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Stability of the solutions of singularly perturbed systems with impulse effect. (English) Zbl 0671.34045

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.
Reviewer: A.Klič

MSC:

34D15 Singular perturbations of ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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