Renormalized Solutions in Optimal Control of Evolution Equations

• Applicant: Professor Dr. Hannes Meinlschmidt
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 551486487

In this research project we are concerned with extending the classical and well known theory of optimal control of evolution equations, that is, time-dependent partial differential equations, to problems where the irregularity of the given data is so severe that the usual notions of solutions to these equations do not lead to a satisfying theory. This may concern lack of uniqueness, but also the more fundamental problem that one cannot even set up a well-defined notion of weak or distributional solution. For certain problem classes, it is then useful to consider so-called renormalized solutions which requires that not only the weak solution itself, but also all nonlinear modulations of the solution within a certain regularity class, solve the partial differential equation, at least in a distributional sense. This allows to recover uniqueness, or, if the original notion of solution could not be well defined to begin with, to set up a well-defined notion of solution in the first place. Such a notion of solution is inherently nonlinear though, and does not lead to the usual formulations via operator theory which makes classical optimal control theory not applicable. This project thus aims at establishing this optimal control theory by different means on the example of Fokker-Planck equations with very irregular drift field, and on strongly nonlinear pabolic equations without growth conditions.

• DFG Programme: Research Grants
• International Connection: Austria
• Cooperation Partner: Professor Dr. Karl Kunisch