Random-Time, State-Dependent Stochastic Drift for Markov Chains and Application to Stochastic Stabilization Over Erasure Channels

It is known that state-dependent, multi-step Lyapunov bounds lead to greatly simplified verification theorems for stability for large classes of Markov chain models. This is one component of the “fluid model” approach to stability of stochastic networks. In this paper we extend the general theory to randomized multi-step Lyapunov theory to obtain criteria for stability and steady-state performance bounds, such as finite moments. These results are applied to a remote stabilization problem, in which a controller receives measurements from an erasure channel with limited capacity. Based on the general results in the paper it is shown that stability of the closed loop system is assured provided that the channel capacity is greater than the logarithm of the unstable eigenvalue, plus an additional correction term. The existence of a finite second moment in steady-state is established under additional conditions.

💡 Research Summary

The paper develops a novel stochastic‑drift framework for Markov chains that allows the Lyapunov verification step to occur at a random, state‑dependent time horizon rather than at every fixed step. Classical fluid‑model approaches rely on deterministic one‑step or fixed‑multi‑step drift conditions; these become cumbersome when the system’s evolution is influenced by random communication events such as packet erasures. To overcome this, the authors introduce the “Random‑Time, State‑Dependent Stochastic Drift” condition. For a non‑negative Lyapunov function (V) and a state‑dependent integer‑valued function (\tau(x)) representing the random number of steps until the next verification, they require

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