Fursikov, Andrei V. (ed.) et al., New directions in mathematical fluid mechanics. The Alexander V. Kazhikhov memorial volume. Boston, MA: Birkhäuser (ISBN 978-3-0346-0151-1/hbk). Advances in Mathematical Fluid Mechanics, 193-221 (2010).
The paper focuses on an optimal control problem for a 2D stationary Navier-Stokes system. The authors consider a domain
consisting of a rectangle without a subset bounded by a smooth curve
. The Navier-Stokes system
is considered in
. Dirichlet or Neumann boundary conditions are imposed on the different pieces of the boundary of this domain. The optimal control problem consists to minimize the functional
under the action of controls
imposed on subintervals of the horizontal parts of the boundary of the domain
through Neumann boundary conditions. Here
is the first unit vector of
is the unit outer normal and
. The applied controls are supposed to satisfy
-norm of their respective domains, for some positive constant
. The first main result of the paper proves the existence of a generalized solution of this optimal control problem. The authors start proving further properties of the generalized solution of the associated Stokes problem. They also prove the existence of a generalized solution of the Navier-Stokes problem assuming the existence of admissible collections
for this Navier-Stokes problem and that the boundary data are small enough. This is done building a contraction operator associated to this Navier-Stokes problem. The proof of the existence of an optimal solution is obtained rewriting the original problem as a minimization problem for a continuous functional on a compact set. The second main result of the paper establishes the optimality system for the optimal solution. The authors here use the abstract Lagrange principle framework. The paper ends with the presentation of some briefly described numerical simulations.