A Distributed Control Problem for Two Coupled Fluids in a Porous Medium
Ruxandra Stavre · SIAM Journal on Control and Optimization · 2015
The purpose of this article is to introduce and study a mathematical model and a distributed control problem associated with the interaction between two nonmiscible fluids in a porous medium, when on their interface there exists a jump of temperature. By means of the Schauder's fixed point theorem we obtain the existence of a solution for the variational formulation of the physical problem. Throughout this work, we study certain variational problems with a priori regularity of the weak solutions. We obtain the existence of less regular solutions and then we prove the desired regularity of these solutions. In order to examine the pressure distribution in the two fluids, we consider a distributed control problem which permits us to determine the radiant sources that provide an optimal configuration for the pressure. Since the solution of the problem is not unique, the control variable does not appear explicitly in the definition of our cost functional. To overcome this difficulty, we introduce a family of penalized control problems which approximates our distributed control problem. The necessary conditions of optimality associated with the proposed control problem are derived by passing to the limit in the penalized optimality conditions.