Delay‐dependent stability analysis and stabilization of stochastic time‐delay systems governed by the Poisson process and Brownian motion

Abstract

AbstractThis article investigates the delay‐dependent stability and stabilization problems for stochastic time‐delay systems (STDSs) governed by the Poisson process and Brownian motion. First, this article exploits the decomposition of semimartingale to transform STDSs governed by the Poisson process and Brownian motion into STDSs governed by martingales, and then one can use effective properties and tools of martingale theory to investigate the stability and stabilization problems. Second, the expectations of stochastic crossterms involving stochastic integrals with respect to (w.r.t.) càdlàg martingale have been studied by martingale theory, and this article proves that the expectations of some particular stochastic crossterms are zero. Third, on the basis of above results on the expectations of stochastic crossterms and the equivalent Itô formula derived, this article uses the free weighting matrix method to give a delay‐dependent stability condition without using bounding techniques. Thus, the conservatism induced by bounding techniques is eliminated, and this method leads to a less conservative condition by a linear matrix inequality. Then, this article designs a state‐feedback controller which can stabilize the STDS governed by the Poisson process and Brownian motion. Finally, the effectiveness of derived results are illustrated by numerical examples.