Abstract In the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with two commensurate delays and a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of the Lyapunov functional for the initial function of time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix.

Abstract In the paper construction of a Lyapunov functional for time delay system with both lumped and distributed delay is presented. The Lyapunov functional is determined by means of the Lyapunov matrix. The method of determination of the Lyapunov matrix for time delay system with both lumped and distributed delay is presented. It is given the example illustrating the method.