Asymptotic stability and instability of the solutions of systems with impulse action (Q881134)

In the present paper the authors consider the following system of ordinary differential equations with impulse action: \[ {dx\over dt}= f(t, x),\quad t\neq\tau_k,\quad\Delta x= I_k(x),\quad t=\tau_k,\tag{1} \] where \(t\in [0,\infty)\) is the time, \(k\in\mathbb{N}\), the \(\tau_k\) are constants, \(x\in\mathbb{R}^n\), \(f: \mathbb{R}^{n+1}\to \mathbb{R}^n\), and \(I_k: \mathbb{R}^n\to \mathbb{R}^n\). The authors study the stability of the zero solution of system (1). Sufficient conditions for asymptotic stability (Theorem 3.1) or instability (Theorem 3.2) are given.