Complete type functionals for homogeneous time delay systems

Abstract

For linear time-invariant time delay systems, the so-called Lyapunov–Krasovskii functionals of complete type (Kharitonov and Zhabko, 2003) are known to be effective in the stability analysis and a number of applications. More precisely, there exist the necessary and sufficient asymptotic stability and instability conditions expressed in terms of these functionals. The case excluded from consideration in the theory (since the functionals either do not exist or are not uniquely defined) is violation of the Lyapunov condition, i.e. the case of systems with the eigenvalues placed symmetrically with respect to the origin of the complex plane. In this paper, an analogue of this theory for a class of nonlinear time delay systems with homogeneous right-hand sides of degree greater than one and a constant delay is developed. An explicit expression for the Lyapunov–Krasovskii functionals as well as necessary and sufficient conditions for the asymptotic stability and instability of the trivial solution based on these functionals are given. An important assumption, which constitutes an analogue of the Lyapunov condition for linear systems, is existence of the Lyapunov function for a delay free system, obtained from the original one setting the delay equal to zero. Furthermore, this Lyapunov function is a key element in the construction of the functional. The functionals are applied to estimating the region of attraction.