The Ergodic Linear-Quadratic Optimal Control Problems for Stochastic Mean-Field Systems with Periodic Coefficients

TL;DR

This paper investigates ergodic linear-quadratic optimal control problems for mean-field stochastic systems with periodic coefficients, transforming the infinite horizon problem into a finite periodic interval using periodic measures and Riccati equations.

Contribution

It introduces a novel approach to ergodic control of mean-field systems with periodic coefficients by employing periodic measures and Riccati equations to simplify the problem.

Findings

Established the existence of periodic measures for the system.

Derived periodic Riccati equations for control synthesis.

Provided an example demonstrating the theoretical results.

Abstract

In this paper, we concern with the ergodic linear-quadratic closed-loop optimal control problems, in which the state equation is the mean-field stochastic differential equation with periodic coefficients. We first study the asymptotic behavior of the solution to the state equation and get a family of periodic measures depending on time variables within a period from the convergence of transition probabilities. Then, with the help of periodic measures and periodic Riccati equations, we transform the ergodic cost functional on infinite horizon into an equivalent cost functional on a single periodic interval without limit, and present the closed-loop optimal controls for our concerned control system. Finally, an example is given to demonstrate the applications of our theoretical results.