Analysis Of Impulsive Dynamic Systems On Time Scales

In this paper,we mainly study the stability properties of the two nonlinear impulsivedynamic systems on time scales as follows:Dynamic systems on time scales can unify the continuous and discrete systems, sothe study of dynamic systems on time scales is more realistic than before. At the sametime, the mathematical model of sudden change phenomena in actual problems can beattributed to impulsive dynamic systems. At present, there are some results on the sta-bility of dynamic systems on time scales[14,19,25], but there are few results on the stabilityof impulsive dynamic systems on time scales. So the study of impulsive dynamic systemson time scales has gained broad applied background and vital practical significance.Impulsive hybrid dynamic systems on time scales, which are defined on time scaleswith the variable structure, are the important class of nonlinear impulsive dynamic sys-tems on time scales. Impulsive hybrid dynamic systems are widely used in hybrid au-tomata, aircraft control systems, automotive control systems, networked control systems,etc. In recent years, there are more and more results on impulsive hybrid dynamic sys-tems, and the research of the stability properties of this systems has achieved fruitfulresults[2628], but the results of impulsive hybrid dynamic systems on time scales is alsoa few.In chapter one, firstly, we introduce the basic concepts of time scale calculus. Second-ly, we treat the vector impulsive hybrid dynamic systems as comparison systems and establish a new variational comparison theorem of impulsive hybrid dynamic systemson time scales. By using the new variational comparison theorem, we study the (h0, h) strict stability of impulsive hybrid dynamic systems on time scales with the perturbationon the right function. At last, we obtain some results of the (h0, h) actual stability ofimpulsive hybrid dynamic systems on time scales using Lyapunov function method. Anexample is given to illustrate the application of the results.In chapter two, Firstly, we study the boundedness properties of impulsive dynamicsystems on time scales by using Lyapunov function method, and obtain some resultsof the boundedness in terms of two measures for impulsive dynamic systems on timescales. Then, we obtain comparision results of the boundedness in terms of two mea-sures for impulsive perturbed dynamic systems on time scales by using perturbationideas. Secondly, we study the ultimate Lipschitz stability in terms of two measures of im-pulsive dynamic systems on time scales, and give an example to verify the application ofthe theorems. Finally, we study some resules of the partial variable stability for impulsivedynamic systems on time scales.