A Port-Hamiltonian Approach to Distributed Parameter Systems

This thesis aims to provide a mathematical framework for the modeling and analysis of open distributed parameter systems. From a mathematical point of view this thesis merges the approach based on Hamiltonian modeling of open distributed-parameter systems, employing the notion of port-Hamiltonian systems, with the semigroup approach of infinite-dimensional systems theory. The Hamiltonian representation provides powerful analysis methods (e.g. for stability), and it enables the use of Lyapunov-stability theory and passivity-based control. The semigroup approach has been widely applied in the analysis of distributed parameter systems and it has facilitated the extension of some notions from finite-dimensional system theory to the infinite-dimensional case. One of the key points of the port-Hamiltonian formulation is the structure of the mathematical model obtained. By exploiting this structure, the port-Hamiltonian approach allows to deal with classes of systems, which provide a relative new point of view in the analysis of distributed parameter systems. In this thesis the port-Hamiltonian formulation is mainly used for the analysis of 1D-boundary control systems. These are systems in which the input (or part of it) acts on the boundary of the spatial domain. In these cases it is possible to parameterize the selection of the inputs (boundary conditions) and outputs by the selection of two matrices in such a way that the resulting system is passive. In this case these matrices determine the supply rate of the passive system, making it easy, in particular, to obtain impedance passive and scattering passive systems. In fact, as it is shown, these matrices can be used to determine further properties of the system, such as stability, controllability, and observability. Furthermore, it is shown that this approach already covers a very large class of 1D-systems. This thesis treats mainly two broad classes of systems. One corresponds to systems where the dissipation phenomena is not present and the other includes systems with some type of dissipation (e.g. heat or mass transfer, damping). These classes can, in turn, be divided into subclasses according to the properties of the structure, to provide further tools for the analysis of such systems. Thus the structure of the resulting models forms the basis for the development of general analysis (and control) techniques. In fact, it is shown that for some classes of systems it is possible to easily determine some of their fundamental properties (e.g. existence of solutions, stability, Riesz basis property). In this thesis we provide simple tools for the analysis of these properties for some classes of systems.