Optimal control of a non-smooth quasilinear elliptic equation
Abstract
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 G\^ateaux differentiable. This leads to a control-to-state operator that is directionally but not G\^ateaux 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 PC^1 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.
Keywords
Cite
@article{arxiv.1810.08007,
title = {Optimal control of a non-smooth quasilinear elliptic equation},
author = {Christian Clason and Vu Huu Nhu and Arnd Rösch},
journal= {arXiv preprint arXiv:1810.08007},
year = {2021}
}