Dynamical shape control in non-cylindrical hydrodynamics

We are interested in a dynamical shape control problem in hydrodynamics. Th e considered cost functional is the kinetic energy of a perfect and incompr essible fluid. We establish the existence of a non-autonomous vector field (V) over bar which constructs the tube followed by the fluid in such a way as to minimize its kinetic energy. By introducing a specific adjoint state closely related to the shape of the tube, we derive the necessary optimalit y condition.