# Non-autonomous, time-varying and switched systems

Systems are called nonautonomous or time-variant if their dynamical laws are explicitly time-dependent. Such systems arise, for instance, when dynamical effects in the modeling of a system are ignored or by linearization of nonlinear systems along non-constant solutions. Switched systems model parametric uncertainties which can vary abruptly as well as designs which model certain dynamics by switching between different autonomous systems. We work on problems of stability and stabilization within these classes of systems. Stability properties are characterized by growth rates, the so-called Bohl or Lyapunov exponents. In the class of switched systems, the corresponding notion leads to the generalized spectral radius of a set of matrices. The characterization of these quantities is achieved via different classes of Lyapunov functions whose investigation was one of our main areas of research in recent years. Switched systems can, in particular, be used to derive models of the Transmission Control Protocol (TCP). This protocol is a central part of the regulation of the data traffic in the internet. Another important characterization of the dynamics of a nonautonomous system is given by its entropy. The entropy of a dynamical system is a real-valued quantity which is a rough measure for the exponential rate of increase in dynamical complexity when the system evolves in time. Depending on the category of dynamical systems there are different notions of entropy. In the measure-theoretic category the measure-theoretic entropy (also called Kolmogorov-Sinai or metric entropy) is the appropriate concept. Here the dynamics in classically modeled by a measure-preserving self-map of a probability space. The entropy is an invariant under appropriately defined isomorphisms and also has an information-theoretic interpretation. It measures how much information about the initial state can be obtained in the spatial and temporal average. Moreover, it is the basis of numerous classification results and it is intimately related to the Lyapunov exponents of the system. In the category of topological dynamical systems one can define the notion of topological entropy. Here the dynamics is classically described by a continu2 ous self-map of a compact metric space. A quantitative relation between both quantities (the measure-theoretic and the topological entropy) is provided by the variational principle. Every continuous self-map of a compact metric space allows for at least one invariant probability measure, defined on the Borel σfield. Hence, in a natural (though not canonical) way the topological system becomes a measure-theoretic system, for which the measure-theoretic entropy is defined. The variational principle says that the topological entropy equals the supremum of all possible measure-theoretic entropies. On the basis of this relation a rich and beautiful theory can be developed. A natural problem is to extend the notions of measure-theoretic and topological entropy to nonautonomous systems in a reasonable way and develop a theory similar to the autonomous case. There are still plenty of open questions in this context. In particular, the general validity of a variational principle so far has neither been proved nor disproved. Many tools of the classical theory are not available for nonautonomous systems which makes the development of this theory a challenging and exciting task.