On the general problem of stability for impulsive differential equations. (Q1421184)

This paper is devoted to criteria for stability, asymptotical stability and instability of nontrivial solutions of the impulsive system \[ \begin{gathered} {dx\over dt}= f(t,x),\quad t\neq \theta_i(x),\\ \Delta x|_{t=\theta_i(x)}= I_i(x),\quad i\in\mathbb{N}= \{1,2,\dots\},\end{gathered}\tag{1} \] with \(\Delta x|_{t=\theta}= x(\theta+)- x(\theta)\), \(x(\theta+)= \lim_{t\to\theta^+}\, x(t)\), obtained by Lyapunov's second method. The author indicates that a construction of a reduced system for (1) with variable time of impulsive action is done for the first time.