On stabilizability and exact observability of stochastic systems with their applications

TL;DR

This paper provides new spectral and algebraic criteria for stabilizability, observability, and robust stabilization of linear stochastic systems, extending deterministic results and introducing novel concepts like unremovable spectrum.

Contribution

It introduces new spectral conditions, concepts, and criteria for stochastic system control, generalizing deterministic results and improving existing algebraic and Lyapunov-based methods.

Findings

Necessary and sufficient conditions for stabilizability and weak stabilizability.

A stochastic PBH criterion for exact observability.

Comparison theorem for generalized algebraic Riccati equations.

Abstract

This paper discusses the stabilizability, weak stabilizability, exact observability and robust quadratic stabilizability of linear stochastic control systems. By means of the spectrum technique of the generalized Lyapunov operator, a necessary and sufficient condition is given for stabilizability and weak stabilizability of stochastic systems, respectively. Some new concepts called unremovable spectrums, strong solutions, and weakly feedback stabilizing solutions are introduced. An unremovable spectrum theorem is given, which generalizes the corresponding theorem of deterministic systems to stochastic systems. A stochastic Popov-Belevith-Hautus (PBH) criterion for exact observability is obtained. For applications, we give a comparison theorem for generalized algebraic Riccati equations (GAREs), and two results on Lyapunov-type equations are obtained, which improve the previous works. Finally, we also discuss robust quadratic stabilization of uncertain stochastic systems, and a necessary and sufficient condition is given for quadratic stabilization via a linear matrix inequality (LMI).