Stability of Impulsive Differential Equation with any Time Delay
Résumé
In this paper, the stability of general impulsive retarded functional differential equations with any time delay has been considered. Many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. Impulsive differential equations, that is, differential equations involving impulse effects, are a natural description of observed evolution phenomena of several real world problems. Impulsive control which based on impulsive differential equations has attracted the interest of many researchers recently. The method of Lyapunov functions and Razumikhin technique have been widely applied to stability analysis of various delay differential equation. When Lyapunov functions are used, it becomes necessary to choose an appropriate minimal class of functionals relative to which the derivative of the Lyapunov function is estimated. This approach is known as the Lyapunov-Razumikhin technique. When Lyapunov functionals are used the corresponding derivative can be estimated without demanding minimal classes of functional. By using Lyapunov functions and analysis technique along with Razumikhin technique, some results for the uniform stability of such impulsive differential equations have been derived. The obtained results extend and generalize some results existing in the literature.