Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls

Abstract

— Optimal control problems for the stationary Navier-Stokes equations are examined from analytical and numerical points ofview. The controls considered are of Dirichlet type, that is, control is effected through the velocity field on (or the mass flux through) the boundary ; the functionals minimized are either the viscous dissipation or the L-distance of candidate flows to some desiredflow. We show that optimal solutions exist and jus t if y the use of Lagrange multiplier techniques to derive a System of partial differential equations from which optimal solutions may be deduced. We study the regularity of solutions of this system. Then, finite element approximations of solutions of the optimality system are defined and optimal error estimâtes are derived. Resume. — On examine quelques problemes de controle optimal des equations de NavierStokes du point de vue a la fois analytique et numerique. Le controle est du type condition de Dirichlet, c'est-a-dire qu'on choisit le champ de vecteurs vitesses sur la frontiere pour minimiser une fonctionnelle. On considere ici des fonctionnelles de type fonction de dissipation qui mesurent Veffet de la trainee et une distance dans l'espace L. On demontre l'existence de solutions optimales et on utilise la methode des multiplicateurs de Lagrange pour obtenir des conditions necessaires d'optimalite. Apres avoir etabli quelques resultats concernant la regularite des solutions optimales, on definit des approximations par des espaces d'elements finis et on presente les majorations d'erreur optimales. (*) Received April 1990, revised September 1990. The work of MDG was supported by the Air Force Office of Scientific Research under grant number AFOSR-88-0197 and partially supported by the U.S. Department of Energy. The work of LSH was supported by the Department of Education of the Province of Quebec, Actions Structurantes Program. O Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA. () Departement de Mathematiques et de Statistiques, Universite Laval, Quebec, G1K 7P4, Canada. Present address : Department of mathematics and statistics, York University, North York, M3J IP3, Canada. () Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, USA. MAN Modelisation mathematique et Analyse numerique 0764-583X/91/06/711/38/$ 5.80 Mathematical Modelling and Numerical Analysis © AFCET Gauthier-Villars 712 M. D. GUNZBURGER, L. S. HOU, TH. P. SVOBODNY