Stability of linear descriptor systems with delay: a Lyapunov-based approach. (English) Zbl 1032.34069

Delay differential-algebraic equations having both delay and algebraic constraints are often called descriptor systems with delay. The author considers a descriptor system with multiple and distributed delays. Using a Lyapunov-Krasovskii functional, sufficient conditions for delay-dependent/delay-independent stability and robustness of stability with respect to small delays are given in terms of matrix inequalities.


34K20 Stability theory of functional-differential equations
Full Text: DOI


[1] Barbalat, I., Systems d’equations differentielles d’oscillations nonlinearies, Rev. roumaine math. pures appl., 4, 267-270, (1959) · Zbl 0090.06601
[2] Boyd, S.; Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, SIAM studies in applied mathematics, 15, (1994), SIAM Philadelphia · Zbl 0816.93004
[3] Brayton, R., Small signal stability criterion for electrical networks containing lossless transmission lines, IBM J. res. develop., 12, 431-440, (1968) · Zbl 0172.20703
[4] Campbell, S., Singular linear systems of differential equations with delays, Appl. anal., 2, 129-136, (1980) · Zbl 0444.34062
[5] Campbell, S., 2-D (differential-delay) implicit systems, (), 1828-1829
[6] Campbell, S.; Nikoukhah, R.; Delebecque, F., Nonlinear descriptor systems, (), 247-282
[7] Dai, L., Singular control systems, (1989), Springer-Verlag Berlin
[8] Charitonov, V.; Melchor-Aguilar, D., On delay-dependent stability conditions, Systems control lett., 40, 71-76, (2000) · Zbl 0977.93072
[9] El’sgol’ts, L.; Norkin, S., Introduction to the theory and applications of differential equations with deviating arguments, Mathematics in science and engineering, 105, (1973), Academic Press New York · Zbl 0287.34073
[10] Fridman, E., Effects of small delays on stability of singularly perturbed systems, Automatica, 38, 897-902, (2002) · Zbl 1014.93025
[11] Fridman, E., New lyapunov – krasovskii functionals for stability of linear retarded and neutral type systems, Systems control lett., 43, 309-319, (2001) · Zbl 0974.93028
[12] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht · Zbl 0752.34039
[13] Halanay, A.; Rasvan, V., Stability radii for some propagation models, IMA J. math. control inform., 14, 95-107, (1997) · Zbl 0873.93067
[14] Hale, J.; Infante, E.; Tsen, F., Stability in linear delay equations, J. math. anal. appl., 105, 533-555, (1985) · Zbl 0569.34061
[15] Hale, J.; Lunel, S., Introduction to functional differential equations, (1993), Springer-Verlag New York · Zbl 0787.34002
[16] Hale, J.; Martinez Amores, P., Stability in neutral equations, Nonlinear anal., 1, 161-172, (1977) · Zbl 0359.34070
[17] Kolmanovskii, V.; Niculescu, S.-I.; Richard, J.P., On the liapunov – krasovskii functionals for stability analysis of linear delay systems, Internat. J. control, 72, 374-384, (1999) · Zbl 0952.34057
[18] Kolmanovskii, V.; Nosov, V., Stability of functional differential equations, (1986), Academic Press New York
[19] Kolmanovskii, V.; Richard, J.-P., Stability of some linear systems with delays, IEEE trans. automat. control, 44, 984-989, (1999) · Zbl 0964.34065
[20] Li, X.; de Souza, C., Criteria for robust stability and stabilization of uncertain linear systems with state delay, Automatica, 33, 1657-1662, (1997)
[21] Lien, C.-H.; Yu, K.-W.; Hsieh, J.-G., Stability conditions for a class of neutral systems with multiple time delays, J. math. anal. appl., 245, 20-27, (2000) · Zbl 0973.34066
[22] Logemann, H., Destabilizing effects of small time delays on feedback-controlled descriptor systems, Linear algebra appl., 272, 131-153, (1998) · Zbl 0986.93056
[23] Mahmoud, M., Robust control and filtering for time-delay systems, (2000), Marcel Decker New York
[24] Niculescu, S.-I., Further remarks on delay-dependent stability of linear neutral systems, ()
[25] Niculescu, S.-I.; Rasvan, V., Delay-independent stability in lossless propagation models with applications, ()
[26] Rasvan, V., Absolute stability of time lag control systems, (1975), Ed. Academiei Bucharest, in Romanian · Zbl 0312.93029
[27] Takaba, K.; Morihira, N.; Katayama, T., A generalized Lyapunov theorem for descriptor system, Systems control lett., 24, 49-51, (1995) · Zbl 0883.93035
[28] Verriest, E.; Niculescu, S.-I., Delay-independent stability of linear neutral systems: A Riccati equation approach, (), 92-100 · Zbl 0923.93049
[29] Zhu, W.; Petzold, L., Asymptotic stability of linear delay differential – algebraic equations and numerical methods, Appl. numer. math., 24, 247-264, (1997) · Zbl 0879.65060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.