**1. Introduction**

Since Kalman published the seminal paper [1], 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 [1–73]. However, even for the controllability of stochastic linear systems, there are still many important problems to be solved. In this paper, we discuss the latest development of controllability of stochastic linear systems and raise some unsolved issues. According to the spatial dimension, equation type and time sequence, the rest of the paper is organized as follows. In Section 2, the following contents are introduced concerning the controllability of stochastic linear systems in finite dimensional spaces: (i) The *<sup>L</sup>p*−exact controllability and exact observability are discussed; (ii) The exact controllability by feedback controller is considered; (iii) The exact controllability of the stochastic linear systems with memory is investigated; (iv) Some theoretical results for these concepts are given and four important problems to be solved are put forward. In Section 3, the controllability of stochastic linear systems in infinite dimensional spaces is considered: (i) The null controllability is investigated by using *C*0−semigroup in the sense of mild solution in Hilbert spaces; (ii) The approximate controllability and approximate null controllability are discussed by using *C*0−semigroup in the sense of mild solution in Hilbert spaces; (iii) The partial approximate controllability is studied by using evolution operator in the sense of mild solution in Hilbert spaces; (iv) According to these theories, three problems that need to be

**Citation:** Ge, Z. Review of the Latest Progress in Controllability of Stochastic Linear Systems and Stochastic GE-Evolution Operator. *Mathematics* **2021**, *9*, 3240. https:// doi.org/10.3390/math9243240

Academic Editors: Mikhail Posypkin, Andrey Gorshenin and Vladimir Titarev

Received: 15 November 2021 Accepted: 11 December 2021 Published: 14 December 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

studied are raised. In Section 4, the controllability of stochastic singular linear systems in finite dimensional spaces is dealt with: (i) The exact controllability is considered by using Gramian matrix; (ii) The exact null controllability is studied by using Gramian matrix; (iii) The impulse controllability and impulse observability are investigated in the sense of impulse solution; (iv) A problem that needs to be discussed is put forward. In Section 5, the controllability of stochastic singular linear systems in infinite dimensional spaces is studied: (i) The exact controllability for a type of time invariant systems is considered by using *C*0−semigroup in the sense of strong solution in Hilbert spaces; (ii) The exact controllability and approximate controllability for a type of time invariant systems are investigated by using GE-semigroup in the sense of mild solution in Banach and Hilbert spaces, respectively; (iii) The exact controllability and approximate controllability for a type of time-varying systems are dealt with by using GE-evolution operator in the sense of mild solution in Hilbert spaces; (iv) The exact controllability and approximate controllability for a type of time invariant systems are considered by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces; (v) The exact controllability, approximate controllability, exact observability, and approximate observability for a type of time-varying systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces. Some necessary and sufficient conditions concerning these concepts are obtained, the dual principle is proved to be true, an example is given to illustrate the validity of the theoretical results obtained in this part, and a problem to be solved is raised.

The main idea of this paper is to introduce the latest progress for the controllability of stochastic linear systems and the mathematical methods applied in this field, including GE-semigroup, GE-evolution operator, stochastic GE-evolution operator and so on. The main purpose of this paper is to facilitate readers to fully understand the latest research results concerning the controllability of stochastic linear systems and the problems that need to be further studied, and attract more scholars to engage in this research.

*Notations.* (Ω, *F*, {*Ft*}, *P*) is a complete probability space with filtration {*Ft*} satisfying the usual condition (i.e., the filtration contains all *P*−null sets and is right continuous); all processes are {*Ft*}−adapted; *w*(*t*) is a standard Wiener process defined on (Ω, *F*, {*Ft*}, *P*); *E* denotes the mathematical expectation; <sup>R</sup>*<sup>n</sup>* is the *<sup>n</sup>*−dimensional real Euclidean space with the standard norm -·-<sup>R</sup>*<sup>n</sup>* , <sup>R</sup>*n*×*<sup>m</sup>* is the space of all (*<sup>n</sup>* <sup>×</sup> *<sup>m</sup>*) real matrices; *In* <sup>∈</sup> <sup>R</sup>*n*×*<sup>n</sup>* denotes the identical matrix; *T* denotes the transpose of a vector or a matrix; *H* = R*n*, R*n*×*m*, etc, and *<sup>p</sup>* <sup>∈</sup> [1, <sup>∞</sup>); *<sup>L</sup>p*([0, *<sup>τ</sup>*]; *<sup>H</sup>*) denotes the set of all functions *<sup>f</sup>* : [0, *<sup>τ</sup>*] <sup>→</sup> *<sup>H</sup>* satisfying *f*(·)-*<sup>L</sup>p*([0,*τ*];*H*) = ( *<sup>τ</sup>* 0 *f*(*t*)*p <sup>H</sup>dt*)1/*<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>; *<sup>L</sup>*∞([0, *<sup>τ</sup>*]; *<sup>H</sup>*) denotes the subset of *Lp*([0, *τ*]; *H*) whose element is essentially bounded; *C*([0, *τ*]; *H*) denotes the set of all functions *f* : [0, *τ*] → *H*, which are continuous on [0, *τ*] in the sense of *f*(·)-*<sup>C</sup>*([0,*τ*];*H*) = max*t*∈[0,*τ*] *f*(*t*)-*<sup>H</sup>*; *<sup>L</sup>p*(Ω, *Ft*, *<sup>P</sup>*, *<sup>H</sup>*) denotes the set of all random variables *<sup>η</sup>* <sup>∈</sup> *<sup>H</sup>*, such that *η* is *Ft*−measurable and *η<sup>p</sup>* = (*E*(*ηp <sup>H</sup>*))1/*<sup>p</sup>* < +∞; *<sup>L</sup>p*([0, *<sup>τ</sup>*], <sup>Ω</sup>, *Ft*, *<sup>H</sup>*) denotes the set of all processes *x*(*t*) ∈ *H* such that *x*(*t*)*<sup>p</sup>* <sup>&</sup>lt; <sup>+</sup>∞, <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *<sup>τ</sup>*]; *<sup>L</sup>p*([0, *<sup>τ</sup>*], <sup>Ω</sup>, *<sup>H</sup>*) denotes the set of all processes *<sup>x</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>p*([0, *<sup>τ</sup>*], <sup>Ω</sup>, *Ft*, *<sup>H</sup>*) such that *<sup>E</sup> τ* 0 *x*(*t*)*p <sup>H</sup>d<sup>τ</sup>* <sup>&</sup>lt; <sup>+</sup>∞; *<sup>L</sup>*∞([0, *<sup>τ</sup>*], <sup>Ω</sup>, *<sup>H</sup>*) is the subset of *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, *<sup>H</sup>*) where each element *<sup>x</sup>*(·) is essentially bounded; Let *<sup>A</sup>* be a linear operator. dom(*A*), ker(*A*) and ran(*A*) denote its domain, kernel and range, respectively; *I* denotes the identical operator. Other mathematical symbols involved in this paper will be properly explained in the discussion.

#### **2. Exact Controllability of Finite Dimensional Stochastic Linear Systems**

In this section, we discuss the latest development of exact controllability of finite dimensional stochastic linear systems.
