Extension of State Space and Lyapunov Matrices

Abstract

One of the main problems in the application of the theory of Lyapunov-Krasovskii functionals is the construction of corresponding Lyapunov matrices. Recently, it was noted that systems with distributed delay and exponential kernel may be rewritten by the introduction of auxiliary variables as systems with one delay. A suggestion was made that one can use the Lyapunov matrix of the latter system to obtain the Lyapunov matrix of the original system. In this article, we establish precise relationships between these two systems and their Lyapunov matrices. We show that if there exists a Lyapunov matrix of the extended system then it can be used to compute a Lyapunov matrix of the nominal system. We demonstrate that this method fails for certain systems and establish necessary and sufficient conditions for the extended system to admit a Lyapunov matrix.