  This article presents the inverse of the kernel operator associated with the
complete quadratic Lyapunov-Krasovskii functional for coupled
differential-functional equations when the kernel operator is separable.
Similar to the case of time-delay systems of retarded type, the inverse
operator is instrumental in control synthesis. Unlike the power series
expansion approach used in the previous literature, a direct algebraic method
is used here. It is shown that the domain of definition of the infinitesimal
generator is an invariant subspace of the inverse operator if it is an
invariant subspace of the kernel operator. The process of control synthesis
using the inverse operator is described, and a numerical example is presented
using the sum-of-square formulation.
