1 Department of Informatics and Mathematical Modeling, Technical University of Denmark2 Scientific Computing, Department of Informatics and Mathematical Modeling, Technical University of Denmark3 Department of Mathematics, Technical University of Denmark4 Department of Applied Mathematics and Computer Science, Technical University of Denmark
Consider a partial di erential equation (PDE) of evolution type, such as the wave equation or the heat equation. Assume now that you can influence the behavior of the solution by setting the boundary conditions as you please. This is boundary control in a broad sense. A substantial amount of literature exists in the area of theoretical results concerning control of partial differential equations. The results have included existence and uniqueness of controls, minimum time requirements, regularity of domains, and many others. Another huge research field is that of control theory for ordinary di erential equations. This field has mostly concerned engineers and others with practical applications in mind. This thesis makes an attempt to bridge the two research areas. More specifically, we make finite dimensional approximations to certain evolution PDEs, and analyze how properties of the discrete systems resemble the properties of the continuous system. A common framework in which the continuous systems are formulated will be provided. The treatment includes many types of linear evolution PDEs and boundary conditions. We also consider di erent types of controllability, such as approximate, null- and exact controllability. We will consider discrete systems with a viewpoint similar to that used for the continuous systems. Most importantly, we study what is required of a discretization scheme in order for computed control functions to converge to the true, continuous, control function. Examples exist for convergent discretization schemes for which divergence of the computed controls occur. We dig deeper for three specific cases: The heat equation, the wave equation, and a linear system of thermoelasticity. Different aspects of the theory are exemplified through these case studies. We finally consider how to e ciently implement computer programs for computing controls in practice.