The Ergodic Linear-Quadratic Optimal Control Problems with Random Periodic Coefficients

TL;DR

This paper studies ergodic linear-quadratic optimal control problems with random periodic coefficients, establishing conditions for stability, solution periodicity, and deriving explicit optimal controls over an infinite horizon.

Contribution

It introduces a new random periodic mean-square stability condition and proves the existence and uniqueness of solutions to stochastic Riccati equations with periodicity, enabling explicit control solutions.

Findings

Established random periodic mean-square stability condition

Proved existence and uniqueness of periodic solutions to stochastic Riccati equations

Derived explicit closed-loop optimal controls based on periodic solutions

Abstract

In this paper, we concern with the ergodic linear-quadratic closed-loop optimal control problems with random periodic coefficients. We put forward the random periodic mean-square exponentially stable condition, and prove the random periodicity of solutions to state equation based on it. Then we prove the existence and uniqueness of random periodic solutions to two types of backward stochastic differential equations which serve as stochastic Riccati equations in the procedure of completing the square. With the random periodicity of state equation and stochastic Riccati equations, the ergodic cost functional on infinite horizon is simplified to an equivalent cost functional over a single periodic interval without limit. Finally, the closed-loop optimal controls are explicitly given based on random periodic solutions to state equation and stochastic Riccati equations.