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arXiv · 1810.08007

Optimal control of a non-smooth quasilinear elliptic equation

Christian Clason·Vu Huu Nhu·Arnd Rösch
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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.

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Christian Clason, Vu Huu Nhu, Arnd Rösch. 2018-10-18. Optimal control of a non-smooth quasilinear elliptic equation. https://doi.org/10.3934/mcrf.2020052

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