Using polynomial semi-separable kernels to construct infinite-dimensional Lyapunov functions

In this paper, we introduce the class of semi-separable kernel functions for use in constructing Lyapunov functions for distributed-parameter systems such as delay-differential equations. We then consider the subset of semi-separable kernel functions defined by polynomials. We show that the set of such kernels which define positive forms can be parameterized by positive semidefinite matrices. In the particular case of linear time-delay systems, we show how to construct the derivative of Lyapunov functions defined by piecewise continuous semi-separable kernels and give numerical examples which illustrate some advantages over standard polynomial kernel functions.