Stability And Control Problems Of Impulsive Systems On Time Scales

Stefan Hilger introduced the theory of time scales in his PhD thesis in1988in order to unify the continuous and discrete analysis. The theory of dynamic systems on time scales provides us a framework to study the continuous and discrete system simultaneously. In terms of dynamic systems, various physical processes undergo abrupt changes of their states at discrete moments, this kind of systems can be modeled by impulsive differential equations. Recently, the theory of impulsive systems has been investigated extensively. From the modeling point of view, it is perhaps more reasonable to model evolution process subjected to impulsive perturbations by a dynamic system which incorporates both continuous and discrete time space, namely impulsive systems on time scales.Stability and control problems of impulsive systems on time scales are inves-tigated. The objectives of this thesis are to analyze the state controllability and observability of linear impulsive time-varying systems on time scales, to estab-lish stability criteria for nonlinear impulsive systems with and without delay on time scales, and to apply the theory and method to real-world applications. The contents of this thesis consist of four parts which are listed as follows.Chapter2considers controllability and observability of linear impulsive time-varying systems on time scales. By employing the method of parameter variation, the form of corresponding system’s solution is obtained, which plays an important role in this chapter. Sufficient and necessary conditions are established for the linear impulsive time-varying systems on time scales, respectively. These results are then used to discuss the systems’time-invariant counterparts. It is shown that when the time scale reduce to the real numbers, our results contains the existing results about the linear impulsive continuous systems, and some other results about linear systems without impulses on time scales can be our results’ special cases.In Chapter3, stability in terms of two measures is investigated for the non-linear impulsive systems on time scales. A new comparison result is obtained by using the mathematical induction method on time scales, which is used to establish the sufficient conditions for the (h0,h)-stability of nonlinear impulsive systems on time scales by comparison method. The asymptotic stability of a class of nonlinear impulsive systems on time scales is discussed by using the obtained (h0, h)-stability criteria. In the second part of this chapter, we use Lyapunov direct method to study stability in terms of two measures for nonlinear impulsive systems on time scales.(h0,h)-(uniform) stability,(h0,h)-(uniform) asymptotic stability,(h0,h)-nstability criteria are established, respectively. Some examples are also discussed to illustrate our theoretical results.Exponential stability problem is considered in Chapter4about nonlinear impulsive functional systems. In order to establish exponential stability criteria for impulsive functional systems on general time scales, the exponential stability problem of impulsive discrete delay systems is concerned. By using Lyapunov-Razumikhin method, the global exponential stability criteria are obtained. Then, Lyapunov functional method is employed to construct sufficient conditions about the exponential stability for impulsive discrete delay systems. Inspired by the results obtained about discrete systems and the method used, the exponential stability criteria about impulsive functional systems on time scales are estab-lished. Two Razumikhin-type global exponential stability results are derived, one of which provides sufficient conditions for maintaining the global exponential sta-bility property of the trivial solution of the functional system without impulsive perturbations, while another of which can be used to impulsively stabilize func-tional systems. These results are also used to discuss the stability problem of some special cases of nonlinear impulsive systems on time scales. The last part of this chapter proposes the Lyapunov functional method in the content of impulsive exponential stabilization of functional systems. Several results are constructed to show that an unstable system can be exponentially stabilized by appropriate se-quence of impulses, impulses can contribute to make a stable system exponentially stable, and to what extent can a well-behaved system preserve its stability prop-erties under impulsive perturbations, respectively. Some examples with numerical simulations are exploited to demonstrate the effectiveness of our results.In Chapter5, several applications of the results obtained in previous parts are concerned. Uniform and nonuniform impulsive controllers are designed to sta-bilize the continuous and discrete chaos systems. As far as the author knows, the impulsive synchronization problem of discrete delay complex dynamic networks is firstly considered in this thesis. By using the Razumikhin-type stability re- sults derived in Chapter4, the impulsive controller is constructed to realize the exponential synchronization of the discrete delay complex dynamic networks. Fi-nally, the complex dynamic networks on time scales is introduced and a hybrid impulsive controller is designed to consensus the networks on time scales to the objective states. It is shown that the state-coupled networks on time scales can be effectively forced to the goal trajectory by adding impulsive control to a small pro-portion of the networks’nodes. Numerical examples and simulations are presented throughout this chapter to illustrate the results.