Mass Transport Equation Optimal Control Problem

The work focuses on the theoretical study of optimal control theory in mass transportation problem, whose physical ground lies in mixing enhancement concernment in real world, i.e. how to achieve greatest homogeneity at the cost of relative low energy. Two physical models have been analyzed:the first one, when the governing equation is Advection-Diffusion form, taking the velocity field as our control variable, we show the existence and uniqueness of an optimal control under relatively weak regularity condition on velocity field, and conclude the first-order necessary conditions for the optimal control and corresponding state variables by Lagrange-Multiplier theorem; the second one, when the velocity field is subject to two-dimension Navier-Stokes system, we design an appropriate cost functional, for which we apply compactness method to prove the existence of optimal solution, furthermore, by the way of computing the derivative of the cost functional in some Banach space, we easily obtain the first order optimal conditions on the basis of the well-known regularity results about the solution of Navier-Stokes equations. At last, we briefly discuss the possible extension of our results.