Multi-objective control problems for parabolic and dispersive systems

Abstract

This thesis is dedicated to the study of some multi-objective control problems for partial di erential equations. Usually, problems containing many objectives are not well-posed, since one objective may completely determine the control, turning the others objectives impossible to reach. For this reason, concepts of equilibrium (or efficiency) are normally applied to nd controls which are acceptable, in the sense they make the best decision possible according to some prescribed goals. By applying the so called Stackelberg-Nash strategy, we consider a hierarchy, in the sense that we have one control which we call the leader, and other controls which we call the followers. Once the leader policy is fixed, the followers intend to be in equilibrium according to their targets, this is what we call Stackelberg's Method. Once this hierarchy is established, we determine the followers in such a way they accomplish their objectives in a optimal way, and to do that a concept of equilibrium is applied. In this work, we apply the concept of Nash Equilibrium, which correspond to a non-cooperative strategy. By combining the Stackelberg's Method and the concept of Nash Equilibrium is what we call Stackelberg-Nash strategy. This thesis is divided into two chapters. In each of them, we solve a multi-objective control problems by following the Stackelberg-Nash strategy. In the rst chapter, we consider a linear system of parabolic equations and prove that the Stackelberg-Nash strategy can be applied under some suitable conditions for the coupling coe cients. In the second one, we consider the nonlinear Korteweg-de Vries (KdV) equation, which has a very di erent nature of parabolic equations, and the same method is applied.