Lyapunov functionals and matrices

In the talk we present a survey of the current state of art in the area of quadratic functionals with prescribed time derivatives. The …rst contribution dedicated to the topic was that by Yu.M. Repin [5]. In this seminal contribution a quadratic functional of a general form was suggested. The time derivative of the functional has been computed, and then, equating the derivative to the prescribed one, a system of equations for the matrices that de…ne the functional has been derived. The system includes a linear matrix partial di¤erential equation, ordinary matrix di¤erential equations, as well as algebraic relations between the matrices. It is worth to be mentioned that many essential elements needed for the computation of Lyapunov matrices, some in an explicit form, and some in an implicit form, can be already found there. This three-pages contribution has had a profound impact on the research in the area. The main target of the paper by R. Datko [1] was to develop an in…nite dimensional version of the Lyapunov-Krasovskii approach to the stability analysis of linear time delay systems. The paper by E.F. Infante and W.B. Castelan, [3], contains very important contributions in the area. Here for the …rst time the three basic properties of Lyapunov matrices have been explicitly indicated. In the contribution we can …nd functionals similar to the complete type ones. The authors have tried to demonstrate that the functionals can be used to derive exponential estimates of the solutions of a time delay system. Next serious break-through in the area has been achieved in the paper by W. Huang [2]. The most important result presented there is the existence theorem. Probably this is the …rst paper where it has been explicitly stated that Lyapunov matrices are completely de…ned by the three basic properties: the dynamic, the symmetry, and the algebraic ones. It is important to say that this result, as well as the other ones presented in this contribution, have been proven for a very general class of linear time delay systems. In the paper by J. Louisell [4] an interesting relation between the spectrum of the time delay system and that of the auxiliary delay free system of matrix equations has been established. Namely, it has been shown that any pure imaginary eigenvalue of the time delay system is also an eigenvalue of the auxiliary system. The