Behaviour of invariant sets of impulsive systems with small perturbation (Q2703297)

Let us consider the system of differential equations with impulsive action at fixed moments \(\tau_{i}\) NEWLINE\[NEWLINEdx/ dt=X(x)+\mu Y(t,x), \quad t\neq \tau_{i}, \quad \Delta x|_{t=\tau_{i}}=\mu I_{i}(x),NEWLINE\]NEWLINE with \(\tau_{i}<\tau_{i+1}\), \(\tau_{i}\to\infty\), \(i\to\infty\); \(\mu\) is a small parameter. Let there be a constant \(C>0\) such that uniformly for \(t\geq 0\): \(i(t,t+T)\leq CT\), where \(i(t,t+T)\) is a number of impulse on the interval \([t,t+T]\). The functions \(X(x), Y(t,x), I_{i}(x)\) are continuous in the domain \(t\geq 0\), \(x\in D\subset \mathbb{R}^{n}\), \(i\in \mathbb{N}\), Lipschitzian on \(x\in D\), \(\|Y(t,x)\|+ \|I_{i}(x)\|\leq K\) for some \(K>0\). If \(M_{0}\) is a compact, asymptotically stable, invariant set of the system \(dx/ dt=X(x)\) and \(M_{0,d}\subset D\) is a \(d\)-neighborhood of \(M_{0}\), then \(\exists \mu_{0}>0\) such that \(\forall\mu<\mu_{0}\) the given system has a closed invariant set \(M_{\mu}^{t}\) for which \(\lim_{\mu\to 0}\sup_{t\geq 0}\rho(M_{0},M_{\mu}^{t})=0\).