Asymptotical Stability of Logic Dynamical Systems With Random Impulsive Disturbances

This article presents an investigation of the asymptotical set stability (ASS) of logic dynamical systems (LDSs) with impulsive disturbances at random instants. Unlike ordinary probabilistic Boolean networks, such an impulsive LDS is not Markovian in general. In this article, an impulsive LDS is described by using a hybrid index model and the sequence of intervals between adjacent impulsive instants is assumed to be independent and identically distributed. Under this assumption, the subsequence of a solution obtained by sampling at impulsive instants is a Markov chain. The initial distribution, the transition probability matrix, and the reachable set of the Markovian subsequence are calculated. Calculations of different invariant subsets are discussed and a necessary and sufficient condition for the convergence of finite Markov chains to subsets is obtained. Based on these results, necessary conditions for the ASS of impulsive LDSs in the hybrid and time domains, respectively, are obtained. In addition, we prove that the necessary conditions become sufficient if the impulsive intervals and the initial impulsive instant are with bounded expectations and variances. An example is provided to show the application of the proposed results to node synchronization of logic dynamical networks.