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Abstract: In this contribution, we study Lyapunov matrices for a time-delay system. The main contribution consists in the demonstration that the set of matrix equations that defines the matrices admit a unique solution if the spectrum of the time-delay system does not contain a point , such that also belongs to the spectrum. [Copyright &y& Elsevier]

Abstract: A necessary and sufficient condition for the existence and uniqueness of Lyapunov matrices for neutral type single delay systems is presented. [Copyright &y& Elsevier]

Abstract: In this contribution we present some basic results concerning the computation of quadratic functionals with prescribed time derivatives for linear time delay systems. Some lower and upper bounds for the functionals are given. The functionals are defined by special matrix valued functions. These functions are called Lyapunov matrices. The theory of the matrices is a rather young topic. Therefore, principal results with respect to the existence and uniqueness of the matrices are included. Some important applications of the functionals are pointed out. A brief historical survey ends the contribution. [Copyright &y& Elsevier]