Riccati Equations in Delay Systems

We give an overview of how the Riccati equation makes its appearance in the stability analysis of linear systems with delays. We avoid complexities as time-invariance, added perturbations, distributed delays etc, to get to the main issues. Most generalizations can be or have been carried out, but do not add substantially to our understanding. We also show the connection between quadratic stability and an optimization problem The notion of Riccati stability is introduced as the existence of a positive definite triple of matrices satisfying a certain Riccati equation. While having its origin in the problem of delay systems, the partial characterization of this Riccati stability is carried out as an independent endeavor. 1 Lyapunov Krasovskii Functionals in Delay Systems The past decade has seen a flurry of activity on systems with delays (See [6,8,9] and references therein). One important direction of this activity involved the stability analysis of delay systems using the Lyapunov-Krasovskii approach. In this connection the Riccati equation made its entrance into the study of delay systems. In this paper we present a stripped down version of the main results on stability and its tangency to the ubiquitous Riccati equation. With stripped down, we mean that we will not be concerned with the time varying, nonlinearly perturbed, or uncertain versions. These inclusions only seem to clutter up the main theory with additional complexities and technicalities, without requiring essentially new techniques or providing greater insight. We shall consider the linear autonomous continuous time retarded system ẋ(t) = Ax(t) + ∑