The Stability Study Of Nonlinear Impulsive Dynamic Systems On Time Scales

Dynamic systems on time scales, which can unify the continuous and discrete systems, have gained more and more important applications in each field of modern technology. For example.the establishing of biology models must consider the continuity of community growing. Differential equations could be programmed when the animals continuously grow and reproduce in some season, and difference equations should be planned when the community have been in the state of the ovums' hatching or dormancy. This kind of issue can be subject to the dynamic systems on time scales.At the same time, the impulsive phenomena widely exists in practical problems of many fields in modern technology, so the study of impulsive dynamic systems on time scales has gained vital practical significance and applied background. In this paper, we mainly study the stability in terms of two measures for impulsive hybrid dynamic systems on time scales as followand we also study the stability in terms of two measures for impulsive dynamic systems on time scales as followIn every time section (t_k,t_k-1) of impulsive hybrid dynamic systems on time scales, f(t,x,λ_k(x(t_k))) = F(t,x) + R(t,x,λ_k(x(t_k))), we consider R(t,x,λ_k(x(t_k))) as the perturbation of x^Δ(t) = F(t,x). It is well known that variational Lyapunov function method is a usefull tool in the investigations of the perturbed systems[19]. Based on the idea, we use variational Lyapunov function method to study the stability theory of impulsive hybrid dynamic systems on time scales (I).In chapter one, firstly, we introduce the basic concepts of time scale calculus. Secondly,we expound the basic ideas of variational Lyapunov function method on time scales, and we establish a new comparison theorem by employing variational Lyapunov function method. At last, by using the comparison theorem, we study the stability of impulsive hybrid dynamic systems on time scales (I), and we obtain some results, such as (h₀,h)-stability,(h₀,h)-asymptotic stability, (h₀,h)-practical stability, (h₀, h)-eventual stability and so on. An example was given to illustrate the application of the theorems.In chapter two, by employing Lyapunov direct method , we study the stability of impulsive dynamic systems (â…£), and we also obtain some results, such as (h₀,h)-stability, (h₀,h)- asymptotic stability, (h₀,h)-practical asymptotic stability. Different as before, because what we are studying now are impulsive dynamic systems on time scales, we should impose conditions everywhere on V(t,x). not only in S(h,ρ) = {(t,x)∈R⁺×Rⁿ: h(t, x) <ρ}for someρ> 0. We also give an example to illustrate the application of the theorems.