Least-Squares Finite Element Methods for Optimality Systems Arising in Optimization and Control Problems

Abstract

The approximate solution of optimization and optimal control problems for systems governed by linear, elliptic partial differential equations is considered. Such problems are most often solved using methods based on applying the Lagrange multiplier rule to obtain an optimality system consisting of the state system, an adjoint-state system, and optimality conditions. Galerkin methods applied to this system result in indefinite matrix problems. Here, we consider using modern least-squares finite element methods for the solution of the optimality systems. The matrix equations resulting from this approach are symmetric and positive definite and are readily amenable to uncoupling strategies. This is an important advantage of least-squares principles as they allow for a more efficient computational solution of the optimization problem. We develop an abstract theory that includes optimal error estimates for least-squares finite element methods applied to optimality systems. We then provide an application of the theory to optimization problems for the Stokes equations.