# Research

## Our main research topics are:

Digital communication technology is increasingly used in control applications. The properties of wireless communication channels such as information loss, time delays and limited band width easily come into conflict with the typical reliability and security standards in control processes. This has led to the development of a very active research field. The focus of our work lies in the modeling and analysis of the dynamics generated by communication protocols such as the internet protocol TCP and particular communication protocols used in control. At the same time, we develop methods for the design of controllers which meet the requirements of digital communication channels. Networked control systems violate the assumption of classical control theory that information in feedback loops can be transmitted instantaneously, lossless and with arbitrary precision. Mathematical models of digital communication and control networks have to take into account general communication constraints, time-delayed transmissions, partial loss of information and variable network topologies. Examples can be found, for instance, in telerobotics, traffic control and the control of unmanned vehicles. A fundamental and natural problem in this field is to determine the smallest channel capacity above which a coder and a controller can be designed which achieve the desired control objective over the given channel. For certain stabilization problems there exists a systematic and general approach for this problem, using the concept of feedback entropy. Under appropriate assumptions, the feedback entropy can be estimated or even characterized through the Lyapunov exponents of the linearized system. Lyapunov exponents also play a central role in the theory of nonautonomous dynamical systems. While in the case of feedback entropy for a single system it is possible to provide analytical estimates and formulas, in the case of real networks the situation appears to be much more difficult. Here it is a desirable goal to develop a numerical scheme for the computation of the relevant quantities. Besides that, there are numerous other interesting open problems such as a generalization of feedback entropy to systems with deterministic or stochastic disturbances which often allow for more realistic models of the networks under consideration.

Partial differential equations describe processes in which temporal as well as spatial derivatives of the state variables are included in the description of the dynamics. Equations of this kind arise in the modeling of many physical, chemical or biological processes. They are also used in the analysis of communication networks, image processing and logistics. Famous examples are the Maxwell equations of electrodynamics as well as wave and heat equations describing various propagation processes. 1 Partial differential equations are an example of infinite-dimensional systems. We are interested in the dynamics of such systems under the influence of external disturbances and/or control inputs. The theory of input-state-stability (ISS) provides a theoretical framework that allows the unified investigation of internal and external stability properties. For infinite-dimensional systems, this theory is so far incomplete and we are working on the further development of these methods.

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.