Weak asymptotic stability of control systems with impulsive actions

We continue investigating the conditions of positive invariance and asymptotic stability of a given set relative to a control system with impulsive actions. We consider the set $\mathfrak M\doteq\{(t,x)\in[t_0,+\infty)\times\mathbb R^n\colon x\in M(t)\}$, given by a function $t\to M(t)$ that is continuous in the Hausdorff metric, where the set $M(t)$ is nonempty and compact for each $t\in\mathbb R$. In terms of the Lyapunov functions and the Clarke derivative, we obtain conditions for weak positive invariance, weak uniform Lyapunov stability and weak asymptotic stability of the set $\mathfrak M$. Also we prove a comparison theorem for solutions of systems and equations with impulses the consequence of which is the conditions for existence of solutions of the system that asymptotically tends to zero. The obtained results are illustrated by the example of model for competition of two species exposed to impulse control at given times.