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

Author

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  • Zhaoqiang Ge

    (School of Mathematics and Statistics, Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an 710049, China)

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.

Suggested Citation

  • Zhaoqiang Ge, 2021. "Review of the Latest Progress in Controllability of Stochastic Linear Systems and Stochastic GE-Evolution Operator," Mathematics, MDPI, vol. 9(24), pages 1-42, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:24:p:3240-:d:702351
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    Citations

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    Cited by:

    1. Zhaoqiang Ge, 2022. "Approximate Controllability of Semilinear Stochastic Generalized Systems in Hilbert Spaces," Mathematics, MDPI, vol. 10(17), pages 1-30, August.
    2. Zhaoqiang Ge, 2023. "Controllability of Semilinear Stochastic Generalized Systems in Hilbert Spaces by GE-Evolution Operator Method," Mathematics, MDPI, vol. 11(3), pages 1-26, February.
    3. Zhaoqiang Ge, 2022. "Linear Quadratic Optimal Control Problem for Linear Stochastic Generalized System in Hilbert Spaces," Mathematics, MDPI, vol. 10(17), pages 1-20, August.

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