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This paper focuses on the analysis of an optimal control problem governed bya nonsmooth quasilinear partial differential equation that models a stationaryincompressible shear-thickening fluid. We start by studying the directionaldifferentiability of the non-smooth term within the state equation as a priorstep to demonstrate the directional differentiability of the solution operator.Thereafter, we establish a primal first order necessary optimality condition(Bouligand (B) stationarity), which is derived from the directionaldifferentiability of the solution operator. By using a local regularization ofthe nonsmooth term and carrying out an asymptotic analysis thereafter, werigourously derive a weak stationarity system for local minima. By combiningthe B- and weak stationarity conditions, and using the regularity of theLagrange multiplier, we are able to obtain a strong stationarity system thatincludes an inequality for the scalar product between the symmetrized gradientof the state and the Lagrange multiplier.

Optimal control of a nonsmooth PDE arising in the modeling of shear-thickening fluids