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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.

49J20 Existence theories for optimal control problems involving partial differential equations
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI EuDML
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