Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jiacheng Wu, Qi Zhang

TL;DR

The paper tackles ergodic linear-quadratic control for stochastic mean-field systems with $\tau$-periodic coefficients. It develops a periodic-measure framework and periodic Riccati equations to convert the infinite-horizon ergodic cost into a finite-horizon problem on one period, enabling explicit closed-loop controls. The main contributions are the existence and characterization of periodic measures, the derivation of periodic Riccati equations and a linear ODE that yield a closed-form optimal control, and an illustrative 2D example confirming the theory. This provides a rigorous, practically applicable method for long-run optimization in large-scale systems subject to periodic environments.

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.

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

TL;DR

The paper tackles ergodic linear-quadratic control for stochastic mean-field systems with -periodic coefficients. It develops a periodic-measure framework and periodic Riccati equations to convert the infinite-horizon ergodic cost into a finite-horizon problem on one period, enabling explicit closed-loop controls. The main contributions are the existence and characterization of periodic measures, the derivation of periodic Riccati equations and a linear ODE that yield a closed-form optimal control, and an illustrative 2D example confirming the theory. This provides a rigorous, practically applicable method for long-run optimization in large-scale systems subject to periodic environments.

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.
Paper Structure (5 sections, 16 theorems, 117 equations)

This paper contains 5 sections, 16 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Notation and Control Problem
  3. Long time asymptotic behavior of stochastic flow
  4. Ergodic optimal control problem
  5. Examples

Key Result

Lemma 2.4

\newlabelwz50 (Corollary 3.4 in ref30) For a given $\tau>0$, assume $A,B,C,D\in\mathscr{B}_\tau$ and that $[A,C;B,D]$ is $\tau$-periodic mean-square exponentially stabilizable. Then there exist $M,\lambda>0$ depending only on given parameters such that for any $0\le s\le t$, the solution $\Phi$ to

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.4
  • Proposition 2.5
  • Proof 1
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proof 2
  • ...and 21 more