Reliable Computing 9: 303–313, 2003.
2003 Kluwer Academic Publishers. Printed in the Netherlands.
Asymptotic Stability of Interval Time-Delay
SVETLANA P. SOKOLOVA
St.Petersburg Institute for Informatics and Automation of the Russian Academy of Sciences,
14-th line, 39, V.O., 199178 St.Petersburg, Russia, e-mail: sokolova
RUSLAN S. IVLEV
Institute of Informatics and Control Problems, 125 Pushkin Str., 480100 Almaty, Republic of
Kazakhstan, e-mail: email@example.com
(Received: 23 December 1999; accepted: 15 April 2003)
Abstract. In this paper we consider the asymptotic stability of linear interval time-delay systems on
the base of using Lyapunov’s direct method and methods of interval analysis. A sufﬁcient condition
of asymptotic stability is obtained using the concept of Lyapunov–Krasovsky functional.
A classical theory of stability was considered the last century and allows to inves-
tigate a rather wide spectrum of processes when having an exact mathematical
description. However, when constructing mathematical models of processes in
practice tolerances in model parameters may appear. One way to take into account
these tolerances is to specify intervals with known bounds. Mathematical models
of such processes can be represented using rules and terminology of intensively
developing interval mathematics.
First formulated in  and then studied and comprehensively solved in  the
investigation of asymptotic stability of a characteristic interval polynomial gave
an impetus to further research in this ﬁeld. A further development of the theory
represented in  generalizes results for the case of interval quasipolynomials ,
occurring when investigating differential time-delay equations. Among the subse-
quent researches considering the asymptotic stability of the equilibrium position
for differential time-delay equations one should mention  and the references
there. In that paper, which is a development of the theory represented in , some
sufﬁcient conditions of the asymptotic stability of an interval quasipolynomial were
obtained on the base of four functions.
The ﬁeld of dynamic properties of interval systems given in the state space is
somewhat different. Attempts to generalize the results of the paper  and to obtain
some analogues of Kharitonov’s theorems for interval matrices  failed. Nowadays