Wolfgang Kliemann

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.

Near invariance and local transience for random diffeomorphisms

For random diffeomorphisms depending on a parameter, nearly invariant sets are described via an associated deterministic control system. Conditions are provided guaranteeing that the system leaves the support of an invariant measure under small perturbations of the parameter and estimates for the exit times are given.

Controllability near a Hopf bifurcation

The authors study controllability properties of control affine systems depending on a parameter and with constrained control values. If the uncontrolled system is subject to a Hopf bifurcation, a continuum of periodic solutions bifurcates from an equilibrium. This, together with an accessibility condition, induces for a small control range a two-parameter bifurcation of the control sets (i.e., the regions of complete controllability) around the equilibria and the periodic solutions, respectively. The proofs are based on methods from dynamical systems theory applied to the associated control flow.<<ETX>>

Computation of almost invariant sets for perturbed systems

Using the relation between the supports of invariant Markov measures and invariant control sets, we discuss the characterization of almost invariant sets for Markov diffusion systems.

Reliability assessment of dynamical systems with random excitation

During recent years, a variety of important interconnections between the theory of nonlinear dynamical systems, nonlinear stochastically excited systems, and nonlinear control systems have been obtained by mathematicians, physicists, and engineers. This paper is the beginning of a systematic study of reliability aspects within the emerging unified view of the different classes of nonlinear dynamical systems. We study the response behavior of dynamical systems with random excitation depending on two parameters: bifurcation parameter /spl alpha//spl epsiv/R, and a parameter /spl rho//spl les/0 that measures the strength of the disturbance. The goal of this paper is to obtain precise stability and reliability diagrams in /spl alpha/-/spl rho/-space. In particular, we characterize for each parameter combination the levers that the (maximal) system response will reach with probability 1, with positive probability, or with probability 0. Emphasis is placed on determining the parameter combinations for which the system behavior changes drastically, i.e., on stochastic bifurcation phenomena, to determine crucial system parameters and their critical values.<<ETX>>