Optimal control of vortices in non-homogeneous Navier–Stokes flows

Optimal control of time dependent fluid flow governed by the incompressible Navier–Stokes equations is considered. A cost functional based on a local dynamical systems characterization of vortices is investigated. The resulting functional is a non-convex function of the velocity gradient tensor. The optimality system based on a Lagrangian formulation and adjoint equations describing first-order necessary optimality conditions is provided. The gradient and the second derivative of the cost functional with respect to the control are derived.