Safety verification for Regime-Switching Jump Diffusions via barrier certificates

Abstract

It is well known that for a stochastic hybrid system, if its failure probability does not exceed a given safety threshold, the safety of the system can be guaranteed. Thus, in this paper, we concern with the upper bound of failure probability for failure analysis of a class of nonautonomous stochastic hybrid systems, denoted as Regime-Switching Jump Diffusions (RSJDs). We start with the definition of RSJDs, which contain not only continuous flows described by stochastic differential equations, but also instantaneous behavior described by Markovian switching and instantaneous behavior described by Lévy jump. Then we decompose the failure probability of RSJDs in [t0,+∞) into two segments: one is defined over [T,+∞), and the other is defined over [t0,T]. For failure probability over [T,+∞), by utilizing multiple vectorial barrier certificates, an asymptotically decreasing bound of failure probability with respect to T is established, where a general nonnegative matrix is used instead of a special nonnegative matrix defined by the exponent of essentially nonnegative matrix for a broader applicability. For failure probability over [t0,T], a generalized c-martingale condition is adopted to obtain a T-dependent failure probability bound, in which two non-negative scalar functions are utilized to relax the conservativeness of infinitesimal generators. Finally, for rational RSJDs, we transform the decomposed failure analysis problems into two semi-definite programming (SDP) problems, and then solve them via sum of squares programming. The applicabilities and effectiveness of our computable decomposition methodology are illustrated through three examples.