Análisis de un problema de control óptimo no-suave asociado a un fluido dilatante.

Abstract

This thesis focuses on the analysis of an optimal control problem governed by a nonsmooth equation that models a shear-thickening fluid. First, we study the directional differentiability of the non-smooth nonlinearity in the state equation. Next, we analyze the directional differentiability of the solution operator. We establish a necessary first order optimality condition, which is deduced from the directional differentiability of the solution operator. This condition is related to the concept of Bouligand stationarity. Also, we derive another necessary optimality condition by using a regularization and passage to the limit technique. The optimality system derived in this way is considered a weak stationarity condition. Finally, we combine both optimality conditions to obtain an optimality system, which is related to a stronger stationarity concept. Furthermore, we show that this stronger stationarity concept is equivalent to the Bouligand stationarity concept, in the case of our problem.