Least-squares Methods for Optimal Control

Optimal control and optimal shape design problems for the Navier-Stokes equations arise in many important practical applications, such as design of optimal profiles [7], drag minimization [9], [11], and heating and cooling [12], among others. Typically, optimal control problems for the NavierStokes equations combine Lagrange multiplier techniques to enforce the constraints and to derive an optimality system (see, e.g., [11]-[12]), with mixed Galerkin discretization for the state equations. Resulting methods are well-studied theoretically, for example, an abstract framework that can be used for the analyses of such optimal control methods has been suggested in [13]. However, the use of Lagrange multipliers and mixed Galerkin discretizations is associated with some complications in the numerical computations which can reduce the overall efficiency and robustness of corresponding algorithms. For example, resulting discrete problems are in general indefinite. Similarly, it is now well-understood that stability of mixed discretizations does not allow one to choose independently the approximation spaces for the velocity and the pressure, and that these spaces are subject to a restrictive stability condition known as the inf-sup (or LBB) condition; see [8]. One possibility to remedy these difficulties is to consider optimal control methods in which the Navier-Stokes constraint is treated by augmented Lagrangian techniques; see [6]. Nevertheless, the use of mixed Galerkin discretization in the method of [6] still requires approximation by finite element spaces that are subject to the inf-sup condition.