Optimal control of nonlinear Fokker-Planck equations with reflecting boundary conditions

Fokker-Planck equations are widely used to describe the evolution of probability density functions in stochastic systems, with applications ranging from physics and biology to finance and engineering. While these equations admit analytical solutions in certain linear or low-dimensional settings, their analysis becomes significantly more challenging for nonlinear systems, especially when combined with reflecting boundary conditions and external control inputs. Such settings are highly relevant in physical contexts, including confined diffusions and controlled transport phenomena. In this work, we study the optimal control of nonlinear Fokker-Planck equations subject to reflecting boundaries and pointwise control constraints. We rigorously establish the existence, uniqueness, and boundedness of solutions. Additionally, we prove regularity properties of the control-to-state mapping and derive first-order necessary optimality conditions in the form of variational inequalities coupled with an adjoint system. Our analysis employs the Schauder fixed-point theorem, De Giorgi's iteration, and functional analysis arguments, adapted to address the nonlinear structure of the problem. These results contribute to the theoretical understanding of controlled stochastic processes in nonlinear physical systems.