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

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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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     title = {A stability criterion for the single delay equation in terms of the {Lyapunov} matrix},
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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/