\left(s\right)-{\xi }_2\left(s\right)\parallel }^{2}ds$$

$$+3(a+tr\left(Q\right))l_1{c}_T^2{\int }_0^t\mathbb{E}{\parallel {\xi }_1\left(s\right)-{\xi }_2\left(s\right)\parallel }^{2}ds$$

$$\le [3a^2{c}_T^2{c}_N^2{p}_1{\lambda }^{-2}+3(a+tr\left(Q\right))l_1{c}_T^2]asu{p}_{s\in [0,a]}\mathbb{E}{\parallel {\xi }_1\left(s\right)-{\xi }_2\left(s\right)\parallel }^2,$$

where $\parallel \lambda {(\lambda I+Q_s^a)}^{-1}\parallel <1$ is used. Then we can easily conclude that if (37) is satisfied, $O_{C_m}$ is a contraction map on $C_m$. Hence, according to the contraction mapping principle, $O_{C_m}$ has a unique fixed point on $C_m$. □

The following proposition gives a formula for a control steering the state ${\xi }_0$ to a neighborhood of ${\xi }_a\in L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{D})$.

Proposition 5.

For any ${\xi }_a\in L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{D})$, the control ${\eta }_{\lambda }(t,\xi )$ in (35) transfers the nonlinear stochastic generalized system (36) from ${\xi }_0$ to some neighborhood of ${\xi }_a$ at time a and

$${\xi }_{\lambda }\left(a\right)={\xi }_a-\lambda {(\lambda I+Q_0^a)}^{-1}[\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0]$$

$$+\lambda {\int }_0^a{(\lambda I+Q_s^a)}^{-1}T(a,s)\alpha (s,\xi \left(s\right),\eta \left(s\right))ds$$

$$+\lambda {\int }_0^a{(\lambda I+Q_s^a)}^{-1}(T(a,s)\beta (s,\xi \left(s\right),\eta \left(s\right))-\psi \left(s\right))dw\left(s\right).$$

Proof. 

By substituting (35) in (36), we can obtain that

$${\xi }_{\lambda }\left(t\right)=T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)N\left(s\right)[N^{\ast }\left(s\right)T^{\ast }(a,s){(\lambda I+Q_0^a)}^{-1}(\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0)$$

$$+N^{\ast }\left(s\right)T^{\ast }(a,s){\int }_0^s{(\lambda I+Q_r^a)}^{-1}\psi \left(r\right)dw\left(r\right)$$

$$-N^{\ast }\left(s\right)T^{\ast }(a,s){\int }_0^s{(\lambda I+Q_r^a)}^{-1}T(a,r)\alpha (r,\xi \left(r\right),\eta \left(r\right))dr$$

$$-N^{\ast }\left(s\right)T^{\ast }(a,s){\int }_0^s{(\lambda I+Q_r^a)}^{-1}T(a,r)\beta (r,\xi \left(r\right),\eta \left(r\right))dw\left(r\right)]ds$$

$$+{\int }_0^{t}T(t,s)\alpha (s,\xi \left(s\right),\eta \left(s\right))ds+{\int }_0^{t}T(t,s)\beta (s,\xi \left(s\right),\eta \left(s\right))dw\left(s\right)$$

$$=T(t,0)E{\xi }_0+{\int }_0^{t}T(t,s)\alpha (s,\xi \left(s\right),\eta \left(s\right))ds+{\int }_0^{t}T(t,s)\beta (s,\xi \left(s\right),\eta \left(s\right))dw\left(s\right)$$

$$+Q_0^t{E}^{\ast }{T}^{\ast }(a,t){(\lambda I+Q_0^a)}^{-1}(\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0)$$

$$-{\int }_0^t{Q}_s^t{E}^{\ast }{T}^{\ast }(a,t){(\lambda I+Q_s^a)}^{-1}T(a,s)\alpha (s,\xi \left(s\right),\eta \left(s\right))ds$$

$$-{\int }_0^t{Q}_s^t{E}^{\ast }{T}^{\ast }(a,t){(\lambda I+Q_s^a)}^{-1}(T(a,s)\beta (s,\xi \left(s\right),\eta \left(s\right))-\psi \left(s\right))dw\left(s\right).$$

