Optimal Ergodic Control of Linear Stochastic Differential Equations with Quadratic Cost Functionals Having Indefinite Weights

TL;DR

This paper studies an ergodic control problem for linear stochastic differential equations with quadratic costs, allowing indefinite weights, and establishes conditions for existence, finiteness, and solutions, including regularization methods.

Contribution

It introduces new solvability conditions for ergodic control with indefinite quadratic weights and connects these to classical stochastic LQ control theory.

Findings

Invariant measure existence under stabilizability

Finiteness conditions for indefinite weights

Regularized problem for optimal value approximation

Abstract

An optimal ergodic control problem (EC problem, for short) is investigated for a linear stochastic differential equation with quadratic cost functional. Constant nonhomogeneous terms, not all zero, appear in the state equation, which lead to the asymptotic limit of the state non-zero. Under the stabilizability condition, for any (admissible) closed-loop strategy, an invariant measure is proved to exist, which makes the ergodic cost functional well-defined and the EC problem well-formulated. Sufficient conditions, including those allowing the weighting matrices of cost functional to be indefinite, are introduced for finiteness and solvability for the EC problem. Some comparisons are made between the solvability of EC problem and the closed-loop solvability of stochastic linear quadratic optimal control problem in the infinite horizon. Regularized EC problem is introduced to be used to obtain the optimal value of the EC problem.