Abstract
In this thesis, we are interested in mathematical analysis and optimal control of diffusion problems where there are pollution terms. In addition, we want to act on the system in a single point of the domain for cost reasons. The systems being studied are parabolic with missing (initial or boundary) data representing pollution, where we introduce a control function. The method of low-regret control of J.-L. Lions, used here for the first time to the pointwise control, seems to be well suited. We then look for the control which can improve the situation instead of doing nothing (no control).Solutions are not regular functions and can only be considered in the weak sense. Two methods are used here: the first one is the method of transposition of Lions-Magenes, detailed in Chapter 3 of the thesis, and the second method consists in regularizing the Dirac mass, presented in chapter 4. Each one of the two methods offers a new point of view. In particular, the functional spaces where the existence of a solution is obtained are different. For both methods, however, a singular optimality system is established for the low-regret pointwise control.A final chapter is devoted to the numerical schemes associated to the low-regret pointwise optimal control, where we obtain error estimates using finite elements method (FEM).