[Paper Review] Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations

Xiongping Dai|arXiv (Cornell University)|Jul 16, 2013

Stability and Control of Uncertain Systems46 references3 citations

TL;DR

This paper establishes that robust periodic stability implies uniform exponential stability for both discrete-time Markovian jump linear systems and random linear ODEs driven by semiflows with the closing by periodic orbits property. Using a novel Gel'fand-Berger-Wang formula derived via Lyapunov exponent approximation by periodic orbits and Shantao Liao’s perturbation techniques, the authors prove equivalence between robust periodic stability and uniform exponential stability, resolving long-standing questions on spectral finiteness and robustness.

ABSTRACT

In this paper we show that if a linear cocycle is robustly periodical stable then it is uniformly stable.

Motivation & Objective

• To establish the equivalence between robust periodic stability and uniform exponential stability in discrete-time Markovian jump linear systems.
• To extend this equivalence to random linear ODEs driven by semiflows with the closing by periodic orbits property.
• To resolve open questions regarding spectral finiteness and robustness conditions in linear cocycle theory.
• To develop a Gel’fand-Berger-Wang formula for Markovian jump systems using periodic orbit approximation of Lyapunov exponents.
• To construct counterexamples to the spectral finiteness conjecture and the robustness condition in linear cocycle theory.

Proposed method

• Derives a Gel’fand-Berger-Wang formula for Markovian jump linear systems by approximating Lyapunov exponents using periodic orbits.
• Applies Shantao Liao’s perturbation technique from differentiable dynamical systems to analyze stability in random linear ODEs.
• Uses semi-uniform ergodic theorems to relate the joint spectral radius to the behavior on nonwandering sets and periodic orbits.
• Employs ergodic theory to analyze the stability of linear cocycles over dynamical systems with the closing by periodic orbits property.
• Constructs counterexamples to the spectral finiteness of linear cocycles and to the robustness condition in stability analysis.
• Establishes equivalence between uniform exponential stability and robust periodic stability via topological and measure-theoretic arguments on symbolic spaces and semiflows.

Experimental results

Research questions

• RQ1 Does robust periodic stability imply uniform exponential stability in discrete-time Markovian jump linear systems?
• RQ2 Can the equivalence between robust periodic stability and uniform exponential stability be extended to random linear ODEs driven by semiflows with the closing by periodic orbits property?
• RQ3 Is the spectral finiteness property valid for general linear cocycles over dynamical systems with periodic orbit structure?
• RQ4 Can a Gel’fand-Berger-Wang formula be established for Markovian jump systems without assuming irreducibility or aperiodicity of the transition matrix?
• RQ5 What are the limitations of the robustness condition in stability analysis of linear switching systems?

Key findings

• A discrete-time Markovian jump linear system is uniformly exponentially stable if and only if it is robustly periodically stable, under no additional assumptions on the Markov transition matrix.
• For random linear ODEs driven by a semiflow with the closing by periodic orbits property, uniform exponential stability is equivalent to robust periodic stability.
• A new Gel’fand-Berger-Wang formula is proven for Markovian jump systems by approximating Lyapunov exponents via periodic orbits.
• The joint spectral radius of a linear cocycle over a system with closing by periodic orbits property equals the infimum over max norms on periodic orbits.
• Counterexamples are constructed showing that spectral finiteness does not hold in general for linear cocycles.
• The robustness condition—uniform boundedness of spectral radii over periodic orbits—is insufficient to guarantee uniform exponential stability, highlighting a subtle distinction in stability criteria.