Optimal control of a non-smooth quasilinear elliptic equation

Abstract

<p style='text-indent:20px;'>This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not Gâteaux differentiable. This leads to a control-to-state operator that is directionally but not Gâteaux differentiable as well. Based on a suitable regularization scheme, we derive C- and strong stationarity conditions. Under the additional assumption that the nonlinearity is a <inline-formula><tex-math id="M1">\begin{document}$ PC^1 $\end{document}</tex-math></inline-formula> function with countably many points of nondifferentiability, we show that both conditions are equivalent. Furthermore, under this assumption we derive a relaxed optimality system that is amenable to numerical solution using a semi-smooth Newton method. This is illustrated by numerical examples.</p>