Convergence Theorems for Stochastic Impulsive Systems With Application to Discrete-Time Stochastic Feedback Control

This article is devoted to stochastic convergence theorems for stochastic impulsive systems (SISs) and their application to discrete-time stochastic feedback control (DTSFC). A general stochastic Barbălat's lemma, which only requires that the concerned stochastic processes are almost surely integrable rather than absolutely integrable in the sense of expectation, for piecewise continuous adapted processes is first proposed, which is truly parallel to the deterministic one. As an extension of this lemma, a general stochastic convergence theorem is established for SISs, which can reveal and sufficiently apply the possible active contribution of the existing noise in the underlying system. To derive easy-to-check stability conditions, a series of LaSalle-type theorems and dwell-time-based conditions are established for stochastic stability/convergence of SISs. In contrast to preceding results, these stability criteria cannot only characterize the stabilizing noise but also be applicable to SISs with both continuous and discrete unstable dynamics. Moreover, supported by the LaSalle-type theorems, the almost sure exponential stabilization problems by DTSFC in both time- and event-triggered control schemes are solved. Particularly, the proposed methods remove the global Lipschitz condition required in the literature and provide an explicit computation of the maximum allowable sampling period. Finally, four numerical examples with comparisons are used to illustrate the theoretical results.