Geodesic
A stability criterion for the single delay equation in terms of the Lyapunov matrix
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2013), pp. 106-115 Cet article a éte moissonné depuis la source Math-Net.Ru

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In case of delay systems the Lyapunov–Krasovskii functional approach plays the role of the second Lyapunov method for the case of ordinary differential equations. To investigate stability of linear systems the so-called complete type functionals are often applied. These functionals depend on special matrix valued functions, named the Lyapunov matrices. It is of interest to find conditions on the Lyapunov matrix guarantees the stability of the system. In the work of A. V. Egorov and S. Mondié (2011) some necessary stability conditions have been obtained for a wide class of delay linear systems. In that contribution it is proved that these necessary conditions become sufficient for the case of a scalar single delay equation. The proof of the result is based on the explicit expression for Lyapunov matrix obtained as the solution of a special difference-differential equation with boundary conditions. Bibliogr. 12. Il. 1.
Keywords: delay systems, linear systems, Lyapunov–Krasovskii functionals, necessary stability conditions. 
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A. V. Egorov; S. Mondié. A stability criterion for the single delay equation in terms of the Lyapunov matrix. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2013), pp. 106-115. http://geodesic.mathdoc.fr/item/VSPUI_2013_1_a11/

[1] Krasovskii N. N., “On the application of the second method of Lyapunov for equations with time delays”, Prikl. Mat. Meh., 20 (1956), 315–327 | MR

[2] Repin M. Yu., “Quadratic Lyapunov functionals for systems with delay”, Prikl. Mat. Meh., 29 (1965), 564–566 | MR | Zbl

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

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

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

[6] Kharitonov V. L., Zhabko A. P., “Lyapunov–Krasovskii approach for robust stability of time delay systems”, Automatica, 39 (2003), 15–20 | MR | Zbl

[7] Kharitonov V. L., Plischke E., “Lyapunov matrices for time-delay systems”, Syst. Contr. Lett., 55 (2006), 697–706 | MR | Zbl

[8] Jarlebring E., Vanbiervliet J., Michiels W., “Characterizing and computing the $\mathcal{H}_2$ norm of time-delay systems by solving the delay Lyapunov equation”, IEEE Trans. on Autom.Contr., 56:4 (2011), 814–825 | MR

[9] Mondié S., Egorov A., “Some necessary conditions for the exponential stability of one delay systems”, 8th Intern. Conference on Electrical Engineering, Computing Science and Automatic Control (Merida, Mexico, 2011), 103–108

[10] Mondié S., “Assessing the exact stability region of the single-delay scalar equation via its Lyapunov function”, IMA J. of Math. Contr. and Inf., 2012 http://imamci.oxfordjournals.org | MR | Zbl

[11] Bellman R., Cooke K., Differential difference equations, Academic Press, New York, 1963, 465 pp. | MR

[12] Hayes N. D., “Roots of the transcendental equation associated with a certain difference-differential equation”, J. London Mathem. Society, 1950, 226–232 | MR | Zbl