Stochastic Stability in Max-Product and Max-Plus Systems with Markovian Jumps

TL;DR

This paper investigates the stochastic stability of Max-Product and Max-Plus systems with Markovian jumps, deriving new Lyapunov-based conditions and applying them to production systems.

Contribution

It introduces novel Lyapunov function techniques for stochastic stability analysis of Max-Product and Max-Plus systems with Markovian jumps, including necessary and sufficient conditions.

Findings

Derived Lyapunov functions for stability analysis.

Established necessary and sufficient conditions for stochastic stability.

Applied results to a production system example.

Abstract

We study Max-Product and Max-Plus Systems with Markovian Jumps and focus on stochastic stability problems. At first, a Lyapunov function is derived for the asymptotically stable deterministic Max-Product Systems. This Lyapunov function is then adjusted to derive sufficient conditions for the stochastic stability of Max-Product systems with Markovian Jumps. Many step Lyapunov functions are then used to derive necessary and sufficient conditions for stochastic stability. The results for the Max-Product systems are then applied to Max-Plus systems with Markovian Jumps, using an isomorphism and almost sure bounds for the asymptotic behavior of the state are obtained. A numerical example illustrating the application of the stability results on a production system is also given.