  We design and compute a class of optimal control problems for
reaction-diffusion systems. They form mean field control problems related to
multi-density reaction-diffusion systems. To solve proposed optimal control
problems numerically, we first apply high-order finite element methods to
discretize the space-time domain and then solve the optimal control problem
using augmented Lagrangian methods (ALG2). Numerical examples, including
generalized optimal transport and mean field control problems between Gaussian
distributions and image densities, demonstrate the effectiveness of the
proposed modeling and computational methods for mean field control problems
involving reaction-diffusion equations/systems.
