A STABILITY CRITERION FOR THE SINGLE DELAY EQUATION IN TERMS OF THE LYAPUNOV MATRIX A stability criterion for the single delay equation in terms of the Lyapunov matrix
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 andMondi´e (2011) some necessary stability conditions have been obtained for a wide class of delaylinear 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.