Linear-Quadratic Control of Discrete-Time Stochastic Systems with Indefinite Weight Matrices and Mean-Field Terms

Abstract

In this paper, the linear-quadratic optimal control problem is considered for discrete-time stochastic systems with indefinite weight matrices in the cost function and mean-field terms in both the cost function and system dynamics. A set of generalized difference Riccati equations (GDREs) is introduced in terms of algebraic equality constraints and matrix pseudo-inverse. It is shown that the solvability of the GDRE is not only sufficient but also necessary for the well-posedness of the indefinite mean-field linear-quadratic optimal control problem and the existence of optimal feedback as well as open-loop controls.