| In this paper, we mainly use Variational Lyapunov Method to study stabilityand boundedness for impulsive hybrid systems as followsWheref(t, x,λk(xk))=F(t, x)+R(t, x,λk(xk)), F∈C[R+×Rn, Rn], R∈C[R+×Rn×Rm, Rn], Ik∈C[Rn, Rn],λk∈C[Rn, Rm], k=0,1,2…,It is known that the phenomena of impulsive widcly exist in thc practicalproblems of many fields in the modern technology, many of them can bc describedby impulsive differential systems. But with the development of the technology,there are a lot of new kinds of mathematical models that can not bc dcscribed bythe impulsive differential systems. In this case, we should switch to a new set ofdifferential equations taking into consideration momentary perturbations of im-pulsive nature. A general description of such systems was called impulsive systemswith variable structure. hnpulsive hybrid system is a special but important caseof it, its characteristic is that its equations in different time periods may be differ-ent and the equation in the latter depends on the former. When the equations inthe different time periods are the same, impulsive hybrid system reduces to theimpulsive differential systems. Impulsive hybrid system is the further extensionof impulsive differential systems.In recent years, impulsive hybrid system gets much attention from many au-thors. But so far, the method of study is only the Lyapunov functiondircct and comparison method. Rencently some authors use cone-valued Lyapunovfunction method and get new achivments.Since f(t, x,λk(xk))=F(t, x)+R(t, x,λk(xk))and wc consider R(t, x,λk(xk) asthe perturbation of x'=F(t,x). We know that variational Lyapunov function method is a useful tool in the investigations of the perturbed systems. Basedon this idea, we use variational Lyapunov function to study the stability the im-pulsive hybrid system.If vector Lyapunov function is used to study the impulsivc hybrid system, anunpleasant fact is the requirement of a quasimontone nondecreasing property ofthe comparison systems and this property is too strong to limit the application.To solve this problem, we can choose suitable cone Z in which quasimontone non-decreasing property satisfied. To pay attention, we should choose the same coneZ in which g(t, s, p, v) satisfy quasimontone nondecreasing property toμandv.The concepts in terms of two measures describe initial value and state ofsolution separately by means of two measures, it enable us to unify a variety ofstability notions found in the literature. When using variational Lya-punov function to study the impulsive hybrid system, we choosc a same initialmeasures h0; When using cone-variational Lyapunov function to study thc impul-sive hybrid system, we should defy measures Q0, Q in the coneZ.Based on the ideas all the above, This paper is divided into two chapters:In chapter one, we use variational Lyapunov function to study the stabil-ity and boundness of the impulsive hybrid system and get (h0, h)-stability andboundness theory.In chapter two, we use cone variational Lyapunov function to study the sta-bility and boundness of the impulsive hybrid system and get (h0. h)-stability andboundness theory.
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