Least-squares methods for optimal control

Pavel Bochev

University of Texas at Arlington Department of mathematics Box 19408 Arlington, TX 76019


Optimal control of the Navier-Stokes equations has wide and valuable applications in problems like drag minimization, computation of optimal profiles, optimal temperature control of engine components and nuclear reactors, design of inlet shapes for jet engines, etc. In general, an optimal control problem can be characterized by a physical objective, e.g., quasiuniform temperature distribution, or drag reduction, and by a control mechanism to achieve the control objective. Mathematically, the objective is stated in terms of a suitable cost functional, and the control mechanisms can be divided into value controls and shape controls. To meet the objective, value controls use data adjustments, such as boundary velocities and fluxes, while shape controls use the shape of the domain as a control mechanism. Here we develop a new finite element method for optimal boundary control of the Navier-Stokes equations based on application of least-squares variational principles. In contrast to augmented Lagrangian methods, our approach utilizes a bona-fide least-squares functional which combines the cost functional describing the control objective with weighted residuals of the state Navier-Stokes equations. =========================================================================== I I I Dr. P. Bochev Phone: (817) 272-5164 I I Department of Mathematics Fax: (817) 272-5802 I I Box 19408 e-mail: bochev@utamat.uta.edu I I University of Texas at Arlington I I Arlington, TX 76019-0408 http://www.uta.edu/math/bochev.html I I USA I | | ===========================================================================