Stochastic Singular Linear Systems and Related Linear-Quadratic Optimal Control Problems under Finite and Infinite Horizons

TL;DR

This paper analyzes stochastic singular linear systems and their linear-quadratic control problems, establishing conditions for well-posedness, controllability, and deriving optimal feedback controls for both finite and infinite horizons.

Contribution

It introduces a comprehensive framework for stochastic singular systems, transforming control problems into normal forms and establishing controllability criteria for infinite-horizon cases.

Findings

Established necessary and sufficient conditions for well-posedness.

Derived feedback optimal control laws.

Provided solutions for illustrative examples.

Abstract

In this paper, we study the necessary and sufficient conditions for ensuring the well-posedness of the stochastic singular systems. Moreover, we investigate the stochastic singular linear-quadratic control problems, considering both finite and infinite horizons, and transform each of these problems into their corresponding normal linear-quadratic control problem. To guarantee the finiteness of the infinite-horizon linear-quadratic control problem, we establish the Popov-Belevitch-Hautus rank criterion for accessing the controllability of the stochastic system. Furthermore, we derive the feedback form of the optimal control. Finally, we provide solutions for illustrated examples of the stochastic singular linear-quadratic control problem in both finite and infinite horizons.