Invariant sets of impulsive differential equations with particularities in \(\omega \)-limit set. (English) Zbl 1238.34022

Perestyuk, Mykola; Feketa, Petro

Abstr. Appl. Anal. 2011, Article ID 970469, 14 p. (2011).

The authors establish sufficient conditions for the existence and asymptotic stability of invariant sets for the following system of differential equations with impulses: \[ \begin{align*}{ \dot{\phi} &= a(\phi), \cr \dot{x} &=A(\phi)x+f(\phi), \quad \phi\not\in \Gamma, \cr \Delta x\vert_{\phi\in\Gamma} &=B(\phi)x+g(\phi.) \cr}\end{align*} \] Here, \(\Gamma\) is a hypersurface regularly embedded into the m-torus \(\mathbb T^{m}:=\mathbb R^{m}/2\pi \mathbb Z^{m}\), \(A,\;B\in C(\mathbb T^{m},\operatorname{Hom}(\mathbb R^{n}))\), \(a\in C(\mathbb T^{m},\mathbb R^{m})\) and satisfies a Lipschitz condition, and the functions \(f,\;g\in C(\mathbb T^{m},\mathbb R^{n})\) (piecewise continuous with first kind discontinuities in the set \(\Gamma\)). The main results are obtained by using the Lyapunov functions method.

Reviewer: Giovanni Giachetta (Camerino)

MSC:

34A37	Ordinary differential equations with impulses
34D35	Stability of manifolds of solutions to ordinary differential equations
34C45	Invariant manifolds for ordinary differential equations

Keywords:

differential equations with impulses; invariant manifold; asymptotic stability

References:

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