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Welcome on the ANR COSS webpage: ANR-22-CE40-0010

You can find here some news on the project, the list of the members, the research activites and also some open positions.

Contact: nicolas.forcadel [at] insa-rouen.fr

News

• Second COSS day on 22th March 2024 !
• PhD offer here !
• Offre de tèsee ici ! (ouvert du15 Juillet au 15 Novembre)
• PhD offer see here !
• Workshop Maathrafic on the 5--8 June 2023 !
• First COSS days on 16th, 17th March 2023 !
• Start of the project on the first of January 2023 !

Description of the project

The central theme of this project lies in the area of control theory and partial differential equations (in particular Hamilton-Jacobi equations), posed on stratified structures and networks. These equations appear very naturally in several applications like traffic flow modeling, energy management in smart-grids networks or sea-land trajectories with different dynamics. These control problems can be studied within the framework of Hamilton Jacobi equations theory. Recently, significant results have been obtained, leading to a good understanding of the notion of viscosity solutions (in particular the questions of existence and uniqueness) on some specific stratified structures. This base of results will be further developed in different directions. It will first be necessary to complete the analysis for more general problems under weaker hypotheses than the one used so far (nature of the stratification, hypotheses on the Hamiltonians, ...). On the other hand, it is necessary to use the already existing base to advance research in other active areas such as homogenization or mean field games. Moreover, all of the theoretical results will be used to achieve progress in the modeling and numerical resolution of some control problems on stratified domains.

More precisely, the aim of this project is to develop the fundamental theory governing optimal control problems, differential games and mean field games in stratified domains and networks, to provide computational methods for their solutions and also to propose a theory of homogenization allowing to pass from microscopic models to macroscopic ones, thereby giving rigorous justifications of the latter. The main objectives include understanding fundamental questions on the structure of optimal trajectories, in particular when moving from one strata to another, the analysis of the value function and its characterization by adequate Hamilton-Jacobi equations, the feedback control, singular perturbations and homogenization. In the particular case of networks, our aim is also to understand the links with conservation laws with discontinuous fluxes. These tools will allow us to tackle a large class of problems in which the dynamics are discontinuous and may depend on the domain where the trajectory takes place.

Our project proposes challenging mathematical and numerical studies for optimal control problems, games and mean-field games, and homogenization on stratified structures. Our approaches are based on nonlinear PDEs theory, non-smooth analysis, and advanced numerical methods. Thanks to the expertise of the team members, and inspired by real-life challenging problems, our project will contribute in advancing the theory and will produce open access academic numerical codes. The project is organized in four major methodological axes: optimal control and optimal trajectories, singular perturbation and homogenization, game theory and mean field games and numerical analysis.