Transient Behavior and Entropy for Dynamical and Control Systems

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The aim of this project is to analyze transient behavior of deterministic control systems and of random dynamical systems. For control systems, the recently introduced notion of (topological) invariance entropy which has been developed in analogy to topological entropy of dynamical systems describes the difficulty to keep a control system in a fixed subset of the state space and gives insight into the required minimal data rates. On the other hand, one may formally replace the control term by a (bounded) random perturbation. Here it is well known that the supports of stationary measures for random dynamical systems can often be described by invariant control sets, i.e., invariant subsets of complete approximate controllability. The proposed project is built on conditionally stationary (probability) measures instead of stationary measures. They describe the transient behavior of random dynamical systems and, via the skew product formalism, they are closely related to conditionally invariant measures of deterministic dynamical systems. It is expected that the supports of conditionally stationary measures can often be described by relatively invariant control sets, instead of invariant control sets. The first part of the project will analyze relatively invariant control sets. Then conditionally stationary measures and the relation of their supports to controllability properties will be analyzed. The third part of the project will be devoted to a metric notion of invariance entropy based on conditionally stationary measures. In particular, the relation to the topological invariance entropy will be explored with a view toward a variational principle. All parts of the project will be considered for systems in discrete and in continuous time. The expected results will be of relevance in the analysis of the connections between the transient behavior of random systems and control systems, as well as in the analysis of minimal data rates for digitally connected control systems.