Stochastic Linear Quadratic Optimal Control Problems in Infinite Horizon

TL;DR

This paper investigates stochastic linear quadratic control problems over an infinite horizon, establishing key equivalences between control set non-emptiness, system stabilizability, and solutions to algebraic Riccati equations, with complete analysis in one dimension.

Contribution

It provides new equivalences linking control set non-emptiness, stabilizability, and Riccati equation solutions for infinite horizon stochastic LQ problems, extending finite horizon results.

Findings

Non-emptiness of control set is equivalent to $L^2$-stabilizability.

Existence of a positive solution to ARE characterizes stabilizability.

Open-loop and closed-loop solvability are equivalent to solutions of a generalized ARE.

Abstract

This paper is concerned with stochastic linear quadratic (LQ, for short) optimal control problems in an infinite horizon with constant coefficients. It is proved that the non-emptiness of the admissible control set for all initial state is equivalent to the $L^2$-stabilizability of the control system, which in turn is equivalent to the existence of a positive solution to an algebraic Riccati equation (ARE, for short). Different from the finite horizon case, it is shown that both the open-loop and closed-loop solvabilities of the LQ problem are equivalent to the existence of a static stabilizing solution to the associated generalized ARE. Moreover, any open-loop optimal control admits a closed-loop representation. Finally, the one-dimensional case is worked out completely to illustrate the developed theory.