Practical Stability For It(?) Stochastic Systems With Markovian Switching

The stability theory in the sense of Lyapunov is well known and has been widely used in many problems of the real world.For a practical system,the desired state may be unstable in the sense of Lyapunov but oscillates in an acceptable area.For example,an aircraft or a missile may be mathematically unstable,However its performance may be acceptable.To deal with such situations,a new concept,called practical stability,was proposed by Lasalle and Lefschetz[1]and then was developed by A.A.Martynyuk[4,5]?V.Laksmikantham[6]and so on.For the nonlinear systems described by de-terministic differential equations,many the results on practical stability can be found,Z.S.Feng[7]and A.H.Tsoi and B.zhang[10].Up to 2001,S.Sathananthan?S.Suthahran studied the practical stability criteria for large-scale nonlinear stochastic.However,up to now,results have rarely been given for the practical stability for large-scale It(?) Stochastic Systems with Markovian switching.In this paper,the notion of practical stability is introduced and extended for a class of large-scale stochastic systems with Markovian switching and a class of large-scale stochastic Markovian Jump system with time-delays.By employing Lyapunov-like functions and the basic comparison principle,some criteria are established for various types of practical stability of nonlinear stochastic systems.The advantage of these results is to convert the problem of practical stability for stochastic systems into the one of practical stability for the corresponding deterministic systems.However,during the research of practical stability for large-scale system,one of the foremost challenges is to overcome the increasing dimension and the complex mathematical models[2,11].However,since the computational effort is enormous,the practical stability of large-scale systems has received relatively little attention.Recently,several important results have been obtained in the area of practical stability for large-scale It(?) stochastic systems,see[16]for example.Owning to the large amount of computational effort of a large scale complex system,it will become simpler and more economical to decompose it into a number of interconnected subsystems.These subsystems,to some extent,can be considered to be independent so that some of the qualitative behaviors of the corresponding subsystems can be combined with interconnection constraints to come up with the qualitative behavior of the overall large scale systems.