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The aim of this paper is to establish necessary optimality conditions for optimal control problems governed by steady, incompressible Navier-Stokes equations with shear-dependent viscosity. The main difficulty derives from the fact that equations of this type may exhibit non-uniqueness of weak solutions, and is overcome by introducing a family of approximate control problems governed by well posed generalized Stokes systems and by passing to the limit in the corresponding optimality conditions.

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an Lp -space (p < ∞). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.