Optimization of the blood pressure with the control in coefficients

Abstract

This article is devoted to the study of a distributed control problem, with the control in coefficients, inspired by a disease that can lead to serious health problems: high blood pressure. We are concerned with the determination of a viscosity function that realizes an optimal blood pressure configuration. Using the mathematical model for viscous fluid-elastic structure interaction problems, we present existence, uniqueness, regularity results and estimates for the three unknown functions of the problem: velocity and pressure of the fluid and displacement of the elastic medium. The weak regularity of the state provided by the variational approach of the problem as well as the choice of the control variable induce some difficulties in the proof of the existence of an optimal control. The choice of the cost functional leads to an adjoint system which is not a divergence free one. For analyzing it, we propose a method based on the construction of several functions with suitable properties. Finally, we establish the necessary conditions of optimality.