The Ergodic Linear-Quadratic Optimal Control Problems with Random Periodic Coefficients
Jiacheng Wu, Qi Zhang
TL;DR
This work tackles ergodic linear-quadratic control problems with random periodic coefficient dynamics. It introduces the tau-random periodic mean-square exponential stability framework, proves the existence of random periodic solutions for both the state SDE and associated stochastic Riccati BSDEs, and uses these to transform the infinite-horizon ergodic cost into a finite-period cost over one cycle. The main contributions are the existence and uniqueness of random periodic solutions, the equivalence between stabilizability and well-posedness of the stochastic Riccati equation, and the derivation of explicit closed-loop optimal controls expressed via the random periodic Riccati and BSDE solutions. These results provide a constructive approach for ergodic control under random periodic coefficients and have implications for systems with periodic randomness in engineering, economics, and finance.
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.