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Exponential estimates for time delay systems

Abstract In this paper we demonstrate how Lyapunov-Krasovskii functionals can be used to obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate, Lyapunov-Krasovskii functional. The objective of this note is to describe a systematic procedure of constructing quadratic Lyapunov functionals for (exponentially) stable linear delay systems in order to obtain exponential estimates for their solutions. | Exponential estimates for time delay systems V. L. Kharitonov Control Automatico CINVESTAV-IPN A.P. 14-740 07000 Mexico D.F. Mexico E-mail khar@ctrl.cinvestav.mx D. Hinrichsen Institut für Dynamische Systeme Universität Bremen D-28334 Bremen Germany E-mail dh@math.uni-bremen.de July 10 2004 Abstract In this paper we demonstrate how Lyapunov-Krasovskii functionals can be used to obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords time delay system exponential estimate Lyapunov-Krasovskii functional. 1 Introduction The objective of this note is to describe a systematic procedure of constructing quadratic Lyapunov functionals for exponentially stable linear delay systems in order to obtain exponential estimates for their solutions. The procedure we propose is a counterpart to the well known method of deriving exponential estimates for stable systems x Ax by means of quadratic Lyapunov functions V x x Ux . Here U - 0 is the solution of a Lyapunov equation A U UA W where W 0 is any chosen positive definite matrix. If 2uU N W for some u 0 then eAi VK U e f t 0 1 where denotes the spectral norm and k U U U-1 is the condition number of U see 6 . Note that this estimate guarantees not only a uniform decay rate u for all solutions of x Ax but also a bound on the transients of the system. It is surprising that a similar constructive method does not exist for delay systems. It is true there exists an operator theoretic version of Lyapunov s equation in the abstract semigroup theory of infinite dimensional time-invariant linear systems see 2 5 but this does not provide us with a constructive procedure. For constructive purposes more concrete Lyapunov functions must be considered. Since the fifties different types of Lyapunov functions have been proposed for the stability analysis of delay systems see the pioneering works of Razumikhin 12 and Krasovskii 11 . Whereas Razumikhin 12 used Lyapunov type functions V x t depending on the current