A stability criterion for Stochastic Hybrid Systems

1
A stability criterion for Stochastic Hybrid
Systems

• A. Abate, L. Shi, S. Simic and S.S. Sastry
• University of California at Berkeley

2
Problem Statement

• n Domains
• Vector fields
• fi ! flows ?i
• Reset maps Rij
• Steady-state distr. pp1,..., pn

• Dynamics ODEs, possibly nonlinear (flows have
  bounded Lipschitz constant)
• Underlying Markov Chain
• Temporal transitions (statistically distributed)
• Single Equilibrium q shared among all domains
• Reset maps with bounded Lip constant

3
Results

• In the case of LTI dynamical systems with jumping
  times ti described by finite-mean r.v. s for
  all i, i1,..,n Eti li .
• Theorem
• Define
• n Pi1,..,nLip(jili)pi
• m Pi,j1,..,nLip(Rij )pi Pij
• If n m lt1, then equilibrium q is stable in
  probability (sufficient condition).
• If jumping times happen at constant
• times, result is valid for general
• NL systems.
• Applicative Example in Finance.