Stochastic Stability in Max-Product and Max-Plus Systems with Markovian Jumps
For researchers in control theory and discrete event systems, this work extends stability analysis to stochastic max-type systems with Markovian jumps, but the results are incremental extensions of existing Lyapunov methods.
The paper derives Lyapunov-based sufficient and necessary conditions for stochastic stability of Max-Product and Max-Plus systems with Markovian jumps, and obtains almost sure bounds on asymptotic behavior. A numerical example on a production system is provided.
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.