Christoph Kawan

Invariance Entropy for Control Systems

For continuous time control systems, this paper introduces invariance entropy as a measure for the amount of information necessary to achieve invariance of weakly invariant compact subsets of the state space. Upper and lower bounds are derived; in particular, finiteness is proven. For linear control systems with compact control range, the invariance entropy is given by the sum of the real parts of the unstable eigenvalues of the uncontrolled system. A characterization via covers and corresponding feedbacks is provided.

A note on topological feedback entropy and invariance entropy

Abstract For discrete-time control systems, notions of entropy for invariance are compared. One is based on feedbacks, and the other one on open-loop control functions. Under a strong invariance condition, it is shown that they are essentially equivalent. Several modifications are also discussed.

Invariance Entropy of Control Sets

Invariance entropy for continuous-time control systems measures how often open-loop controls have to be updated in order to render a controlled invariant subset of the state space invariant. A special type of a controlled invariant set is a control set, i.e., a maximal set of approximate controllability. In this paper, we investigate the properties of the invariance entropy of such sets. Our main result gives an upper bound of this quantity in terms of the positive Lyapunov exponents of a periodic solution in the interior of the control set. Moreover, for one-dimensional control-affine systems with a single control vector field we provide an analytical formula for the invariance entropy of a control set in terms of the drift vector field, the control vector field, and their derivatives. As an application, we study a controlled bilinear oscillator.

Metric Entropy of Nonautonomous Dynamical Systems

Abstract We introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence (Xn; μn) of probability spaces and a sequence of measurable maps fn : Xn → Xn+1 with fnμn = μn+1. This notion generalizes the classical concept of metric entropy established by Kolmogorov and Sinai, and is related via a variational inequality to the topological entropy of nonautonomous systems as defined by Kolyada, Misiurewicz, and Snoha. Moreover, it shares several properties with the classical notion of metric entropy. In particular, invariance with respect to appropriately defined isomorphisms, a power rule, and a Rokhlin-type inequality are proved

Invariance entropy for outputs

For continuous time control systems, this paper analyzes output invariance entropy as a measure for the information necessary to achieve invariance of compact subsets of the output space. For linear control systems with compact control range, relations to controllability and observability properties are studied. Furthermore, the notion of asymptotic output invariance entropy is introduced and characterized for these systems.

Lower bounds for the strict invariance entropy

In this paper, we present a new method for obtaining lower bounds of the strict invariance entropy by combining an approach from the theory of escape rates and geometric methods used in the dimension theory of dynamical systems. For uniformly expanding systems and for inhomogeneous bilinear systems we can describe the lower bounds in terms of uniform volume growth rates on subbundles of the tangent bundle. In particular, we obtain criteria for positive entropy. We also apply the estimates to bilinear systems on projective space.

Exponential state estimation, entropy and Lyapunov exponents

In this paper we study the notion of estimation entropy recently established by Liberzon and Mitra. This quantity measures the smallest rate of information about the state of a dynamical system above which an exponential state estimation with a given exponent is possible. We show that this concept is closely related to the

Expanding and expansive time-dependent dynamics

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On expanding maps and topological conjugacy

-entropy introduced by Thieullen and we give a lower estimate in terms of Lyapunov exponents assuming that the system preserves an absolutely continuous measure with a bounded density, which includes in particular Hamiltonian and symplectic systems. Although in its current form mainly interesting from a theoretical point of view, our result could be a first step towards a more practical analysis of state estimation under communication constraints.

On the structure of uniformly hyperbolic chain control sets

In this paper, time-dependent dynamical systems given by sequences of maps are studied. For systems built from expanding -maps on a compact Riemannian manifold M with uniform bounds on expansion factors and derivatives, we provide formulas for the metric and topological entropy. If we only assume that the maps are , but act in the same way on the fundamental group of M, we can show the existence of an equi-conjugacy to an autonomous system, implying a full variational principle for the entropy. Finally, we introduce the notion of strong uniform expansivity that generalizes the classical notion of positive expansivity, and we prove time-dependent analogues of some well-known results. In particular, we generalize Reddys result which states that a positively expansive system locally expands distances in an equivalent metric.