Let $t=a$. Then

$${\xi }_{\lambda }\left(a\right)=T(a,0)E{\xi }_0+{\int }_0^{a}T(a,s)\alpha (s,\xi \left(s\right),\eta \left(s\right))ds+{\int }_0^{a}T(a,s)\beta (s,\xi \left(s\right),\eta \left(s\right))dw\left(s\right)$$

$$+(-\lambda I+\lambda I+Q_0^a){(\lambda I+Q_0^a)}^{-1}(\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0)$$

$$-{\int }_0^a(-\lambda I+\lambda I+Q_s^a){(\lambda I+Q_s^a)}^{-1}T(a,s)\alpha (s,\xi \left(s\right),\eta \left(s\right))ds$$

$$-{\int }_0^a(-\lambda I+\lambda I+Q_s^a){(\lambda I+Q_s^a)}^{-1}(T(a,s)\beta (s,\xi \left(s\right),\eta \left(s\right))-\psi \left(s\right))dw\left(s\right).$$

Therefore, we have that

$${\xi }_{\lambda }\left(a\right)={\xi }_a-\lambda {(\lambda I+Q_0^a)}^{-1}[\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0]$$

$$+\lambda {\int }_0^a{(\lambda I+Q_s^a)}^{-1}T(a,s)\alpha (s,\xi \left(s\right),\eta \left(s\right))ds$$

$$+\lambda {\int }_0^a{(\lambda I+Q_s^a)}^{-1}(T(a,s)\beta (s,\xi \left(s\right),\eta \left(s\right))-\psi \left(s\right))dw\left(s\right),$$

where Proposition 3 is used. □

Theorem 5.

Suppose that Hypotheses 6–9 and Theorem 4 hold. If $\alpha ,\beta$ are uniformly bounded, then semilinear stochastic generalized system (32) is approximately controllable on $[0,a]$.

Proof. 

By Theorem 4, $O_{C_m}$ has a unique fixed point ${\xi }_{\lambda }\in C_m.$ According to the Fubini Theorem and Proposition 5, we can easily see that

$${\xi }_{\lambda }\left(a\right)={\xi }_a-\lambda {(\lambda I+Q_0^a)}^{-1}[\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0]$$

$$+\lambda {\int }_0^a{(\lambda I+Q_s^a)}^{-1}T(a,s)\alpha (s,{\xi }_{\lambda }\left(s\right),{\eta }_{\lambda }(s,{\xi }_{\lambda }\left(s\right)))ds$$

$$+\lambda {\int }_0^a{(\lambda I+Q_s^a)}^{-1}(T(a,s)\beta (s,{\xi }_{\lambda }\left(s\right),{\eta }_{\lambda }(s,{\xi }_{\lambda }\left(s\right)))-\psi \left(s\right))dw\left(s\right).$$

From the property of $\alpha$ and $\beta$, we have

$$\mathbb{E}\parallel \alpha (s,{\xi }_{\lambda }\left(s\right),{\eta }_{\lambda }(s,{\xi }_{\lambda }\left(s\right))){\parallel }^2+\mathbb{E}{\parallel \beta (s,{\xi }_{\lambda }\left(s\right),{\eta }_{\lambda }(s,{\xi }_{\lambda }\left(s\right)))\parallel }^2\le l_3.$$

Therefore, there exists a subsequence, still denoted by

$$\{\alpha (s,{\xi }_{\lambda }\left(s\right),{\eta }_{\lambda }(s,{\xi }_{\lambda }\left(s\right))),\beta (s,{\xi }_{\lambda }\left(s\right),{\eta }_{\lambda }(s,{\xi }_{\lambda }\left(s\right)))\},$$

which converges weakly to $\left\{\alpha \right(s),\beta (s\left)\right\}$ in $E\left(\mathbb{D}\right)\times L_Q(\mathbb{V},E\left(\mathbb{D}\right))$. On the other hand, by Hypotheses 6 and 9, $\lambda {(\lambda I+Q_s^a)}^{-1}\to 0$ strongly as $\lambda \to 0^{+}$ and $\parallel \lambda {(\lambda I+Q_s^a)}^{-1}\parallel \le 1$ together with the Lebesgue-dominated convergence theorem, we have

