Lyapunov functional for a linear system with two delays

Lyapunov quadratic functionals are used to test the stability of systems, to compute the critical delay values for time delay systems, to compute the exponential estimates for the solutions of time delay systems, to calculate the robustness bounds for uncertain time delay systems, and to calculate a quadratic performance index for the process of parametric optimization for time delay systems. We construct the Lyapunov functionals for the systems with time delay with a given time derivative. For the first time such Lyapunov functional was introduced by Repin (1965) for the case of retarded time delay linear systems with one delay. Repin (1965) provided also the procedure for determining the coefficients of functional. Duda (1986) used the Lyapunov functional for the calculation of the value of a quadratic performance index in the process of parametric optimization for systems with time delay of retarded type and extended the results to the case of neutral type time delay systems in Duda (1988). In Infante and Castelan (1978), construction of the Lyapunov functional is based on a solution of a matrix differential-difference equation on a finite time interval. This solution satisfies symmetry and boundary conditions. Kharitonov and Zhabko (2003) extended the results of Infante and Castelan (1978) and proposed a procedure of constructing quadratic functionals for linear retarded type delay systems, which could be used for the robust stability analysis of time delay systems. This functional was expressed by means of the