Stability Analysis For Impulsive Hybrid Systems On Time Scales

In this paper, we study the stability properties of the two nonlinear impulsive dy-namic systems on time scales as follows:(1) impulsive hybrid systems on time scales(2) impulsive dynamic systems on time scaleswhere I0(x0)=0, xΔ(t) is the delta derivative of x(t) at t, We get the results of stability in terms of two measures for impulsive hybrid systems on time scales, the results of exponential stability and stability in terms of two measures for impulsive Dy-namic systems on time scales. Examples are also discussed to illustrate the theorems, respectively.Dynamic systems on time scales can unify the continuous and discrete systems. On the one hand, the theory on time scales was put up a bridge between continuous analysis and discrete analysis; on the other hand, Some mathematical models in actual problems established on time scales are more realistic than before, For example, it can be modeled insect populations that are continuous in some season by differential equations, however in other season the community have been in the state of the ovums'hatching or dormancy, So it only can be modeled insect populations by difference equations at this time. This kind of issue can be subject to the dynamic systems on time scales.But there is often accompanied by sudden change phenomena in physics, biology, medicine and other areas of modern technology, the mathematical model of these phe-nomena in actual problems can be attributed to impulsive differential systems, at present, this research has achieved fruitful results, but the results of impulsive dynamic systems on time scale is also a few, so the study of impulsive dynamic systems on time scales has gained vital practical significance and applied background.In chapter one, firstly, we introduce the basic concepts of time scale calculus. Then, we establish a new comparison theorem of impulsive hybrid systems on time scales. Finally, we studied the stabilities of impulsive hybrid systems on time scales by the new comparison theorem and vector comparison systems corresponding to impulsive hybrid systems on time scales, and we obtain some results, such as (h0,h)- stability, (h0,h) -asymptotic stability, (h0,h)- actual stability. An example is given to illustrate the application of the theorems.In the first part of chapter two, we mainly built a impulsive differential inequality, and obtain some results of exponential stability of impulsive dynamic systems on time scales by using Lyapunov function method, and regard the impulsive linear perturbation differential system as a special kind of system, without the aid of Lyapunov functions, we directly obtain it's exponential stability by using perturbation ideas; In the second part, using Lyapunov direct method functions, we get the stability of impulsive dynamic systems on time scales by setting weaker conditions for discrete and continuous part of Lyapunov function. Finally, an example is given to verify the application of the theorems.