zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Novel robust stability criterion for a class of neutral systems with mixed delays and nonlinear perturbations. (English) Zbl 1065.34076
This paper is concerned with the asymptotic stability of the delay differential systems of neutral type $$x'(t) - C x'(t-\tau_2)= A x(t) + B x(t-\tau_1)+ f_1(t,x(t))+f_2(t,x(t-\tau_1), \quad t \ge 0, \tag $*$ $$ where $ \tau_1, \tau_2$ are positive constant delays, $ A,B,C$ constant real matrices of appropriate dimensions, and the nonlinear functions $ f_i$ satisfy $ \Vert f_i(t,u) \Vert \le \alpha_i \Vert u \Vert$, $i=1,2$, for some scalars $ \alpha_i$. For system $(*)$, the author gives sufficient conditions on the matrix coefficients that imply the asymptotic stability of the zero solution. The main advantage of the present result is that the sufficient conditions for stability can be checked by means of a convex optimization algorithm whereas other stability criteria are expressed in terms of matrix norms and turn out to be more conservative. Some examples of two-dimensional problems are presented in which the stability is tested by using Matlab’s LMI Control Toolbox.

34K20Stability theory of functional-differential equations
34K40Neutral functional-differential equations
Full Text: DOI
[1] Hale, J.; Lunel, S. M. Verduyn: Introduction to functional differential equations. (1993) · Zbl 0787.34002
[2] Kolmanovskii, V.; Myshkis, A.: Applied theory of functional differential equations. (1992) · Zbl 0917.34001
[3] Hu, G. D.; Hu, G. D.: Some simple stability criteria of neutral delay-differential systems. Applied mathematics and computation 80, 257-271 (1996) · Zbl 0878.34063
[4] Park, J. H.: Stability criterion for neutral differential systems with mixed multiple time-varying delay arguments. Mathematics and computers in simulation 59, 401-412 (2002) · Zbl 1006.34072
[5] Park, J. H.; Won, S.: A note on stability of neutral delay-differential systems. Journal of the franklin institute 336, 543-548 (1999) · Zbl 0969.34066
[6] Park, J. H.; Won, S.: Asymptotic stability of neutral systems with multiple delays. Journal of optimization theory and applications 103, 187-200 (1999) · Zbl 0947.65088
[7] J.H. Park, Delay-dependent criterion for asymptotic stability of a class of neutral equations, Applied Mathematics Letters, in press · Zbl 1122.34339
[8] Park, J. H.: On the design of observer-based controller of linear neutral delay-differential systems. Applied mathematics and computations 150, 195-202 (2004) · Zbl 1043.93032
[9] Fan, K. K.; Chen, J. D.; Lien, C. H.; Hsieh, J. G.: Delay-dependent stability criterion for neutral time-delay systems via linear matrix inequality approach. Journal of mathematical analysis and applications 273, 580-589 (2002) · Zbl 1010.93084
[10] Lien, C. H.: New stability criterion for a class of uncertain nonlinear neutral time-delay systems. International journal of systems science 32, 215-219 (2001) · Zbl 1018.34069
[11] Han, Q. L.: Robust stability of uncertain delay-differential systems of neutral type. Automatica 38, 719-723 (2002) · Zbl 1020.93016
[12] Yue, D.; Won, S.; Kwon, O.: Delay-dependent stability of neutral systems with time delay: an LMI approach. IEE Proceedings----control theory and applications 150, 23-27 (2003)
[13] Boyd, B.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V.: Linear matrix inequalities in systems and control theory. (1994) · Zbl 0816.93004
[14] K. Gu, An integral inequality in the stability problem of time-delay systems, in: Proceedings of 39th IEEE CDC Sydney, Australia (2000) 2805--2810
[15] P. Gahinet, A. Nemirovski, A. Laub, M. Chilali, LMI Control Toolbox User’s Guide, The Mathworks, Natick, Massachusetts, 1995