$$\mathbb{E}\parallel {\xi }_{\lambda }\left(a\right)-{\xi }_a{\parallel }^2\le \mathbb{E}[\parallel \lambda {(\lambda I+Q_0^a)}^{-1}(\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0)\parallel$$

$$+\parallel {\int }_0^a\lambda {(\lambda I+Q_s^a)}^{-1}T(a,s)(\alpha (s,{\xi }_{\lambda }\left(s\right),{\eta }_{\lambda }(s,{\xi }_{\lambda }\left(s\right)))-\alpha \left(s\right))ds\parallel$$

$$+\parallel {\int }_0^a\lambda {(\lambda I+Q_s^a)}^{-1}T(a,s)\alpha \left(s\right)ds\parallel$$

$$+\parallel {\int }_0^a\lambda {(\lambda I+Q_s^a)}^{-1}T(a,s)(\beta (s,{\xi }_{\lambda }\left(s\right),{\eta }_{\lambda }(s,{\xi }_{\lambda }\left(s\right)))-\beta \left(s\right))dw\left(s\right)\parallel$$

$$+\parallel {\int }_0^a\lambda {(\lambda I+Q_s^a)}^{-1}T(a,s)\beta \left(s\right)dw\left(s\right)\parallel$$

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

$$\le 6\mathbb{E}\parallel \lambda {(\lambda I+Q_0^a)}^{-1}[\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0]{\parallel }^2$$

$$+6a\mathbb{E}{\int }_0^a{\parallel \lambda {(\lambda I+Q_s^a)}^{-1}T(a,s)(\alpha (s,{\xi }_{\lambda }\left(s\right),{\eta }_{\lambda }(s,{\xi }_{\lambda }\left(s\right)))-\alpha \left(s\right))\parallel }^{2}ds$$

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

$$+6{\int }_0^a\mathbb{E}{\parallel \lambda {(\lambda I+Q_s^a)}^{-1}T(a,s)(\beta (s,{\xi }_{\lambda }\left(s\right),{\eta }_{\lambda }(s,{\xi }_{\lambda }\left(s\right)))-\beta \left(s\right))\parallel }^{2}ds$$

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

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

$$\le 6\mathbb{E}\parallel \lambda {(\lambda I+Q_0^a)}^{-1}[\mathbb{E}{\xi }_a-T(a,0)E{\xi }_0]{\parallel }^2$$

$$+6a\mathbb{E}{\int }_0^a{\parallel T(a,s)(\alpha (s,{\xi }_{\lambda }\left(s\right),{\eta }_{\lambda }(s,{\xi }_{\lambda }\left(s\right)))-\alpha \left(s\right))\parallel }^{2}ds$$

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

$$+6{\int }_0^a\mathbb{E}{\parallel T(a,s)(\beta (s,{\xi }_{\lambda }\left(s\right),{\eta }_{\lambda }(s,{\xi }_{\lambda }\left(s\right)))-\beta \left(s\right))\parallel }^{2}ds$$

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

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

as $\lambda \to 0^{+}$, which implies that semilinear stochastic generalized system (32) is approximately controllable on $[0,a]$. □

3.3. Approximate Controllability of Linear Case

In this subsection, we investigate the approximate controllability for a class of linear stochastic generalized system up to whole space in the sense of impulse solution.

