Basic concepts for estimations of domains of attraction in time-delay systems

With respect to equilibria of autonomous retarded functional differential equations, concepts for inner estimations of domains of attraction are derived. These are based on generalizations of Lyapunov's direct method and LaSalle's invariance principle. In delay-free ordinary differential equations, subsets of the domain of attraction can be described by sublevel sets of Lyapunov functions. In contrast, in time-delay systems it may be impossible to inscribe a non-empty sublevel set of a respective Lyapunov-Krasovskii functional into the monotonicity domain. The present paper presents admissible restrictions of the sublevel sets to solve this problem. In addition, numerical methods for upper bounds on the radius of attraction are described.