Review of the Latest Progress in Controllability of Stochastic Linear Systems and Stochastic GE-Evolution Operator

Abstract

According to the spatial dimension, equation type, and time sequence, the latest progress in controllability of stochastic linear systems and some unsolved problems are introduced. Firstly, the exact controllability of stochastic linear systems in finite dimensional spaces is discussed. Secondly, the exact, exact null, approximate, approximate null, and partial approximate controllability of stochastic linear systems in infinite dimensional spaces are considered. Thirdly, the exact, exact null and impulse controllability of stochastic singular linear systems in finite dimensional spaces are investigated. Fourthly, the exact and approximate controllability of stochastic singular linear systems in infinite dimensional spaces are studied. At last, the controllability and observability for a type of time-varying stochastic singular linear systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces, some necessary and sufficient conditions 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 introduced. 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.

Approximate Controllability of Semilinear Stochastic Generalized Systems in Hilbert Spaces

“…Similar to the proof ideas of ( [79], Th.50) and ( [89], Th.4), we can obtain the following theorem. Theorem 7.…”

“…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.…”

“…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]).…”

Preface to the Special Issue on “Control, Optimization, and Mathematical Modeling of Complex Systems”

“…A review of Ge [18] is devoted to the latest progress in the controllability of stochastic linear systems and some unsolved problems. Firstly, the exact controllability of stochastic linear systems in finite-dimensional spaces is discussed.…”

“…Numerical simulation in physical, social, and life sciences [1][2][3][4]; • Modeling and analysis of complex systems based on mathematical methods and AI/ML approaches [5,6]; • Control problems in robotics [3,[7][8][9][10][11][12]]; • Design optimization of complex systems [13]; • Modeling in economics and social sciences [4,14]; • Stochastic models in physics and engineering [1,[15][16][17][18]; • Mathematical models in material science [19]; • High-performance computing for mathematical modeling [20].…”