Geodesic
A Lyapunov matrix based stability criterion for a class of time-delay systems
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 13 (2017) no. 4, pp. 407-416 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper is devoted to the stability analysis of linear time-invariant systems with multiple delays. First, we recover some basic elements of our research. Namely, we introduce the complete type functionals, the delay Lyapunov matrix, and a space of special functions that allow to present a family of necessary stability conditions. Then, we prove a sufficient stability condition (instability condition) in terms of a quadratic Lyapunov–Krasovskii functional. Summarizing these results, we finally obtain an exponential stability criterion for a class of linear time-delay systems. The criterion requires only a finite number of mathematical operations to be tested and depends uniquely on the delay Lyapunov matrix. Refs 15.
Keywords: time-delay system, Lyapunov matrix, stability criterion. 
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M. Gomez; A. V. Egorov; S. Mondié. A Lyapunov matrix based stability criterion for a class of time-delay systems. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 13 (2017) no. 4, pp. 407-416. http://geodesic.mathdoc.fr/item/VSPUI_2017_13_4_a6/

[1] Krasovskii N. N., “On the application of the second method of Lyapunov for equations with time delays”, Applied Mathematics and Mechanics, 20 (1956), 315–327 (In Russian) | MR

[2] Datko R., “An algorithm for computing Lyapunov functionals for some differential difference equations”, Ordinary Differential Equations, ed. L. Weiss, Academic Press, New York, 1972, 387–398 | DOI | MR

[3] Infante E. F., Castelan W. B., “A Liapunov functional for a matrix difference-differential equation”, Journal of Differential Equations, 29 (1978), 439–451 | DOI | MR | Zbl

[4] Huang W., “Generalization of Liapunov's theorem in a linear delay systems”, Journal of Mathematical Analysis and Applications, 142 (1989), 83–94 | DOI | MR | Zbl

[5] Kharitonov V. L., Zhabko A. P., “Lyapunov–Krasovskii approach to the robust stability analysis of time-delay systems”, Automatica, 39:1 (2003), 15–20 | DOI | MR | Zbl

[6] Kharitonov V. L., Time-delay systems. Lyapunov functionals and matrices, Birkhäuser Press, Basel, 2013, 311 pp. | MR | Zbl

[7] Egorov A. V., Mondié S., “Necessary stability conditions for linear delay systems”, Automatica, 50:12 (2014), 3204–3208 | DOI | MR | Zbl

[8] Cuvas C., Mondié S., “Necessary stability conditions for delay systems with multiple pointwise and distributed delays”, IEEE Transactions on Automatic Control, 61:7 (2016), 1987–1994 | DOI | MR | Zbl

[9] Gomez M. A., Egorov A. V., Mondié S., “Necessary stability conditions for neutral type systems with a single delay”, IEEE Transactions on Automatic Control, 62:9 (2016), 4691–4697 | DOI | MR

[10] Mondié S., “Assessing the exact stability region of the single-delay scalar equation via its Lyapunov function”, IMA Journal of Mathematical Control and Information, 29:4 (2012), 459–470 | DOI | MR | Zbl

[11] Egorov A. V., “A stability criterion for the neutral type time-delay equation”, 2016 Intern. Conference on “Stability and Oscillations of Nonlinear Control Systems” (Moscow, Russia, V. A. Trapeznikov Institute of Control Sciences, 2016), 1–4 | MR

[12] Egorov A. V., “A new necessary and sufficient stability condition for linear time-delay systems”, 19th IFAC Workshop on Time Delay Systems (Cape Town, South Africa, 2014), 11018–11023

[13] Egorov A. V., Cuvas C., Mondié S., “Necessary and sufficient stability conditions for linear systems with pointwise and distributed delays”, Automatica, 80 (2017), 218–224 | DOI | MR | Zbl

[14] Egorov A. V., “A finite necessary and sufficient stability condition for linear retarded type systems”, Conference on Decision and Control (Las Vegas, USA, 2016), 3155–3160

[15] Medvedeva I. V., Zhabko A. P., “Synthesis of Razumikhin and Lyapunov–Krasovskii approaches to stability analysis of time-delay systems”, Automatica, 51 (2015), 372–377 | DOI | MR | Zbl