# Dynamics and control of distributed parameter systems

In our research on infinite dimensional systems we focus on the theory as well as on the applications. Linear partial differential equations form one of the important model classes within infinite-dimensional systems theory. If the eigenfunctions of the associated linear operator are a basis of the state space, then it is easy to write down the solution of the partial differential equation. However, systems theoretic properties, like stabilizability and controllability become also easy to characterize and hence to be checked. Unfortunately, it can be a cumbersome analysis to prove the basis property. We have shown that if the eigenvalues satisfy a uniform gap condition, and the partial differential equation possesses a solution for negative and positive time, then the basis property is automatically satisfied.

The Hautus test is a famous test for checking controllability of finite-dimensional systems. For decades it has been tried to extend this test to infinite dimensional systems. For normal groups it has been showed that a Hautus type test is possible. As can be seen from the above research questions the fundamental research on infinite dimensional systems focuses on solutions which are easily applicable in applications.

An important subclass of infinite-dimensional systems is formed by port-Hamiltonian systems. This subclass contains models which an energy function, i.e., the Hamiltonian. Although controller design has been done for non-linear (finite dimensional) systems, the analysis of the infinite dimensional counterpart is much harder. For linear, infinite dimensional, port-Hamiltonian systems this analysis is now almost complete. Properties like existence of solutions, stabilizability and well-posedness are now well understood.

One of our main applications is controlling the temperature in a bulk storage room. The model is given by a partial differential equation describing the temperature distribution in the storage room. A cooling installing that can be switched on or off controls the product temperature. We were able to design a controller that robustly controls these switching times.

Exponential stability for infinite-dimensional systems on a Hilbert space is well understood. An unsolved question is to give bounds on its overshoot. In the Ph.D. project “Bounds on Stable Semigroups” (funded by NWO) it is shown that it is possible to identify classes of discrete- and continuous-time systems with similar overshoot behavior. If two continuous-time systems are in the same class, then their discretization obtained via the Crank-Nicolson scheme are in the same discrete-time class. If the operators are matrices, i.e., the state space is finite-dimensional, then a characterization of these classes is obtained. The system property that has been studied is the concepts of (exact) observability. It is shown that relative compactness of the output operator together with exact observability implies that the inverse of the systems operator is compact. Among others this implies that if a pde on a bounded spatial domain is observability with a observation acting on the boundary of this spatial domain, then the system can only have eigenvalues.