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This paper focuses on the stability analysis and invariant set stability theorems for nonlinear impulsive systems. A set of Lyapunov-based sufficient conditions are established for these convergent properties. These results do not require the Lyapunov function to be positive definite. Inequalities relating the righthandside of the differential equation and the Lyapunov function derivative are involved for these results. These inequalities make it possible to deduce properties of the functions and thus leads to sufficient conditions for convergence and stability.

It is well known that integral input-to-output stability(IIOS) is weaker than input-to -output stability(IOS).In this paper,IIOS problems for one class of discontinuous dynamical systems are considered.Piecewise smooth IIOS-Lyapunov functions are adopted.Furthermore,interconnecti ons of discontinuous dynamical systems are studied.The results for a class of discontinuous dynam ical systems and interconnection of discontinuous dynamical systems' IIOS character are shown,res pectively.