Lyapunov functions and comparison theorems for control systems with impulsive actions

We extend the results of E. L. Tonkov and E. A. Panasenko to differential equations and control systems with impulsive actions. In terms of Lyapunov functions and the Clarke derivative we obtain comparison theorems for systems with impulsive effect. We consider the set $\mathfrak M\doteq\bigl\{(t,x)\in[t_0,+\infty)\times\mathbb R^n\colon x\in M(t)\bigr\}$, defined by continuous function $t\to M(t)$, where for every $t\in\mathbb R$ the set $M(t)$ is nonempty and compact. We obtain conditions for the positive invariance of this set, the uniform Lyapunov stability and the uniform asymptotic stability. We make a comparison with the researches of other authors who have considered the zero solution stability for similar systems.