Stabilization of regime-switching processes by feedback control based on discrete time observations II: state-dependent case

TL;DR

This paper advances the stabilization of regime-switching systems by providing new conditions for almost sure stability using discrete observations, focusing on state-dependent processes and employing spectral theory and coupling methods.

Contribution

It extends previous work by establishing almost sure stability criteria for state-dependent regime-switching systems using spectral analysis and coupling techniques.

Findings

Derived sufficient conditions for almost sure stability.

Developed estimation methods for exponential functionals of Markov chains.

Constructed coupling processes to control state-dependent switching.

Abstract

This work investigates the almost sure stabilization of a class of regime-switching systems based on discrete-time observations of both continuous and discrete components. It develops Shao's work [SIAM J. Control Optim., 55(2017), pp. 724--740] in two aspects: first, to provide sufficient conditions for almost sure stability in lieu of moment stability; second, to investigate a class of state-dependent regime-switching processes instead of state-independent ones. To realize these developments, we establish an estimation of the exponential functional of Markov chains based on the spectral theory of linear operator. Moreover, through constructing order-preserving coupling processes based on Skorokhod's representation of jumping process, we realize the control from up and below of the evolution of state-dependent switching process by state-independent Markov chains.