Mean-Field Stochastic Linear Quadratic Optimal Control Problems: Open-Loop Solvabilities

TL;DR

This paper investigates mean-field linear quadratic optimal control problems with deterministic coefficients, establishing conditions for problem finiteness, solvability, and providing explicit feedback control representations.

Contribution

It characterizes the finiteness and open-loop solvability of mean-field LQ problems via convexity conditions and Riccati equations, offering new theoretical insights.

Findings

Convexity of the cost functional is necessary for finiteness.

Uniform convexity guarantees open-loop solvability and feedback control.

Solutions are characterized through coupled Riccati equations.

Abstract

This paper is concerned with a mean-field linear quadratic (LQ, for short) optimal control problem with deterministic coefficients. It is shown that convexity of the cost functional is necessary for the finiteness of the mean-field LQ problem, whereas uniform convexity of the cost functional is sufficient for the open-loop solvability of the problem. By considering a family of uniformly convex cost functionals, a characterization of the finiteness of the problem is derived and a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Then, it is proved that the uniform convexity of the cost functional is equivalent to the solvability of two coupled differential Riccati equations and the unique open-loop optimal control admits a state feedback representation in the case that the cost functional is uniformly convex. Finally, some examples are presented to illustrate the theory developed.