Discrete-time mean-field Stochastic linear–quadratic optimal control problems, II: Infinite horizon case

This paper first presents results on the equivalence of several notions of L2L2-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 L2L2-stabilizability is proved to be equivalent to the closed-loop L2L2-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.