Solvability and optimal stabilization controls of discrete-time mean-field stochastic system with infinite horizon

Abstract

Abstract The paper addresses the optimal control and stabilization problems for the indefinite discrete-time mean-field system over infinite horizon. Firstly, we show the convergence of the generalized algebraic Riccati equations (GAREs) and establish their compact form GARE. By dealing with the GARE, we derive the existence of the maximal solution to the original GAREs along with the fact that the maximal solution is the stabilizing solution. Then, the maximal solution is employed to design the linear-quadratic (LQ) optimal controller and the optimal value of the control problem. Specifically, we deduce that under the assumption of exact observability, the mean-field system is L 2 $L^{2}$ -stabilizable if and only if the GAREs have a solution, which is also the maximal solution. By semi-definite programming (SDP) method, the solvability of the GAREs is discussed. Our results generalize and improve previous results. Finally, some numerical examples are exploited to illustrate the validity of the obtained results.

Keywords

• Indefinite stochastic LQ control
• Discrete-time system
• Mean-field theory
• Stabilization
• Generalized algebraic Riccati equation