Analysis On Stochastic Differential Equations By Using Linear Matrix Inequality

Mean square stability has been intensively investigated with respect to stochastic differential equations(SDE).Recently,the weaker concepts of stability,such as pth moment exponential stability and almost surely exponential stability have drawn much attention.On the other hand,linear matrix inequality(LMI) has been widely used in the analysis of multidimensional systems.How to apply LMI to analyze almost surely exponential stability of SDE,may have significance in the field of analysis of multidimensional stochastic system and stochastic control.This paper contains five chapters.Chapter 1 is the preface.we introduce the history of stochastic stability,the rela-tions between different stochastic stabilities,the preliminaries of LMI and the structure of the paper.In chapter 2,we investigate the estimation of quadratic function of the solution to SDE.For the purpose investigating the stability of the solution,it is very important to estimate quadratic function of the solution of SDE.Although the estimation of the Euclid norm form of solution has been widely discussed,little literature has been published on the estimation of other forms.This paper considers quadratic function of the solution,and put forward pth moment exponential estimation and almost surely exponential estimation. In chapter 3,we explore the conditions of almost surely stability with respect to SDE with Markovian switching(MSDE).To our knowledge,no criterion in the form of LMI is reported yet with respect to the almost surely stability of MSDE.We obtain sufficient condition of almost surely exponential stability of MSDE.Moreover,criteria in the form of LMI on pth moment exponential stability and almost surely exponential stability are derived.Examples are given to illustrate the criteria.In chapter 4,we discuss exponential instability of MSDE.Sufficient conditions about qth moment exponential instability and almost surely exponential instability are given,and criteria on qth moment exponential instability and almost surely exponential instability of MSDE are derived.In chapter 5,some conclusions and future problem study are addressed.