On the stability of integral sets of impulse systems of differential equations (Q2768856)

Here, a nonautonomous impulse system of the form NEWLINE\[NEWLINE\dot{x}=f(t,x),\quad t\neq\tau_i,\quad \Delta x|_{t=\tau_i}=I_i(x),\tag{1}NEWLINE\]NEWLINE with \(x\in \mathbb{R}^n\), \(t\in \mathbb{R}_+ \), \(0=\tau_0<\tau_1<\tau_2\ldots\), \(\lim_{i\to \infty}\tau_i= \infty \) is considered. Based on ideas of the direct Lyapunov method, the authors establish sufficient conditions for stability and asymptotic stability of (positively) integral sets to (1). Namely, let \({\mathcal H}\subset C(\mathbb{R}_+ \mapsto \mathbb{R}_+)\cap C((0,\infty) \mapsto(0,\infty))\) be the set of functions vanishing at the origin of \(\mathbb{R}_+\), and let there exist a set \(M\subset\mathbb{R}_+\times \mathbb{R}^n\) and functions \(V(t,x):\mathbb{R}_+\times \mathbb{R}^n \mapsto \mathbb{R}_+\), \(\varphi,\psi_{i}\in {\mathcal H}\), \(\alpha \in C(\mathbb{R}_+ \mapsto \mathbb{R}_+)\) which satisfy the following conditions: NEWLINE\[NEWLINEV(t,x)=0\;\forall (t,x)\in M; \quad V(t,x)>a(\text{dist}(x,M_t)) \forall (t,x)\in \mathbb{R}_+\times \mathbb{R}^n,NEWLINE\]NEWLINE NEWLINE\[NEWLINEV'_t+\langle \text{grad}_x V,f\rangle \leq-\alpha(t)\varphi(V),\;t\neq\tau_i, \quad V(\tau_i+0,x+I_i(x))\leq\psi(V(\tau_i,x)),\;i=1,2,\ldotsNEWLINE\]NEWLINE Here, \(M_t\) denotes the section of \(M\) by the ``vertical'' hyperplane passing through the point \((t,0)\). The authors impose conditions on the functions \(\alpha,\varphi,\psi_i\) under which the set \(M\) is integral and stable (asymptotically stable).