Internship M2 2014-2015 1. Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction PLACE: Institut de Recherche Math´ematique de Rennes CNRS UMR6625 ADVISORS: Yves Achdou (Univ. Paris 7), Olivier Ley (INSA Rennes) Nicoletta Tchou (Univ. Rennes 1) CONTACT:
[email protected] SUBJECT: An optimal control problem is an optimization problem where one tries to minimize a cost which depends on the solution of a controlled ordinary differential equation (ODE). The ODE is controlled in the sense that it depends on a fonction called the control. By changing the control, one modifies the solution. The goal is to find the best control in order to minimize the given cost. When trying to write the necessary condition for a solution to the optimal control problem, we obtain a partial differential equation called Hamilton-Jacobi-Bellman (HJB) equation. If one can solve this HJB equation, one can solve the optimal control problem. This problem is extensively studied when the solution of the ODE lies in IRN or in an open subset of IRN . Here we consider optimal control problems where the solution of the ODE is constrained to lie in a network of IR2 (a collection of edges and vertices). The goal of the internship is to study an optimal control problem in a network Ω ⊂ IR2 as a limit of an optimal control problem where the solution of the ODE is contrained to remain in an open set Ω ⊂ IR2 whose thickness tends to 0 and Ω approaches Ω when → 0. REFERENCES: [1] G. Galise, C. Imbert, and R. Monneau, A junction condition by specified homogenization and application to traffic lights. Preprint 2014, https://hal.archives-ouvertes.fr/hal-01010512 [2] Y. Achdou and N.A. Tchou Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction, To appear in Comm. Partial Differential Equations. [3] G.Barles, A. Briani, E. Chasseigne, N.A. Tchou. Homogenization Results for a Deterministic Multi-domains Periodic Control Problem, Preprint 2014, https://hal.archives-ouvertes.fr/hal-00986700 1
