Least-squares finite element methods for optimization and control problems for the Stokes equations

The approximate solution of optimization and control problems for systems governed by the Stokes equations is considered. Modern computational techniques for such problems are predominantly based on the application of the Lagrange multiplier rule, while penalty formulations, even though widely used in other settings, have not enjoyed the same level of popularity for this class of problems. A discussion is provided that explains why naively defined penalty methods may not be practical. Then, practical penalty methods are defined using methodologies associated with modern least-squares finite element methods. The advantages, with respect to efficiency, of penalty/least-squares methods for optimal control problems compared to methods based on Lagrange multipliers are highlighted. A tracking problem for the Stokes system is used for illustrative purposes.