Discrete Time Mean-Field Stochastic Linear-Quadratic Optimal Control Problems

TL;DR

This paper establishes conditions for solving discrete-time mean-field stochastic linear-quadratic control problems, deriving explicit optimal feedback controls through operator and matrix optimization methods.

Contribution

It introduces a novel operator-based framework transforming the control problem into a matrix optimization, providing explicit solutions and validation methods.

Findings

Necessary and sufficient conditions for solvability derived

Explicit optimal feedback control obtained

Validation through completing the square method

Abstract

This paper first presents necessary and sufficient conditions for the solvability of discrete time, mean-field, stochastic linear-quadratic optimal control problems. Then, by introducing several sequences of bounded linear operators, the problem becomes an operator stochastic LQ problem, in which the optimal control is a linear state feedback. Furthermore, from the form of the optimal control, the problem changes to a matrix dynamic optimization problem. Solving this optimization problem, we obtain the optimal feedback gain and thus the optimal control. Finally, by completing the square, the optimality of the above control is validated.