Consider the following linear stochastic generalized system:

$$E_{1}d\xi \left(t\right)=[M_1\left(t\right)\xi \left(t\right)+N_1\left(t\right)\eta \left(t\right)]dt+M_{21}\left(t\right)\xi \left(t\right)dw\left(t\right),$$

$$t\in [0,a],\xi \left(0\right)={\xi }_0,$$

(38)

$$0=\sigma \left(t\right)+N_2\left(t\right)\eta \left(t\right),t\in [0,a],\sigma \left(0\right)={\sigma }_0.$$

(39)

Let

$$E=\left[\begin{array}{cc}{E}_1& 0\\ 0& 0\end{array}\right],M\left(t\right)=\left[\begin{array}{cc}{M}_1\left(t\right)& 0\\ 0& I_2\end{array}\right],N\left(t\right)=\left[\begin{array}{c}{N}_1\left(t\right)\\ N_2\left(t\right)\end{array}\right],$$

$$M_2\left(t\right)=\left[\begin{array}{cc}{M}_{21}\left(t\right)& 0\\ 0& 0\end{array}\right],$$

$\mathbb{H}=\mathbb{D}\oplus {\mathbb{H}}_2,x\left(t\right)=\left[\begin{array}{c}\xi \left(t\right)\\ \sigma \left(t\right)\end{array}\right]\in \mathbb{H},I_2$ is an identical operator on ${\mathbb{H}}_2$. Then, we can obtain the following linear stochastic generalized system:

$$Edx\left(t\right)=[M\left(t\right)x\left(t\right)+N\left(t\right)\eta \left(t\right)]dt+M_2\left(t\right)\xi \left(t\right)dw\left(t\right),$$

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

(40)

Here, $E,M\left(t\right),N\left(t\right)$ satisfies Hypotheses 6 and 7, respectively.

$$M_2\left(t\right)\in P([0,a],L(\mathbb{D},E\left(\mathbb{D}\right))),$$

$w\left(t\right)$ is a one-dimensional standard Wiener process, and (40) satisfies the condition of ([79], Th.5).

Definition 7.

If $\xi \left(t\right)$ is the mild solution of linear stochastic generalized system (38), $\sigma \left(t\right)=-N_2\left(t\right)\eta \left(t\right)$, then $x\left(t\right)=\left[\begin{array}{c}\xi \left(t\right)\\ \sigma \left(t\right)\end{array}\right]$ is called to be the impulse solution of linear stochastic generalized system (40).

Linear stochastic generalized system (40) is called to be approximately controllable on [0,a] up to $L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{H})$ if, for any state $x_a\in L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{H})$ and any $ϵ>0$, there exists a control $\eta \in L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{U})$ such that the impulse solution of linear stochastic generalized system (40) satisfy $(\mathbb{E}\parallel x\left(a\right)-x_a{{\parallel }^2)}^{1/2}<ϵ.$

Similar to the proof of ([89], Th.3), we can obtain the following theorem.

Theorem 6.

Subsystem (39) is approximately controllable on [0,a] up to $L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},{\mathbb{H}}_2)$ if and only if $N_2\left(a\right):L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{U})\to L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},{\mathbb{H}}_2)$ satisfies $\overline{ran{N}_2\left(a\right)}=L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},{\mathbb{H}}_2).$

Similar to the proof ideas of ([79], Th.50) and ([89], Th.4), we can obtain the following theorem.

Theorem 7.

Linear stochastic generalized system (40) is approximately controllable on $[0,a]$ up to $L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{H})$ if and only if linear stochastic system (38) and system (39) are approximately controllable on [0,a] up to $L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},{\mathbb{H}}_1)$ and $L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},{\mathbb{H}}_2)$, respectively.

4. Application Examples

In order to study the controllability of the actual system by mathematical method, the mathematical model describing the actual system must be established first. Many practical system can be described by stochastic generalized systems, and their controllability can be reduced to the controllability of stochastic generalized systems. The application of controllability of stochastic linear system in the sense of the mild solution has been discussed in detail in reference [79] by the author. In the following, three application examples are given to illustrate the effectiveness of the theoretical results obtained in this paper, in which mathematical models have been established.

Example 1.

