Extremal boundary value problems of the dynamics of a viscous incompressible fluid

We consider the control problem for stationary Navier–Stokes equations. The boundary values of the total head of the flow are taken as control parameter. We prove a solvablity theorem for the extremal problem and obtain conditions that guarantee existence for a singular optimality system and a homeomorphism between the solution set and some finite-dimensional compact set. We study the structure of the solution set of the extremal problem and present the asymptotic expansion of the optimal control.