Stability Checking of Markov Jump Linear Systems via Probabilistic Temporal Logic (Extended Version)

2026-06-23

AI Summary

Traditional notions of mean and mean-square stability are insufficient to accurately characterize the local stability behavior of Markov jump linear systems over specific sets of initial states. To address this limitation, this work proposes the first extension of Probabilistic Computation Tree Logic (PCTL) to formally express moment stability properties alongside temporal specifications, incorporating constraints on initial states to enable fine-grained, formal descriptions and verification of stability within regions of interest. By integrating probabilistic temporal logic, linear algebraic techniques, and model checking methods, the proposed framework achieves precise logical characterization and efficient verification of stability under prescribed initial conditions.

Abstract

Markov jump linear systems (MJLSs) model dynamical phenomena subject to random switching among multiple linear modes, driven by an underlying Markov chain. Classical notions such as mean and mean-square stability characterize the long-term asymptotic behaviour of the first and second moments of an MJLS, but they can be overly conservative or even misleading when only a specific subset of initial conditions is of interest. We tackle this challenge through the lens of model checking, where reasoning about specific sets of initial conditions is intrinsic to the approach. We begin by formalizing probabilistic computation tree logic (PCTL) on MJLSs, enabling the specification of state-based temporal properties for these systems. Building on this foundation, we extend the logic to capture moment-based stability properties relative to a prescribed set of initial states. While we ultimately do not obtain a decision procedure for the entire base logic, the logical extensions can be handled -- albeit with some technical subtleties -- by exploiting linear-algebraic techniques.

Problem

• Markov jump linear systems
• stability
• initial conditions
• probabilistic temporal logic
• mean-square stability

Innovation

• Markov jump linear systems
• probabilistic temporal logic
• model checking
• moment-based stability
• initial state specification