Impulsive Differential Systems Of Two Measures For The Boundedness And Stability Theorem

In this dissertation, mainly by virtue of Lyapunov direct method, the houndedness and stability in terms of two measures of impulsive differential system are discussed. There are three chapters altogether. First, in chapter 1, we discuss the boundedness in terms of two measures of impulsive differential system. For the effect of impulse, the derivative of Lyapunov function doesn need to be negative along the trajectory. which is necessary in the discussion on ordinary differential system. So the Lyapunov function may increase or decrease along the continuous parts and switch back and forth at the impulsive times. Through an example, we indicate that, after being applied impulsive effect, a former continuous system with unbounded solution may satisfis boundedness. In chapter 2, we investigate the stability in terms of two measures of impulsive differential system. First, we give some results on attractiveness in the large. Together with the theorems on boundedness in chapter 1, the Lagrange stabilities of impulsive differential system are discussed. Next, we discuss the stability of impulsive differential system. The theorem 2.3.5 is superior to the according result in [18] and the demand on Lyapunov function is obviously weaker. Single Lyapunov function is a useful tool for studying the stability. When we use a family of Lyapunov functions, the demand on each Lyapunov function can be weakened. This idea is well exhibited in theorem 2.3.6. In the literature, the perturbation problem is one of the popular problems. In chapter 3, we make an elementary investigation. Through the Lyapunov function applying on impulsive differential system, we can get the stability of its according perturbing system directly.