Quelques problèmes de transport et de contrôle en économie: aspects théoriques et numériques

In this thesis we explore some uses of optimal control and mass transport in economic modeling. We thus catch the opportunity to bring together some works involving both tools, sometimes mixing them. First, we briefly present the recent mean field games theory introduced by Lasry & Lions and focus on the optimal control of Fokker-Planck setting. We take advantage of this aspect in order to obtain both existence results and numerical methods to approximate solutions. We test the algorithms on two complementary settings, namely the convex setting (crowd aversion, two populations dynamics) and the concave one (attraction, externalities and scale effect for a stylized technology switch model). Secondly, we study matching problems combining optimal transport and optimal control. The planner looks for an optimal coupling, fixed during the considered time period (commitment), knowing that the marginals evolve (possibly randomly) and that she can control the evolution. Finally we reformulate a risk-sharing problem between d agents (for whose we prove an existence result) into an optimal control problem with comonotonic constraints. This enables us to write optimality conditions that we use to build a simple convergent algorithm.