Some stability properties for impulsive differential equations with respect to initial time difference

The main goal of the paper is studying stability properties of the difference of solutions of a system of nonlinear impulsive differential equations with different initial data. The initial data differs both in initial time and initial position. In many real repeated experiments it is difficult to keep both the initial position and the initial time unchanged because of all kinds of disturbed factors. It requires the changing of initial time to be taken into consideration. In the paper various types of stability with respect to initial time difference are studied. The investigations are based on piecewise continuous Lyapunov functions and comparison results for scalar impulsive differential equations. A special type of appropriately generalized derivative of Lyapunov functions is defined and applied. It is proved that the equistability/uniform stability/Lipschitz stability property of the zero solution of the comparison scalar equation implies the corresponding stability properties with respect to difference initial time of the given nonlinear system. The obtained sufficient conditions significantly depend on the impulses.