Topological, Smooth, and Control Techniques for Perturbed Systems

The theory of dynamical systems has become a center piece in the systematic study of systems with deterministic or stochastic perturbations, based on measurable, topological, and smooth dynamics. Recent developments also forge a close connection between control theory and topological and smooth dynamics. On the other hand, the support theorem of Stroock and Varadhan shows how control theoretic techniques may aid in the Markovian analysis of systems perturbed by diffusion processes. This paper presents an overview of topological, smooth, and control techniques and their interrelations, as they can be used in the study of perturbed systems. We concentrate on global analysis and parameter dependent perturbation systems, where we emphasize comparison of the Markovian and the dynamical structure of systems with Markovian diffusion perturbation process. A series of op en problems highlights the areas in which the interconnections between different techniques and system classes are not (yet) well understood.

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