Consider the semilinear generalized heat equation, which comes from the temperature distribution in the composite material:

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

$$+\left[\begin{array}{c}{\alpha }_1(t,{\xi }_1(t,\tau ))\\ 0\end{array}\right]dt+\left[\begin{array}{c}1\\ {\xi }_2(t,\tau )\end{array}\right]dw\left(t\right),$$

$${\xi }_1(t,0)={\xi }_1(t,\pi )=0,0\le t\le a,0<\tau <\pi ;{\xi }_2(t,0)={\xi }_2(t,\pi )=0.$$

(41)

Let

$${\mathbb{H}}_1=L^2(0,\pi ),\mathbb{H}={\mathbb{H}}_1\oplus {\mathbb{H}}_1,E=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],$$

$M\left(t\right):domM\left(t\right)\subseteq \mathbb{H}\to \mathbb{H}$ be an operator defined by

$$M\left(t\right)=\left[\begin{array}{cc}{M}_1& 0\\ 0& 1+t^2\end{array}\right],M_1\xi =\frac{d^2\xi }{d{\tau }^2},$$

with domain $dom{M}_1=\{\xi \in {\mathbb{H}}_1,\xi ,\frac{d\xi }{d\tau }$ are absolutely continuous, $\frac{d^2\xi }{d{\tau }^2}\in {\mathbb{H}}_1,\xi \left(0\right)=\xi \left(\pi \right)=0\}$. Then $\mathbb{D}={\mathbb{H}}_1$ and

$$M_1\xi =\sum _{n=1}^{\infty }(-n^2)\langle \xi ,e_n\rangle e_n\left(\tau \right),\xi \in \mathbb{D},$$

where $e_n\left(\tau \right)={(2/\pi )}^{1/2}sinn\pi ,0\le \tau \le \pi ,n=1,2,\cdots .$ It is known that $M\left(t\right)$ is a generator of compact GE-evolution operator $T(t,s)$ induced by E in $\mathbb{H}$ and is given by

$$T(t,s)\left[\begin{array}{c}{\xi }_1\\ {\xi }_2\end{array}\right]=\left[\begin{array}{c}\sum _{n=1}^{\infty }{e}^{-n^2(t-s)}\langle {\xi }_1,e_n\rangle e_n\left(\tau \right)\\ 0\end{array}\right],$$

where $\left[\begin{array}{c}{\xi }_1\\ {\xi }_2\end{array}\right]\in {\mathbb{H}}_1\oplus {\mathbb{H}}_1$.

Now define an infinite-dimensional space

$$\mathbb{U}=\{\eta =\sum _{n=2}^{\infty }{\eta }_n{e}_n\left(\tau \right):\sum {\eta }_n^2<\infty \}$$

with norm defined by $\parallel \eta \parallel ={\left({\sum }_{n=2}^{\infty }{\eta }_n^2\right)}^{1/2}$ and a linear continuous mapping $N\left(t\right)$ from $\mathbb{U}$ to $E\left(\mathbb{D}\right)$ as follows:

$$N\left(t\right)\eta =(1+t)\left[\begin{array}{c}2{\eta }_2{e}_1\left(\tau \right)+\sum _{n=2}^{\infty }{\eta }_n{e}_n\left(\tau \right)\\ 0\end{array}\right].$$

It is obvious that for

$$\eta (t,\tau ,\omega )=\sum _{n=2}^{\infty }{\eta }_n(t,\omega )e_n\left(\tau \right)\in L^2([0,a],\Omega ,{\mathbb{F}}_t,\mathbb{U}),$$

$$N\left(t\right)\eta \left(t\right)=(1+t)\left[\begin{array}{c}2{\eta }_2\left(t\right)e_1\left(\tau \right)+\sum _{n=2}^{\infty }{\eta }_n\left(t\right)e_n\left(\tau \right)\\ 0\end{array}\right]\in L^2([0,a],\Omega ,{\mathbb{F}}_t,E\left(\mathbb{D}\right)).$$

