Domains of attraction

In this chapter we will discuss domains of attraction for strongly and weakly attracting sets as well as. reachable sets. We introduce and discuss suitable robustness properties for these sets and investigate their behavior under discretization and perturbation. In addition we formulate and discuss algorithms for their computation. Domains of attraction (and their close relatives reachable sets, cf. Section 7.7) play an important role in the analysis of nonlinear dynamical systems. For systems without input domains of attraction were extensively investigated in the late 1960s, see for instance Zubov [129], Coleman [21], Wilson [125] and Bhatia [13] or the textbooks by Hahn [61] or Khalil [71]. For controlled and perturbed nonlinear systems one should in particular mention the monograph by Colonius and Kliemann [22] which presents an approach for the analysis of such systems where domains of attraction and reachable sets play a prominent role. Many of the mentioned results for systems without input could be transferred for systems with (control or perturbation) input u, see the papers by Camilli, Wirth and the author [16, 17] and by Wirth and the author [58]. In particular, it was shown that Zubov's method for the characterization of domains of attraction can be extended to systems with input, a result that we will investigate in more detail in Section 7.2, below. In order to simplify the presentation, throughout this chapter we assume that (2.8) or (2.24) holds, even without explicitly mentioning it.