Approximations of Lyapunov-Krasovskii functionals of complete type with given cross terms in the derivative for the stability of time delay systems

Sufficient conditions expressed in terms of the piecewise linear approximation of the Lyapunov matrix of the time delay system for the derivative condition and the bound condition of the Lyapunov-Krasovskii stability Theorem are provided. These conditions can be utilized to verify if a given constructed Lyapunov matrix or its approximation gives rise to a functional that satisfies the Lyapunov-Krasovskii theorem. They can also be employed in a constructive manner.