Probabilistic bisimulations of switching and resetting diffusions

This contribution presents sufficient conditions for the existence of probabilistic bisimulations between two diffusion processes that are additionally endowed with switching and resetting behaviors. A probabilistic bisimulation between two stochastic processes is defined by means of a bisimulation function, which induces an approximation metric over the distance between the two processes. The validity of the proposed sufficient conditions results in the explicit characterization of one such bisimulation function. The conditions depend on contractivity properties of the two stochastic processes.

[1]  Alessandro Abate,et al.  A contractivity approach for probabilistic bisimulations of diffusion processes , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[2]  S. Sastry.  Nonlinear Systems: Analysis, Stability, and Control , 1999 .

[3]  R. Durrett.  Probability: Theory and Examples , 1993 .

[4]  Antoine Girard.  Approximately Bisimilar Finite Abstractions of Stable Linear Systems , 2007, HSCC.

[5]  S. Shankar Sastry,et al.  O-Minimal Hybrid Systems , 2000, Math. Control. Signals Syst..

[6]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[7]  M. K. Ghosh,et al.  Ergodic Control of Switching Diffusions , 1997 .

[8]  Arunabha Bagchi,et al.  Modeling Stochastic Hybrid Systems , 2003, System Modelling and Optimization.

[9]  Mark H. Davis.  Markov Models and Optimization , 1995 .

[10]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Nonlinear Systems Analyzing stability differentially leads to a new perspective on nonlinear dynamic systems , 1999 .

[11]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[12]  Abbas Edalat,et al.  A logical characterization of bisimulation for labeled Markov processes , 1998, Proceedings. Thirteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.98CB36226).

[13]  David Angeli,et al.  A Lyapunov approach to incremental stability properties , 2002, IEEE Trans. Autom. Control..

[14]  Abbas Edalat,et al.  Bisimulation for Labelled Markov Processes , 2002, Inf. Comput..

[15]  Robin Milner,et al.  Communication and concurrency , 1989, PHI Series in computer science.

[16]  Tomomichi Hagiwara,et al.  Popov-Type Criterion for Stability of Nonlinear Sampled-Data Systems , 1998, Autom..

[17]  Kim G. Larsen,et al.  Bisimulation through Probabilistic Testing , 1991, Inf. Comput..

[18]  George J. Pappas,et al.  Approximations of Stochastic Hybrid Systems , 2009, IEEE Transactions on Automatic Control.

[19]  Antoine Girard,et al.  Approximation Metrics for Discrete and Continuous Systems , 2006, IEEE Transactions on Automatic Control.

[20]  Paulo Tabuada,et al.  Approximately Bisimilar Symbolic Models for Incrementally Stable Switched Systems , 2008, IEEE Transactions on Automatic Control.

[21]  Nicolas Tabareau,et al.  A Contraction Theory Approach to Stochastic Incremental Stability , 2007, IEEE Transactions on Automatic Control.

[22]  George J. Pappas.  Bisimilar linear systems , 2003, Autom..