Moreover

$$N^{\ast }\left(t\right)\theta =(1+t)\left[\begin{array}{c}(2{\theta }_1+{\theta }_2)e_2\left(\tau \right)+\sum _{n=3}^{\infty }{\theta }_n{e}_n\left(\tau \right)\\ 0\end{array}\right]$$

$$N^{\ast }\left(t\right)T^{\ast }(t,0)\xi =(1+t)\left[\begin{array}{c}(2{\xi }_1{e}^{-t}+{\xi }_2{e}^{-4t})e_2\left(\tau \right)+\sum _{n=3}^{\infty }{\xi }_n{e}^{-n^{2}t}{e}_n\left(\tau \right)\\ 0\end{array}\right],$$

for $\theta ={\sum }_{n=1}^{\infty }{\theta }_n{e}_n\left(\tau \right),\xi ={\sum }_{n=1}^{\infty }{\xi }_n{e}_n\left(\tau \right)$. Let

$$\parallel N^{\ast }\left(t\right)T^{\ast }\left(t\right)\left[\begin{array}{c}\xi \\ 0\end{array}\right]\parallel =0,t\in [0,a].$$

Then

$$\parallel 2{\xi }_1{e}^{-t}+{\xi }_2{e}^{-4t}{\parallel }^2+\sum _{n=3}^{\infty }{\parallel {\xi }_n\parallel }^2{e}^{-2n^{2}t}=0.$$

Therefore, ${\xi }_n=0(n=1,2,\cdots )$, i.e., $\xi =0$. Thus, by ([100], Th.4.1.7), the deterministic linear generalized system corresponding to semilinear stochastic generalized system (41) is approximately controllable on $[0,a]$ and by Theorem 3, the semilinear stochastic generalized system (41) is approximately controllable on $[0,a]$ provided that $\alpha =\left[\begin{array}{c}{\alpha }_1\\ 0\end{array}\right]$ satisfies Hypothesis 4.

Example 2.

Consider the semilinear stochastic generalized Ornstei–Uhlenbeck equation, which is abstracted from [101], many practical problems can be classified as such mathematical model, such as diffusion problems, economic problems, epidemic problems, etc.:

$$\left[\begin{array}{cc}I& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}d{\xi }_1(t,y)\\ d{\xi }_2(t,y)\end{array}\right]=\left[\begin{array}{cc}\sum _{k=1}^n[\frac{1}{2}\frac{{\partial }^2}{\partial y_k^2}-y_k\frac{\partial }{\partial y_k}]& 0\\ 0& 1+t\end{array}\right]\left[\begin{array}{c}{\xi }_1(t,y)\\ {\xi }_2(t,y)\end{array}\right]dt$$

$$+\left[\begin{array}{c}1+2t\\ 0\end{array}\right]\eta (t,y)dt+\left[\begin{array}{c}{\alpha }_1(t,{\xi }_1(t,y),\eta (t,y))\\ 0\end{array}\right]dt$$

$$+\left[\begin{array}{c}{\beta }_1(t,{\xi }_1(t,y),\eta (t,y))\\ {\xi }_2(t,y)\end{array}\right]dw\left(t\right),[0,a]\times S$$

$${\xi }_1(0,y)={\xi }_{10}\left(y\right),{\xi }_2(0,y)={\xi }_{20}\left(y\right),$$

(42)

where S is a bounded domain in ${\mathbb{R}}^n,n\ge 1,y=(y_1,y_2,\cdots ,y_n)\in S$, the distributed control

$$\eta \in L^2([0,a],\Omega ,{\mathbb{F}}_t,L^2(S,m)),$$

$m\left(y\right)=\frac{1}{{\pi }^{n/2}}{\Pi }_{k=1}^n{e}^{-|y_k{|}^2}dy$ is the Gaussian measure in S, w is the given one dimensional Brownian motion on S. The nonlinear operator ${\alpha }_1:[0,a]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ satisfies

$$su{p}_{(t,\xi ,\eta )\in [0,a]\times \mathbb{R}\times \mathbb{R}}\left|{\alpha }_1(t,{\xi }_1,\eta )\right|<\infty .$$

