On the Global Existence of Solution of the Comparison System and Vector Lyapunov Asymptotic Eventual Stability for Nonlinear Impulsive Differential Systems

In this paper, the existence of maximal solution of the comparison system as well as the asymptotic eventual stability of nonlinear impulsive differential equations with fixed moments of impulse is examined using the vector Lyapunov functions which is generalized by a class of piecewise continuous Lyapunov functions. It was distinctly established that the maximal solution of the comparison system majorizes the vector form of the Lyapunov functions. The novelty in the use of the vector Lypunov functions lies in the fact that the "restrictions" encountered by the scalar Lyapunov function is safely handled especially for large scale dynamical systems, since it involves splitting the Lyapunov functions into components so that each of the components can easily describe the behavior of the solution state. Together with comparison results, sufficient conditions for eventual stability and asymptotic are presented.