Optimal control of nonlocal conservation laws and the singular limit problem

• Applicant: Dr. Alexander Keimer
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 547096773

We suggest the study of optimal control problems subject to nonlocal conservation laws and the related singular limit. Nonlocal, in this context, refers to the fact that a part of the flux of the conservation law, i.e., the density-dependent velocity, is spatially averaged over a small neighborhood of a given space-time point. The singular limit then refers to the question of what happens if the nonlocal kernel converges to a Dirac distribution. We aim for answering whether a limit of the sequence of minimizers of the optimization problem subject to nonlocal conservation laws converges to an optimizer of the optimization problem subject to the corresponding local conservation laws. Recently, nonlocal modeling has drawn more interest as it is more "correct", reasonable and intuitive for particular (semi-)physical processes and possesses a significantly different mathematical structure than the typical local models. We are convinced that rather classical differentiability results hold for the corresponding nonlocal dynamics. We aim, among other things, to approximate "local optimal control problems" by their nonlocal counterparts and to establish suitable convergence results. We conclude with an application: Calibrating nonlocal conservation laws as traffic flow models by means of accurate high resolution data.

• DFG Programme: Research Grants
• International Connection: Italy, Switzerland, USA
• Co-Investigators: Professor Dr. Hannes Meinlschmidt; Dr. Lukas Pflug
• Cooperation Partners: Professorin Dr. Debora Amadori; Professor Dr. Alexandre Bayen; Professor Dr. Gianluca Crippa