$\mathbb{D}=L^2\left(S\right),E=\left[\begin{array}{cc}I& 0\\ 0& 0\end{array}\right],{\beta }_1:[0,a]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is the nonlinear operator and

$$\mathbb{E}|{\beta }_1(t,{\xi }_1,\eta ){|}^2<\infty .$$

The nonlinear operators ${\alpha }_1,{\beta }_1$ satisfy Hypothesis 8. In this case $N\left(t\right)\eta =\left[\begin{array}{c}1+2t\\ 0\end{array}\right]\eta$; therefore, Hypothesis 7 is true. The operator $M_1={\sum }_{k=1}^n[\frac{1}{2}\frac{{\partial }^2}{\partial y_k^2}-y_k\frac{\partial }{\partial y_k}]$ is the generator of compact semigroup $T_1\left(t\right)$ (see [101] for the details). Thus $\left[\begin{array}{cc}{T}_1(t-s)& 0\\ 0& 0\end{array}\right]$ is the compact GE-evolution operator with generator $M\left(t\right)=\left[\begin{array}{cc}{M}_1& 0\\ 0& 1+t\end{array}\right]$ induced by E; therefore, Hypothesis 6 holds true. The deterministic linear generalized system corresponding to semilinear stochastic generalized system (42) is

$$\left[\begin{array}{cc}I& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}d{\xi }_1(t,y)\\ d{\xi }_2(t,y)\end{array}\right]=\left[\begin{array}{cc}\sum _{k=1}^n[\frac{1}{2}\frac{{\partial }^2}{\partial y_k^2}-y_k\frac{\partial }{\partial y_k}]& 0\\ 0& 1+t\end{array}\right]\left[\begin{array}{c}{\xi }_1(t,y)\\ {\xi }_2(t,y)\end{array}\right]dt$$

$$+\left[\begin{array}{c}1+2t\\ 0\end{array}\right]\eta (t,y)dt,{\xi }_1(0,y)={\xi }_{10}\left(y\right),{\xi }_2(0,y)={\xi }_{20}\left(y\right).$$

(43)

Similar to the proof of [77], it can be easily seen that the deterministic linear generalized system (43) corresponding to semilinear stochastic generalized system (42) is approximately controllable, this implies that Hypothesis 9 holds true. Hence, all the conditions stated in Theorem 5 have been satisfied for semilinear stochastic generalized system (42). This implies that semilinear stochastic generalized system (42) is approximately controllable on $[0,a]$.

Remark 1.

The approximate controllability of system (1) can also be studied by using the method of studying the approximate controllability of system (32). In order to show the richness of mathematical methods in control theory, we use the Nussbaum fixed-point theorem to study the approximate controllability of system (1).

Remark 2.

It is obvious that Theorem 5 is true for the following semilinear stochastic generalized system with control only acting on drift terms:

$$Ed\xi \left(t\right)=[M(t\left)\xi \right(t)+N(t\left)\eta \right(t)+\alpha (t,\xi \left(t\right),\eta \left(t\right)\left)\right]dt+\beta (t,\xi (t\left)\right)dw\left(t\right),$$

$$t\in [0,a],\xi \left(0\right)={\xi }_0.$$

(44)

This implies that the open problem concerning the approximate controllability of semilinear stochastic generalized system of type (44) has been concluded under the condition of Theorem 5.

Remark 3.

The approximate controllability of the following nonlinear stochastic generalized system can be solved by using semilinear stochastic generalized system (32):

$$Ed\xi \left(t\right)=\alpha (t,\xi (t),\eta (t\left)\right)dt+\beta (t,\xi (t),\eta (t\left)\right)dw\left(t\right),$$

$$t\in [0,a],\xi \left(0\right)={\xi }_0,$$

(45)

where ξ is the state process valued in Euclidean space $\mathbb{H}$. Since $E\in L(\mathbb{H},\mathbb{H})$, there exist invertible operators $P,Q$ such that $PEQ=\left[\begin{array}{cc}{I}_1& 0\\ 0& 0\end{array}\right]$. Let $V\left(t\right)$ be a compact semigroup with generator A, i.e.,

$$A\xi =li{m}_{h\to 0^{+}}\frac{V\left(h\right)-I_1}{h}\xi .$$

Then

$$T(t,s)=Q\left[\begin{array}{cc}V(t-s)& 0\\ 0& 0\end{array}\right]P$$

is a compact GE-evolution operator induced by E with generator

$$M=P^{-1}\left[\begin{array}{cc}A& 0\\ 0& I_2\end{array}\right]Q^{-1}.$$

In the above, $I_1,I_2$ are the corresponding identical operators, respectively. Let

$$\mathbb{D}=\{\xi :T(s,s)E\xi =\xi ,\exists li{m}_{h\to 0^{+}}\frac{ET(h+s,s)E-ET(s,s)E}{h}\xi \},$$

