Examples for Global Behavior
The study of the global behavior of control systems serves two purposes. First, it describes the possible responses of the system under all admissible controls by identifying the regions of controllability and their domains of attraction. This knowledge is necessary, e.g., for control, tracking, or stabilization. This is obvious for the first two problems, and stabilization is only possible in the closures of control sets and from points in the corresponding domain of attraction. Furthermore, a study of the limit sets of controlled trajectories shows which limit sets are actually contained in the interior of control sets where the system can be steered from any point to any other point. The second benefit of global theory for control systems is the study of systems under bounded, time-varying perturbations, as explained in Chapter 2. Orbits, invariant sets, and domains of attraction describe the worst-case behavior of perturbed systems, and genericity results indicate, for “how many” perturbations the worst case behavior actually occurs.
KeywordsHopf Bifurcation Phase Portrait Homoclinic Orbit Stable Manifold Global Behavior
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