Positivity of kernel functions for systems with communication delay

The purpose of this paper is to provide further results on a method of constructing Lyapunov functionals for infinite-dimensional systems using semideflnite programming. Specifically, we give a necessary and sufficient condition for positivity of a positive integral operator described by a polynomial kernel. We then show how to combine this result with multiplier operators in order to obtain positive composite Lyapunov functionals. These types of functionals are used to prove stability of linear time-delay systems.