Stability of functional differential equations with variable impulsive perturbations via generalized ordinary differential equations
type: Item
instance of: scholarly article
title: Stability of functional differential equations with variable impulsive perturbations via generalized ordinary differential equations
zbMATH Open document ID: 1273.34085
author: S. M. Afonso
author: M. Federson
author: L. P. Gimenes
author: E. M. Bonotto
published in: Bulletin des Sciences Mathématiques
publication date: 2013-03-15T00:00:00Z
review text: The authors consider impulsive functional differential equations of the form \[ \begin{aligned} y'(t)&=f(t,y_t)\;\quad\text{ for } t\not=\tau_k(y(t)),\;k\in\mathbb N,\\ \Delta y(t)&=I_k(y(t))\quad\text{ for } t=\tau_k(y(t)),\;k\in\mathbb N,\end{aligned}\tag{1} \] where \(y\) and \(f\) take values in \(\mathbb R^n\), the functions \(\tau_k:\mathbb R^n\to\mathbb R\) correspond to hypersurfaces in \(\mathbb R^n\), and the impulses described by the operators \(I_k:\mathbb R^n\to\mathbb R^n\) take place whenever the solution \(y\) intersects one of the hypersurfaces. The main results of the paper are Lyapunov-type theorems related to the uniform stability and uniform asymptotic stability for the trivial solution of (1). The main idea is to convert (1) to an equivalent generalized ordinary differential equation (as introduced by \textit{J. Kurzweil}, see, e.g., [Czech. Math. J. 7(82), 418--449 (1957; Zbl 0090.30002); Generalized ordinary differential equations. Not absolutely continuous solutions. Hackensack, NJ: World Scientific (2012; Zbl 1248.34001)] and \textit{Š. Schwabik} [Generalized ordinary differential equations. Singapore: World Scientific (1992; Zbl 0781.34003)]) with values in an infinite-dimensional Banach space, and then apply the existing stability results for generalized equations. Reviewer's remark: Although the authors say that they consider equations of form (1) with state-dependent impulses, it seems that their method is applicable only to equations with preassigned moments of impulse action which are of the form \[ \begin{aligned} y'(t)&=f(t,y_t)\;\quad\text{ for } t\not=t_k^i,\;k\in\mathbb N,\;i\in\{1,\dotsc,m(\tau_k)\} \\ \Delta y(t)&=I_k(y(t))\quad \text{ for }t=t_k^i,\;k\in\mathbb N,\;i\in\{1,\dotsc,m(\tau_k)\}.\end{aligned}\tag{2} \] The reason is that, in (1), the moments \(t_k^i\) at which the solution \(y\) intersects the hypersurface given by \(\tau_k\) depend on the choice of \(y\). However, the \(t_k^i\) appear in the definition of the fun…
reviewed by: Antonín Slavík
Mathematics Subject Classification ID: 34K45
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 6145431
zbMATH Keywords: generalized ordinary differential equations
zbMATH Keywords: impulsive functional differential equations
zbMATH Keywords: stability
zbMATH Keywords: Lyapunov functionals
MaRDI profile type: Publication
full work available at URL: j.bulsci.2012.10.001
OpenAlex ID: W2033572723
DOI: 10.1016/J.BULSCI.2012.10.001
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description: scientific article; zbMATH DE number 6145431