Asymptotic Properties of Stochastic Delay Systems

Abstract

A gentle and elementary introduction to the theory of stochastic lime delay systems is presented in this contribution. First an introduction to the stability problem in the discrete case is given. This class of systems is simpler from a dynamical point of view, as they remain finite dimensional, and provide thus a ‘wann-up’ for what is to come. Since continuous time stochastic systems are analyzed in the language of Itô-calculus, an elementary introduction to the latter is included to make these notes self-contained. Delay-independent and delay dependent conditions for stochastic stability arc derived. Some of these are new. In the absence of equilibria, invariant distributions may still exist. Existence conditions, and the stationary Fokker-Planck equation are discussed. These results are further extended to the class of stochastic neutral systems. Finally. a new realistic design procedure is suggested for dynamic controllers in the absence of precise delay information.