Instabilizability Conditions for Continuous-Time Stochastic Systems Under Control Input Constraints

TL;DR

This paper establishes conditions under which continuous-time stochastic systems with control input constraints cannot be stabilized, highlighting fundamental limitations in control policy design for such systems.

Contribution

It introduces new instability conditions for constrained control of stochastic systems, demonstrating that divergence can be unavoidable under certain parameter and constraint settings.

Findings

Constrained control policies may fail to stabilize certain stochastic systems.

Divergence of the second moment is inevitable under specific system parameters.

Control input bounds can fundamentally limit stabilization capabilities.

Abstract

In this paper, we investigate constrained control of continuous-time linear stochastic systems. We show that for certain system parameter settings, constrained control policies can never achieve stabilization. Specifically, we explore a class of control policies that are constrained to have a bounded average second moment for Ito-type stochastic differential equations with additive and multiplicative noise. We prove that in certain settings of the system parameters and the bounding constant of the control constraint, divergence of the second moment of the system state is inevitable regardless of the initial state value and regardless of how the control policy is designed.