Resumen de Free boundary problems: qualitative behaviour and control results
Tommaso Mingazzini
This thesis has the objective to explore different aspects of Control Theory. Control Theory is by now a very huge area of research which includes very different problems and various techniques. Inside Control Theory two types of problems can mainly be found: Optimal Control problems and Exact Control problems. We deal with both of them.In this thesis a central role is played by a specific property which some solutions of PDEs display: the Free Boundary or Dead Core. This itself is a huge field of study in Mathematics which has experienced many contributions. This property is intrinsically connected with the nonlinear character of such PDEs since the linear theory does not present this aspect. In chapter one we start by modelling a decision-making process about the policy of an industry. In this situation the owner has to decide how to regulate the production facing the consecutive emission of pollution, which we can suppose directly connected to the amount of polluting substances generated during the activity. We also assume that the pollution is discharged through a pipe in a pond of water where there is a special area which cannot be infected due to some regulation law: the penalty for any infringement of this restriction is punished with a fine proportional to the volume invaded. The point is to decide how to regulate the rate of production in order to get the best balance between the gain obtained by a high rate of production and the loss due to the fine which needs to be paid for the contamination of the protected area.Mathematically, this translates into an optimization problem via a cost functional, which represents the loss of the industry and has to be minimized. The aim is to decide whether a best solution exists. Although the theory of minimization is already well studied, we adopt a functional which depends on the measure of the positivity set of solutions of nonlinear PDEs. This type of functional is not common in the literature though it is highly useful.In chapter 2, we focus on the global null controllability and stabilization of the Porous Medium Equation (PME) in one space dimension under different boundary type conditions. This equation may be used to model different processes, from nonlinear filtration in porous media to nonlinear heat transfer.We underline the degenerate character of this equation which is what makes the problem of controllability and stabilization not easy at all. Actually, there are really few works in the literature concerningcontrollability of such type of equations. To prove the global null controllability for this equation with appropriate boundary and interior controls, we applied different techniques borrowed from the theory of nonlinear, nondegenerate PDEs and from Control Theory for finite dimensional systems (Return Method). We then pass to analyze a simple procedure to stabilize the zero state for the porous medium equation with homogeneous Neumann boundary condition through a feedback control which is constant in space. Chapter 3 does not focus on direct aspects of Control Theory but goes to study very useful tools for this field. In fact, the topic is the existence and behaviour of very weak solutions of certain semilinear elliptic and parabolic problems and of their free boundary when the data are very irregular. We first prove the well-posedness in the sense of Hadamard of very weak solutions in any dimension for multivalued problems and then show the two types of behaviours (Expansion on the boundary of the support and No-diffusion on the boundary of the support) that these solutions display when the domain is the upper half plane.
