Lyapunov matrices for a class of time-delay systems with piecewise-constant kernel

In this contribution, we study time-delay systems with distributed delay whose kernel is piecewise constant. It is shown that the problem of finding a Lyapunov matrix for this class of time-delay systems can be reduced to the computation of solutions to an auxiliary system of linear differential equations with boundary conditions. The problem of solving the auxiliary system is then reduced to the solution of a system of linear algebraic equations. It is established that the auxiliary system with boundary conditions admits a unique solution if and only if there exists a unique Lyapunov matrix.