Asymptotic Synchronization Analysis in Drive-Response Markovian Jump Boolean Networks

This paper explores the asymptotic synchronization of drive-response Markovian jump Boolean networks (MJBNs) using an algebraic state space representation. For two switching signals, an augmented variable is introduced, recasting the synchronization problem into an asymptotic set stability challenge. Initially, a criterion is developed to identify all nonnegative solutions of a related equation via stochastic processes. Then, an alternative criterion based on invariant subsets and the largest singular value is proposed, where the equivalence of the proposed two criteria is demonstrated. Contrasting with deterministic BNs, where synchronization methods depend on finite power of state transition matrices, the criteria for MJBNs involve a more intricate approach. Numerical examples are included to validate the effectiveness of the proposed theoretical concepts.