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}-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.
Highlights
1 Introduction We are curious about the indefinite discrete-time mean-field LQ (MF-LQ) optimal control problems over infinite horizon
Remark 2.1 Compared with most of previous works, the maximum principle for MF-LQ optimal control problem was based on the mean-field backward stochastic differential equation, while Proposition 2.1 provides a convenient calculation method and can be reduced to the standard stochastic LQ case
3 generalized algebraic Riccati equations (GAREs) we present several results about the GAREs which play a key role in deriving our main results
Summary
We are curious about the indefinite discrete-time mean-field LQ (MF-LQ) optimal control problems over infinite horizon. Ni et al [14] considered the indefinite mean-field stochastic LQ optimal control problem with finite horizon. Song and Liu [15] derived the necessary and sufficient solvability condition of the finite horizon MF-LQ optimal control problem. It is a critical condition to study the MF-LQ optimal control problems It is, natural to ask whether similar results can be derived if Q, Q , R, Rare just assumed to be symmetric, which is of particular and significant mathematical interest. To the best of our knowledge, the study on the necessary and sufficient stabilization conditions of the discrete-time MF-LQ stochastic optimal control problems, especially.
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