Lyapunov functionals and matrices

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.