Asymptotically stable sets of control systems with impulse actions

We get sufficient conditions for asymptotic stability and weak asymptotic stability of a given set $\mathfrak M\doteq\bigl\{(t,x)\in [t_0,+\infty)\times\mathbb{R}^n: x\in M(t)\bigr\}$ with respect to the control system with impulse actions. We assume that the function $t\mapsto M(t)$ is continuous in the Hausdorff metric and for each $t \in [t_0,+\infty)$ the set $M(t)$ is nonempty and closed. Also, we obtain conditions under which for every solution $x(t,x_0)$ of the control system that leaves a sufficiently small neighborhood of the set $M(t_0)$ there exists an instant $t^*$ such that point $(t,x(t,x_0))$ belongs to $\mathfrak M$ for all $t\in[t^*,+\infty).$ Some of the statements presented here are analogues of the results obtained by E.A. Panasenko and E.L.Tonkov for systems with impulses, and in other statements the specificity of impulse actions is essentially used. The results of this paper are illustrated by the “pest–bioagents” model with impulse control and we assume that the addition of bioagents (natural enemies of the given pests) occur at fixed instants of time and the number of pests consumed on average by one biological agent per unit time is given by the trophic Holling function. We obtain conditions for asymptotic stability of the set $\mathfrak M=\bigl\{(t,x)\in \mathbb R^3_+: x_1\leqslant C_1\bigr\},$ where $x_1=y_1/K,$ $y_1$ is the size of the population of pests and $K$ is the capacity of environment.