zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Delay-dependent exponential stabilization for uncertain linear systems with interval non-differentiable time-varying delays. (English) Zbl 1241.34080

The problem of exponential stability and stabilization is studied for a class of uncertain linear systems with time-varying delay. The time delay is a continuous function belonging to a given interval, which means that the lower and the upper bounds for the time-varying delay are available.

The distinctive features of the presented results are, by the author’s opinion, the following: the delay function is not necessary to be differentiable and the lower bound of the delay is not restricted to be zero. Notice that, in the most delay-dependent stability results for systems with time-varying delay, the time delay function is required to be differentiable and, moreover, the upper bound of the derivative is restricted to a number less than unity.

Based on the construction of some Lyapunov-Krasovskii functionals, new delay-dependent sufficient conditions for the exponential stabilization of the systems are established in terms of LMIs. Numerical examples are given to demonstrate the effectiveness of the derived conditions.

MSC:
34K20Stability theory of functional-differential equations
34K06Linear functional-differential equations
34K27Perturbations of functional-differential equations
34K35Functional-differential equations connected with control problems
References:
[1]Gu, K.; Kharitonov, V. L.; Chen, J.: Stability of time-delay system, (2003)
[2]Hale, J. K.; Lunee, S. M. Verduyn: Introduction to functional differential equations, (1993)
[3]Dacunha, J. J.: Stability for time-varying linear dynamical systems on time scale, J. comput. Appl. math. 176, 381-410 (2000) · Zbl 1064.39005 · doi:10.1016/j.cam.2004.07.026
[4]Kharitonov, V. L.; Hinrichsen, D.: Exponential estimates for time-delay systems, Syst. control lett. 53, 395-405 (2004) · Zbl 1157.34355 · doi:10.1016/j.sysconle.2004.05.016
[5]Kolmanovskii, V. B.; Niculescu, S.; Richard, J. P.: On the Lyapunov – Krasovskiĭ functionals for stability analysis of linear delay systems, Int. J. Control 72, 374-384 (1999) · Zbl 0952.34057 · doi:10.1080/002071799221172
[6]Phat, V. N.; Savkin, A. V.: Robust state estimation for a class of linear uncertain time-delay systems, Syst. control lett. 47, 237-245 (2002) · Zbl 1106.93339 · doi:10.1016/S0167-6911(02)00203-7
[7]Phat, V. N.; Nam, P. T.: Exponential stability criteria of with multiple delays, Elect. J. Diff. eqn. 58, 1-9 (2005)
[8]Kwon, O. M.; Park, J. H.: On improved delay-dependent robust control for uncertain time-delay systems, IEEE trans. Automat. control 49, 1991-1995 (2004)
[9]Qian, W.; Cong, S.; Sun, Y.; Fei, S.: Novel robust stability criteria for uncertain systems with time-varying delay, Appl. math. Comput. 215, 866-872 (2009) · Zbl 1192.34087 · doi:10.1016/j.amc.2009.06.022
[10]Park, J. H.; Kwon, O.: Novel stability criterion of time delay systems with nonlinear uncertainties, Appl. math. Lett. 18, 683-688 (2005) · Zbl 1089.34549 · doi:10.1016/j.aml.2004.04.013
[11]Li, X.; De Souza, C. E.: Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach, IEEE trans. Automat. control 42, 1144-1148 (1997) · Zbl 0889.93050 · doi:10.1109/9.618244
[12]Kwon, O. M.; Park, J. H.: Delay-range-dependent stabilization of uncertain dynamic systems with interval time-varying delays, Appl. math. Comput. 208, 58-68 (2009) · Zbl 1170.34054 · doi:10.1016/j.amc.2008.11.010
[13]Wang, Y.; Xie, L.; De Souza, C. E.: Robust control of a class of uncertain nonlinear systems, Syst. control lett. 19, 139-149 (1992) · Zbl 0765.93015 · doi:10.1016/0167-6911(92)90097-C
[14]Zhanga, X. -M.; Wua, M.; Shec, J. -H.; He, Y.: Delay-dependent stabilization of linear systems with time-varying state and input delays, Automatica 41, 1405-1412 (2005) · Zbl 1093.93024 · doi:10.1016/j.automatica.2005.03.009
[15]I. Amri, D. Soudani, M. Benrejeb, Exponential stability and stabilization of linear systems with time varying delays, in: 6th International Multi-Conference on Systemsm Signals and Devices, 2009, pp. 1 – 6.
[16]Hien, L. V.; Phat, V. N.: Exponential stability and stabilization of a class of uncertain linear time-delay systems, J. franklin inst. 346, 611-625 (2009) · Zbl 1169.93396 · doi:10.1016/j.jfranklin.2009.03.001
[17]Nam, P. T.; Phat, V. N.: Robust exponentially stability and stabilization of linear uncertain polytopic time-delay systems, J. control theory appl. 6, No. 2, 163-170 (2008)
[18]Botmart, T.; Niamsup, P.: Robust exponential stability and stabilizability of linear parameter dependent systems with delays, Appl. math. Comput. 217, 2551-2566 (2010) · Zbl 1207.93087 · doi:10.1016/j.amc.2010.07.068
[19]Kwon, O. M.; Park, J. H.: Robust exponential stability of uncertain dynamic systems including state delay, Appl. math. Lett. 19, 901-907 (2006) · Zbl 1220.34095 · doi:10.1016/j.aml.2005.10.017
[20]Park, J. H.; Jung, H. Y.: On the exponential stability of a class of nonlinear systems including delayed perturbations, J. comput. Appl. math. 159, 467-471 (2003) · Zbl 1033.93055 · doi:10.1016/S0377-0427(03)00550-8
[21]Shao, H. -Y.: New delay-dependent stability criteria for systems with interval delay, Automatica 45, No. 3, 744-749 (2009) · Zbl 1168.93387 · doi:10.1016/j.automatica.2008.09.010
[22]Han, Q. -L.: Robust stability for a class of linear systems with time varying delay and nonlinear perturbations, Comput. math. Appl. 47, 1201-1209 (2004) · Zbl 1154.93408 · doi:10.1016/S0898-1221(04)90114-9
[23]Han, Q. -L.; Gu, K.: Stability of linear systems with time-varying delay: a generalized discretized Lyapunov functional approach, Asian J. Control 3, 170-180 (2001)
[24]Tian, J.; Zhou, X.: Improved asymptotic stability criteria for neural for networks with interval time-varying delay, Expert syst. Appl. 37, 7521-7525 (2010)
[25]Yu, K. -Y.; Lien, C. -H.: Stability criterion for uncertain neutral systems with interval time-varying delays, Chaos solitons fract. 38, 650-657 (2008) · Zbl 1146.93366 · doi:10.1016/j.chaos.2007.01.002
[26]Wang, D.; Wang, W.: Delay-dependent robust exponential stabilization for uncertain systems with interval time-varying delays, J. control theory appl. 7, No. 3, 257-263 (2009)
[27]Zhang, Y.; Yue, D.; Tian, E.: New stability criteria of neural networks with interval time-varying delay: a piecewise delay method, Appl. math. Comput. 208, 249-259 (2009) · Zbl 1171.34048 · doi:10.1016/j.amc.2008.11.046