Renormalized solutions for optimal control of the drift in Fokker-Planck equations

Hannes Meinlschmidt (FAU Erlangen-Nürnberg) #

In this talk I will consider an optimal control problem subject to a Fokker-Planck equation, where the control acts in the (temporal amplitude of the) drift vector. There will be only very mild assumptions on the regularity of the data in the problem; thus, in order to obtain uniqueness of solutions, we will consider the concept of renormalized solutions. These were popularized by diPerna and Lions in 1989 in their seminal work on transport equations and have since become a renowed tool to deal with low regularity data in transport and parabolic equations; there are only few results on associated optimal control problems, though. It will turn out that we will have to consider the sensitivity equation for the optimal control problem in a „fully renormalized“ form, which is nonlinear, but I will show how we can still obtain Fréchet differentiability of the objective function and a few more results.