Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization

L. Grüne: Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization
in: Lecture Notes in Mathematics 1783, Springer Verlag, 2002,

Keywords: dynamical systems; control systems; discretization
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Abstract:

This monograph (based on the author's habilitation thesis) contributes to the numerical analysis of dynamical systems and control systems. It lies in the intersection of perturbation theory of dynamical systems, numerical dynamics, control theory and (dynamic) game theory. Concepts and methods of all these areas are interwoven. The general theme is the behavior of attracting sets and attractors under discretization. This contribution is not, as many others, based on a priori structural assumptions like hyperbolicity. This assumption is already violated for most relevant dynamical systems from applications; if controls or perturbations are present, they are restricted to small amplitudes. Thus, although it is possible (and also interesting) to develop those theories, their relevance, however, is very restricted.
Grüne uses the classical method of comparison functions for stability analysis, going back to W. Hahns work. They allow for a quantitative description of stability properties, and in the last few years, they have witnessed a renaissance in control theory. He considers two main classes of problems: systems with internal disturbances and controlled systems. In order to understand the behavior of numerical approximations to these systems, further (external) disturbances are introduced, which are of two different types: for systems with internal disturbances, Grüne takes time dependent disturbances; for control systems he considers external disturbances, generated by nonanticipatory strategies, a concept from game theory. In the first case, so-called strong concepts are obtained, in the second case, weak concepts. Nonanticipatory perturbations make it possible to model numerical errors for controlled systems.
After the introduction and the preparatory Chapter 2 (mainly on the considered systems classes), Chapters 3 and 4 study attracting sets. Here the new concept of dynamical Input-to-State-Stability is central. In Chapter 5 the connection between time and space dicretization on the one hand and disturbed systems on the other is established. Thus the theory developed in Chapters 3 and 4 becomes applicable to the analysis of numerical methods. This is performed in detail in Chapter 6 for attracting sets and for attractors. In Chapter 7 domains of attraction and reachable sets are discussed. The text is completed by 3 appendices: on viscosity solutions, on comparison functions and on numerical examples; finally, it contains a list of notation, a subject index and 129 references.
The monograph contains many new insights and results which cannot be reported here. An important complement is Appendix C which illustrates the main results by some -- very successful -- numerical experiments. This monograph lays foundations for a systematic treatment of the numerical analysis of control systems with nonlinear dynamics; this is achieved by embedding this theory into numerical stability theory of dynamical systems. In my opinion this monograph will be a milestone in our understanding of numerical methods for nonlinear control systems and of perturbed systems and promises to be very influential for the further work in this area.

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