Stability of the solutions of singularly perturbed systems with impulse effect (Q1119806)

The exponential stability of the trivial solution \(x=y=0\) of the system ODE's with impulse effect is investigated. The system in question has a form: \(\dot x=f(t,x,y)\), \(\mu\) \(\dot y=g(t,x,y)\), \(t\neq \tau_ k\), \(\Delta x|_{t=\tau_ k}=I_ k(x,y)\), \(\Delta y|_{t=\tau_ k}=J_ k(x,y),\) \(k=1,2,...\), \(x\in {\mathbb{R}}^ n\), \(y\in {\mathbb{R}}^ m\) and \(\mu\) is a small positive parameter. The authors make use of partially continuous auxiliary functions, which are analogues of Lyapunov functions. Also the conditions are given, under which the Lyapunov partially continuous functions exist.