Some transport and control problems in economics: theoretical and numerical aspects.

Summary In this thesis we explore the use of optimal control and mass transport for economic modeling. We thus take the opportunity to bring together several works involving these two tools, sometimes interacting with each other. We first briefly introduce the recent mean-field game theory introduced by Lasry and Lions and focus on the control aspect of the Fokker-Planck equation. We exploit this aspect both to obtain equilibrium existence results and to develop numerical solution methods. We test the algorithms in two complementary cases, namely the convex framework (crowd aversion, two population dynamics) and the concave framework (attraction, externalities and scale effects in a stylized model of technological transition). In a second step, we study a matching problem mixing optimal transport and optimal control. The planner seeks an optimal matching, fixed for a given period (commitment), given that the margins evolve (possibly randomly) in a controlled manner. Finally, we reformulate a risk sharing problem between d agents (for which we prove an existence result) into an optimal control problem with comonotonicity constraints. This allows us to obtain optimality conditions with which we construct a simple and convergent algorithm.