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

This paper focuses on the analysis of an optimal control problem governed by a nonsmooth quasilinear partial differential equation that models a stationary incompressible shear-thickening fluid. We start by studying the directional differentiability of the non-smooth term within the state equation as a prior step 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 directional differentiability of the solution operator. By using a local regularization of the nonsmooth term and carrying out an asymptotic analysis thereafter, we rigourously derive a weak stationarity system for local minima. By combining the B- and weak stationarity conditions, and using the regularity of the Lagrange multiplier, we are able to obtain a strong stationarity system that includes an inequality for the scalar product between the symmetrized gradient of the state and the Lagrange multiplier.