Uniform Stability with Respect to the Impulse Hypersurfaces of the Solutions of Differential Equations with Impulses

The systems of differential equations with impulses are divided into several classes according to the way in which the moments of the impulses are determined. In the present paper the initial value problem is considered for systems of differential equations for which the impulses are realized at the moments when the integral curve of the problem meets certain hypersurfaces, called impulse hypersurfaces, of the extended phase space. The notions of strong uniform stability of the solutions of systems without impulses and uniform stability with respect to the impulse hypersurfaces of the solutions of systems with impulses are introduced. The main results are given in two theorems. The first one contains sufficient conditions under which strong uniform stability of the zero solution of the respective system without impulses implies uniform stability with respect to the impulse hypersurfaces of the zero solution of the initial system with impulses. In the second theorem sufficient conditions are given under which uniform Lipschitz stability of the zero solution of the corresponding system without impulses implies uniform stability with respect to the impulse hypersurfaces of the zero solution of the initial system with impulses.