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Some optimal control problems of multistate equations appearing in fluid mechanics. (English) Zbl 0769.49002
Two optimal control problems associated to the steady-state Navier-Stokes equations are considered. These problems consist in minimizing a cost functional involving the vorticity in the fluid. The controls are the body forces or the heat flux on the boundary and the state is the velocity of the fluid.
Existence of an optimal control is proved and some optimality conditions are derived. In both problems, the relation $$\text{control}\to\text{state}$$ is multivalued. To overcome this difficulty, an approximate family of optimal control problems governed by a well-posed linear elliptic system is introduced and the optimality conditions for these problems are obtained.

##### MSC:
 49J20 Existence theories for optimal control problems involving partial differential equations 76D05 Navier-Stokes equations for incompressible viscous fluids
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##### References:
 [1] ABERGEL and R. TEMAM, 1990, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynamics, 1, 303-325. Zbl0708.76106 · Zbl 0708.76106 [2] F. ABERGEL and R. TEMAM, 1992, Optimal control of turbulent flows, in Optimal control of viscous flows, S. S. Sritharan ed., Frontiers in Applied Mathematics Series, SIAM, Philadelphia. MR1632423 [3] E. CASAS and L. FERNANDEZ, 1989, A Green’s formula for quasilinear elliptic operators, J. of Math. Anal. & Appl., 142, 62-72. Zbl0704.35047 MR1011409 · Zbl 0704.35047 [4] H. CHOI, J. KIM, P. MOIN, R. TEMAM, à paraître, Methods of feedback controlfor distributed Systems and applications to Burgers equations. [5] M. GAULTIER and M. LEZAUN, 1989, Equations de Navier-Stokes couplées à des équations de la chaleur : résolution par une méthode de point fixe endimension infinie, Ann. Sc. Math. Québec, 13, 1-17. Zbl0716.35064 MR1006500 · Zbl 0716.35064 [6] [6] M. GUNZBURGER, L. Hou and T. SVOBODNY, 1991, Analysis and finite element approximations of optimal control problems for the stationary Navier-Stokes equations with Dirichlet conditions, M2AN, 25, 711-748. Zbl0737.76045 MR1135991 · Zbl 0737.76045 [7] M. GUNZBURGER, L. Hou and T. SVOBODNY, 1991, Boundary velocity controlof incompressible flow with an application to viscous drag reduction, SIAM J. on Control & Optimization. Zbl0756.49004 · Zbl 0756.49004 [8] A. IOFFE and V. TIKHOMOROV, 1979, Extremal Problems, North-Holland, Amsterdam. [9] J. LIONS, 1968, Contrôle de Systèmes Gouvernés pat des Equations aux Dérivées Partielles, Dunod, Paris. Zbl0179.41801 · Zbl 0179.41801 [10] J. LIONS, 1969, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris. Zbl0189.40603 · Zbl 0189.40603 [11] J. NEČAS, 1967, Les Méthodes Directes en Théorie des Equations Elliptiques, Editeurs Academia, Prague. MR227584 · Zbl 1225.35003 [12] P. RABINOWITZ, 1968, Existence and nonuniqueness of rectangular solutions of the Benard problem, Arch Rational Mech. Anal., 29, 32-57. Zbl0164.28704 MR233557 · Zbl 0164.28704 [13] R. TEMAM, 1979, Navier-Stokes Equations, North-Holland, Amsterdam. Zbl0426.35003 MR603444 · Zbl 0426.35003
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