Recurrence for linearizable switching diffusion with past dependent switching and countable state space

TL;DR

This paper studies the recurrence and stabilization of linearizable switching diffusions with past-dependent switching in countable state spaces, providing verifiable conditions and control strategies.

Contribution

It extends previous work by analyzing recurrence and stabilization of systems with countable state spaces and past dependence, offering new verifiable conditions and control methods.

Findings

Established verifiable conditions for recurrence and positive recurrence.

Provided feasible criteria based on system coefficients for positive recurrence.

Developed linear feedback controls for weak stabilization.

Abstract

This work continues and substantially extends our recent work on switching diffusions with the switching processes that depend on the past states and that take values in a countable state space. That is, the discrete components of the two-component processes take values in a countably infinite set and its switching rates at current time depend on the current value of the continuous component. This paper focuses on recurrence, positive recurrence, and weak stabilization of such systems. In particular, the paper aims to providing more verifiable conditions on recurrence and positive recurrence and related issues. Assuming that the system is linearizable, it provides feasible conditions focusing on the coefficients of the systems for positive recurrence. Then linear feedback controls for weak stabilization are considered. Some illustrative examples are also given.