A criterion of existence and uniqueness of the Lyapunov matrix for a class of time delay systems

The Lyapunov matrix for systems of linear time-delay equations is a matrix-valued function which is a solution of a special dynamic system with some additional boundary conditions. This matrix allows to construct the complete type Lyapunov–Krasovskii functionals with a prescribed derivative, which are used successfully in analysis of behavior of time-delay systems. It was shown in works of Kharitonov and Chashnikov that the Lyapunov condition, that is the absence of opposite eigenvalues of the system, guarantees the existence and uniqueness of the Lyapunov matrix for systems of retarded type with multiple delays and for systems of neutral type with a single delay. In this contribution, we consider a linear time-invariant system of neutral type with two delays. It is shown that under a certain constraint the Lyapunov condition, that is the absence of opposite eigenvalues of the system, is also a criterion of the existence and uniqueness of the Lyapunov matrix. Refs 14.