The Stability And Stabilization Of Stochastic Systems With Markovian Switching

The practical models are often disturbed by internal uncertainty and external random factors.Stochastic differential equations have the ability to simulate these complex models. The stochasticdifferential equations with Markovian switching have been used to model many practical systemswhere they may experience abrupt changes in their structure and parameters. Stochastic modelshas come to play an important role in many branches of science and industry. In recent years,one of the important issues of the stochastic systems is the stability of solutions. This paper isconcerned with the stability and stabilization of stochastic systems with Markovian switching.The main contents and results obtained in this thesis are as follows:1. Chapter1sums up the recent development and several hot topics of stochastic systemswith Markovian switching，introduces the main existing methods and the basic conclusions forstochastic systems with Markovian switching.2. Chapter2introduces the basic knowledge of mathematics which will be used in this paper,including the concepts and related properties of Markov chain, the existence and uniqueness ofsolution to stochastic differential equations, and some basic inequalities that are necessary forfurther investigation.3. Chapter3studies the stability of stochastic systems with Markovian switching. It is ob-vious that the classical and powerful technique applied in the study of stability is the stochasticversion of the Lyapunov direct method. Most of the existing conditions for stability implicitlyrequire the assumption that the transition jump rates are completely known. In many situations,however, it is difficult to acquire the exact transition jump rates. This chapter will explore thethemes of the stability of the nonlinear stochastic systems with Markovian switching. The sys- tems under consideration are more general, whose transition jump rates matrix Q is not preciselyknown. The main idea is to use the switching process jump times to subdivide the”time” and theninvestigate the related Lyapunov functions which we assume the function candidates are compati-ble, and we provide sufficient conditions for asymptotic stability and moment exponential stabilityof stochastic systems with Markovian switching.4. Chapter4deals with the robust stabilization problem for uncertain Markovian jump linearstochastic systems. One of the uncertain Markovian jump stochastic systems under considerationinvolves parameter uncertainties both in the system matrices and in the mode transition jump ratematrix. New criteria for testing the robust stability of such systems are established in terms of bi-linear matrix inequalities (BLMIs), and sufficient conditions are proposed for the design of robuststate-feedback controllers. Based on stationary distribution theory, we also discuss the almost sureexponential stabilization of a class of uncertain Markovian jump linear stochastic systems whenwe acquire the exact transition jump rates.5. Chapter5consider nonlinear stochastic differential equations and use Lyapunov functionsto study the boundedness of solutions. Sufficient conditions for stochastic boundedness are estab-lished. We provide two examples to illustrate our results.6. Chapter6summarizes the work presented in this thesis and provides several problemswhich are worth studying further.The main contributions of this thesis can be summarized as the following three aspects:1. For nonlinear stochastic systems with Markovian switching whose transition jump ratesmatrix Q is not precisely known，we establish sufficient conditions for stability. This work doesnot only extend the application of stochastic systems with Markovian switching, but also improvesthe theory of stochastic systems with Markovian switching.2. We discuss the robust stability problem for uncertain Markovian jump linear stochasticsystems. The systems under consideration are more general, which involve parameter uncertaintiesboth in the system matrices and in the mode transition jump rate matrix. Based on the bi-linearmatrix inequalities, the sufficient conditions for robust stability of this systems are presented.3. We design a more general robust state-feedback controllers for uncertain Markovian jumplinear stochastic systems, and solve the stabilization problem of the system effectively.