Impulse systems with fixed moments of shocks of general position: the structure of the set of shock moments (Q919156)

The authors continue their study on the impulse system (1) \(dx/dt=f(t,x)\), \(t\neq t_i\), \(t\in [a,b]\), \(x\in \Omega \subset\mathbb{R}^n\), (2) \(\Delta x/t_ i=h_ i(x)\), \(i\in \mathbb N\), when the set \(T=\{t_i, \ i\in \mathbb N\}\) has a finite number of limit points. A topological structure of the set \(T\) of shock moments is pointed out. It is assumed that \(\sum_{a\leq t_i\leq b}H_i = \sum^{\infty}_{i=1}H_i\), and \(\displaystyle\sum_{\substack{t_i; \vert t-t_i\vert <\varepsilon;\\ t\notin T}} H_i= o(\varepsilon)\), \(\varepsilon\to 0\), where \(\displaystyle H_i = \sup_{x\in \Omega}\Vert h_i(x)\Vert >0\).