On the necessity of barrier certificates

Abstract A methodology for safety verification of nonlinear systems using barrier certificates has been proposed recently. The condition was stated in a sufficiency form: if there exists a barrier certificate, then the system is safe, in the sense that there is no trajectory starting from a given set of initial states that reaches a given unsafe region. Using the concepts of convex duality and density functions, in this paper we derive a converse statement for barrier certificates, showing that in a quite general setting the existence of a barrier certificate is also necessary for safety.

[1]  Pravin Varaiya,et al.  Ellipsoidal Techniques for Reachability Analysis , 2000, HSCC.

[2]  M. Jirstrand.  Invariant sets for a class of hybrid systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[3]  Anders Rantzer,et al.  Primal-Dual Tests for Safety and Reachability , 2005, HSCC.

[4]  George J. Pappas,et al.  Stochastic safety verification using barrier certificates , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[5]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[6]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .

[7]  A. Rantzer.  A dual to Lyapunov's stability theorem , 2001 .

[8]  Rajeev Alur,et al.  Progress on Reachability Analysis of Hybrid Systems Using Predicate Abstraction , 2003, HSCC.

[9]  Ali Jadbabaie,et al.  Safety Verification of Hybrid Systems Using Barrier Certificates , 2004, HSCC.

[10]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[11]  Pablo A. Parrilo,et al.  Introducing SOSTOOLS: a general purpose sum of squares programming solver , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[12]  A. Rantzer,et al.  Duality between cost and density in optimal control , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[13]  Stephan Merz,et al.  Model Checking , 2000 .

[14]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.