$N\in L(\mathbb{U},E(\mathbb{D}\left)\right)$. Then, nonlinear stochastic generalized system (45) is equivalent to semilinear stochastic generalized system:

$$Ed\xi \left(t\right)=[M\xi (t)+N\eta (t)+\alpha (t,\xi \left(t\right),\eta \left(t\right))-M\xi (t)-N\eta (t\left)\right]dt+\beta (t,\xi (t),\eta (t\left)\right)dw\left(t\right),$$

$$t\in [0,a],\xi \left(0\right)={\xi }_0.$$

(46)

Mark

$$\alpha (t,\xi (t),\eta (t\left)\right)-M\xi \left(t\right)-N\eta \left(t\right)$$

as $\alpha (t,\xi (t),\eta (t\left)\right)$. Then, semilinear stochastic generalized system (46) can be written as semilinear stochastic generalized system:

$$Ed\xi \left(t\right)=[M\xi (t)+N\eta (t)+\alpha (t,\xi \left(t\right),\eta \left(t\right)\left)\right]dt+\beta (t,\xi (t),\eta (t\left)\right)dw\left(t\right),$$

$$t\in [0,a],\xi \left(0\right)={\xi }_0.$$

(47)

This is the form of semilinear stochastic generalized system (32); therefore, the approximate controllability of nonlinear stochastic generalized system (45) can be transformed into that of semilinear stochastic generalized system (46). According to Theorem 5, we can obtain the result concerning the approximate controllability of nonlinear stochastic generalized system (45).

Example 3.

Consider the linear stochastic generalized equation in input–output economics, which is abstracted from [72]:

$$d\xi \left(t\right)=-(2t+1)\xi \left(t\right)dt+(1+t^2)\eta \left(t\right)dt+{\left(2t\right)}^{1/2}\xi \left(t\right)dw\left(t\right),$$

$$t\in [0,a],{\xi }_1(0,y)={\xi }_{10}\left(y\right),$$

(48)

$$0=\sigma \left(t\right)dt+(1+t)\eta \left(t\right)dt,t\in [0,a],\sigma \left(0\right)={\sigma }_0.$$

(49)

Here, $\xi ,\eta \in H_1,H_1$ is a Hilbert space, $\mathbb{H}=H_1\oplus H_1$. According to ([79], Th.50) and Theorem 6, we can obtain that linear stochastic system (48) and (49) are approximately controllable on $[0,a]$ up to $L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},{\mathbb{H}}_1)$, respectively. These imply that linear stochastic generalized system (48) and (49) is approximately controllable on $[0,a]$ up to $L^2(\Omega ,{\mathbb{F}}_a,\mathbb{P},\mathbb{H})$ by Theorem 7.

Remark 4.

There has been special research on the GE-evolution operator, see [92,93,95,102] for details. From these documents, we can see the properties, construction methods, and applications of the GE-evolution operator. Because this paper only uses the basic concept of the GE-evolution operator, it is only briefly introduced.

Remark 5.

Since the main results of this paper use the orthogonality of Hilbert space in the discussion and proof, these results are still true in Euclidean space, but not necessarily in Banach space.

5. Conclusions

The approximate controllability for some types of semilinear and nonlinear stochastic generalized systems have been investigated by using the GE-evolution operator in the sense of a mild solution in Hilbert spaces, some sufficient conditions have been obtained, and some application examples have been given to illustrate the validity of the theoretical results obtained in this paper. The next research direction is to consider the controllability of semilinear stochastic generalized time delay systems. In particular, we should study the controllability of nonlinear stochastic generalized time delay heat equation, nonlinear stochastic generalized time delay wave equation, and so on—thus, providing a theoretical basis for solving practical problems.