MEAN SQUARE EXPONENTIAL DISSIPATIVITY OF SINGULARLY PERTURBED STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

Abstract

Abstract. This paper investigates mean square exponential dissipativ-ity of singularly perturbed stochastic delay differential equations. TheL-operator delay differential inequality and stochastic analysis techniqueare used to establish sufficient conditions ensuring the mean square expo-nential dissipativity of singularly perturbed stochastic delay differentialequations for sufficiently small e > 0. An example is presented to illus-trate the efficiency of the obtained results. 1. IntroductionSingularlyperturbed delaydifferentialequationsisordinarydifferentialequa-tions in which the highest derivative are multiplied by a small parameter andinvolving at least one delay term. These equations arise naturally in a widevariety of engineering applications, representative examples include catalyticcontinuous stirred-tank reactors [1], biochemical reactors [3], fluidized catalyticcrackers [15], flexible mechanical systems [5], electromechanical networks [2],etc. In recent years, in a number of papers [4, 8, 10, 18, 19, 20], the stabil-ity, dissipativity and other behaviors of singularly perturbed delay differentialequations are considered.However, in addition to delay effects and singular perturbation, stochasticeffects likewise exist in real systems. Many dynamical systems have variablestructures subject to stochastic abrupt changes, which may result from abruptphenomena such as stochastic failures and repairs of the components, changesin the interconnections of subsystems, sudden environment changes, etc. Mucheffort has been devoted to extend many fundamental results for determinis-tic systems to stochastic systems [9, 11, 12, 13, 14, 22]. In the past decades,increasing attention has been devoted to the problems of stability and other