Wasserstein metric convergence method for Fokker-Planck equations with point controls

In this work we present a result on linear diffusion equations with point controls. This is the first partial result obtained along the completion of [5] which includes more general nonlinear diffusion equations. The main focus here is showing how recent variational principles based on Wasserstein metric, used to solve homogeneous diffusion equations, can actually be extended to solving nonhomogeneous equations as well. In particular, we study here an initial value problem from control theory. We consider a Fokker-Planck Equation (FPE) in one dimension with a time dependent point control of this form: 8