Vector Lyapunov functions: Nonlinear, time-varying, ordinary and functional differential equations. (English) Zbl 1039.34066

Borne, P.; Dambrine, M.; Perruquetti, W.; Richard, J. P.

Martynyuk, A. A. (ed.), Advances in stability theory at the end of the 20th century. London: Taylor & Francis (ISBN 0-415-26962-8/hbk). Stab. Control Theory Methods Appl. 13, 49-73 (2003).

The authors consider the system of functional-differential equations \[ dx(t)/dt=f(t,x(t),x_t,d),\qquad x_{t_0}=\varphi, \tag{1} \] where \(t\in {\mathbb R}\) is the time variable, \(d\in S_d\) is either a vector or a function representing disturbances or parameter uncertainties of the system, \(S_d\) is a set of vector or functions for which some bounds are usually supposed to be known, \(x(t)\in {\mathbb R}^n\) is a set of internal variables, \(x_t\) is the map defined by \[ x_t: [-\tau,0]\to {\mathbb R}^n. \] A comparison system to (1) and a vector Lyapunov function are considered, and then some stability properties of some compact set \({ A}\) for system (1) are stated.
Reviewer’s Remark: Definition 5.3 of this paper is in contradiction to the classical definition of attractivity [see Definition 2.5 of the book Stability theory by Lyapunov’s direct method by N. Rouche, P. Habets and M. Laloy, New York etc.: Springer-Verlag (1977; Zbl 0364.34022)]. To define the attractivity correctly, in Definition 5.3 \(\delta\) should not depend on \(\varepsilon\), and \(T\) should depend not only on \(t_0, \varepsilon\) but also on \(\varphi\).
For the entire collection see [Zbl 1020.00004].

Reviewer: Alexander O. Ignatyev (Donetsk)

MSC:

34K20

Stability theory of functional-differential equations

Keywords:

stability; vector Lyapunov functions

Citations:

Zbl 0364.34022