Algebraic stability of non-homogeneous regime-switching diffusion processes

TL;DR

This paper establishes conditions for the algebraic stability of non-homogeneous regime-switching diffusion processes, showing how stability properties in different states influence overall system stability and applications in SDE stabilization.

Contribution

It introduces new sufficient conditions for algebraic stability in regime-switching diffusions using a common Lyapunov function, extending stability analysis to systems with mixed decay rates.

Findings

Finite-state regime-switching processes can be exponentially stable despite algebraic stability in some states.

Stability results are applicable to feedback control for stochastic differential equations.

The decay rate depends on the switching behavior and stability properties of individual states.

Abstract

Some sufficient conditions on the algebraic stability of non-homogeneous regime-switching diffusion processes are established. In this work we focus on determining the decay rate of a stochastic system which switches randomly between different states, and owns different decay rates at various states. In particular, we show that if a finite-state regime-switching diffusion process is $p$-th moment exponentially stable in some states and is $p$-th moment algebraically stable in other states, which are characterized by a common Lyapunov function, then this process is ultimately exponentially stable regardless of the jumping rate of the random switching between these states. Moreover, these results can be applied to the stabilization of SDE by feedback control to reduce the observation times.