期刊
AUTOMATICA
卷 57, 期 -, 页码 65-77出版社
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.automatica.2015.04.002
关键词
Stochastic linear quadratic optimal control; Mean-field theory; Generalized algebraic Riccati equation
资金
- National Natural Science Foundation of China [11101303, 11471242]
- China Postdoctoral Science Foundation [2014M560128]
- China Scholarship Council [201308120011]
- Australian Research Council [DP0877639]
- Natural Sciences and Engineering Research Council of Canada
- Hong Kong RGC [GRF520412, GRF15209614]
- Australian Research Council [DP0877639] Funding Source: Australian Research Council
This paper first presents results on the equivalence of several notions of L-2-stability for linear mean-field stochastic difference equations with random initial value. Then, it is shown that the optimal control of a mean-field linear quadratic optimal control with an infinite time horizon uniquely exists, and the optimal control can be expressed as a linear state feedback involving the state and its mean, via the minimal nonnegative definite solution of two coupled algebraic Riccati equations. As a byproduct, the open-loop L-2-stabilizability is proved to be equivalent to the closed-loop L-2-stabilizability. Moreover, the minimal nonnegative definite solution, the maximal solution, the stabilizing solution of the algebraic Riccati equations and their relations are carefully investigated. Specifically, it is shown that the maximal solution is employed to construct the optimal control and value function to another infinite time horizon mean-field linear quadratic optimal control. In addition, the maximal solution being the stabilizing solution, is completely characterized by properties of the coefficients of the controlled system. This enriches the existing theory about stochastic algebraic Riccati equations. Finally, the notion of exact detectability is introduced with its equivalent characterization of stochastic versions of the Popov-Belevitch-Hautus criteria. It is then shown that the minimal nonnegative definite solution is the stabilizing solution if and only if the uncontrolled system is exactly detectable. (C) 2015 Elsevier Ltd. All rights reserved.
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