A boundary control problem for the blood flow in venous insufficiency. The general case

The purpose of this article is to introduce and study an optimal control problem with medical applications. When a vein loses its elasticity, phenomena such as stagnation and recirculation of the blood may appear; these phenomena produce medical complications. We propose an optimization model in order to diminish the negative consequences of the lack of vein elasticity. We extend a previous model involving the interaction between a viscous fluid and an elastic boundary to the case when both the fluid and the elastic medium occupy three dimensional domains. After establishing the existence and uniqueness of the solution for the coupled problem, we present a boundary control problem in order to determine an exterior compression that realizes a blood flow without recirculation. Since it is not possible to find such a compression directly, we consider a sequence of cost functionals and we study the corresponding optimal control problems. The existence and uniqueness of the optimal controls are proved and the optimality conditions that characterize the optimal controls are derived. Finally, we establish the relation between the control problem with physical meaning and the sequence of optimal controls already constructed.