Feedback theory extended for proving generation of contraction semigroups

Recently, the following novel method for proving the existence of solutions for certain linear time-invariant PDEs was introduced: The operator associated to a given PDE is represented by a (larger) operator with an internal loop. If the larger operator (without the internal loop) generates a contraction semigroup, the internal loop is accretive, and some non-restrictive technical assumptions are fulfilled, then the original operator generates a contraction semigroup as well. Beginning with the undamped wave equation, this general idea can be applied to show that the heat equation and wave equations with damping are well-posed. In the present paper we show how this approach can benefit from feedback techniques and recent developments in well-posed systems theory, at the same time generalising the previously known results. Among others, we show how well-posedness of degenerate parabolic equations can be proved.Comment: 33 page

The linear wave equation on N-dimensional spatial domains

We study the wave equation on a bounded Lipschitz set, characterizing all homogeneous boundary conditions for which this partial differential equation generates a contraction semigroup in the energy space L2(Omega) . The proof uses boundary triplet techniques

Approximate robust output regulation of boundary control systems

We extend the internal model principle for systems with boundary control and boundary observation, and construct a robust controller for this class of systems. However, as a consequence of the internal model principle, any robust controller for a plant with infinite-dimensional output space necessarily has infinite-dimensional state space. We proceed to formulate the approximate robust output regulation problem and present a finite-dimensional controller structure to solve it. Our main motivating example is a wave equation on a bounded multidimensional spatial domain with force control and velocity observation at the boundary. In order to illustrate the theoretical results, we construct an approximate robust controller for the wave equation on an annular domain and demonstrate its performance with numerical simulations.Comment: 29 pages, 4 figure

The duality between the gradient and divergence operators on bounded Lipschitz domains

This report gives an exact result on the duality of the divergence and gradient operators, when these are considered as operators between $L^2$-spaces on a bounded $n$-dimensional Lipschitz domain. The necessary background is described in detail, with a self-contained exposition

De Branges-Rovnyak realizations of operator-valued Schur functions on the complex right half-plane

We give a controllable energy-preserving and an observable co-energy-preserving de Branges-Rovnyak functional model realization of an arbitrary given operator Schur function defined on the complex right-half plane. We work the theory out fully in the right-half plane, without using results for the disk case, in order to expose the technical details of continuous-time systems theory. At the end of the article, we make explicit the connection to the corresponding classical de Branges-Rovnyak realizations for Schur functions on the complex unit disk.Comment: 68 pages: General polishing; no essential change

Functional model realizations for Schur functions on C+

For an arbitrary given operator Schur function defined on the complex right-half plane, we give a controllable energy-preserving and an observable co-energy-preserving de Branges-Rovnyak functional model realization. Topics appearing only in the right-half-plane setting, such as the extrapolation space, are also discussed

Towards input/output-free modelling of linear infinite-dimensional systems in continuous time

This dissertation describes a networking approach to infinite-dimensional systems theory, where there is a minimal distinction between inputs and outputs. We introduce and study two closely related classes of systems, namely the state/signal systems and the port-Hamiltonian systems, and describe how they relate to each other. Some basic theory for these two classes of systems and the interconnections of such systems is provided. The main emphasis lies on passive and conservative systems, and the theoretical concepts are illustrated using the example of a lossless transfer line. Much remains to be done in this field and we point to some directions for future studies as well