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Mathematics > Optimization and Control

arXiv:1810.08007 (math)
[Submitted on 18 Oct 2018 (v1), last revised 5 Dec 2018 (this version, v2)]

Title:Optimal control of a non-smooth quasilinear elliptic equation

Authors: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 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 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.
Subjects: Optimization and Control (math.OC)
Cite as: arXiv:1810.08007 [math.OC]
  (or arXiv:1810.08007v2 [math.OC] for this version)
  https://doi.org/10.48550/arXiv.1810.08007
Journal reference: Mathematical Control and Related Fields 11 (2021), 521-554
Related DOI: https://doi.org/10.3934/mcrf.2020052

Submission history

From: Christian Clason [view email]
[v1] Thu, 18 Oct 2018 12:18:17 UTC (205 KB)
[v2] Wed, 5 Dec 2018 10:57:44 UTC (207 KB)
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