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No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation

This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is finitely PC2 (continuous and C2 apart from finitely many points). We prove that the control-to-state operator is continuously differentiable...

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Main Authors:	Christian Clason, Vu Huu Nhu, Arnd Rösch
Format:	Bài trích
Language:	eng
Published:	ESAIM: COCV 27 2021
Subjects:	Optimal control non-smooth optimization second-order necessary optimality condition
Online Access:	https://www.esaim-cocv.org/articles/cocv/abs/2021/02/cocv200066/cocv200066.html https://dlib.phenikaa-uni.edu.vn/handle/PNK/2823 https://doi.org/10.1051/cocv/2020092
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spelling	oai:localhost:PNK-28232022-08-17T05:54:38Z No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation Christian Clason Vu Huu Nhu Arnd Rösch Optimal control non-smooth optimization second-order necessary optimality condition This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is finitely PC2 (continuous and C2 apart from finitely many points). We prove that the control-to-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish “no-gap” second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i.e., for which the sufficient condition only differs from the necessary one in the fact that the inequality is strict. A condition that is equivalent to the second-order sufficient optimality condition and could be useful for error estimates in, e.g., finite element discretizations is also provided. 2021-09-13T04:24:48Z 2021-09-13T04:24:48Z 2021 Bài trích https://www.esaim-cocv.org/articles/cocv/abs/2021/02/cocv200066/cocv200066.html https://dlib.phenikaa-uni.edu.vn/handle/PNK/2823 https://doi.org/10.1051/cocv/2020092 eng application/pdf ESAIM: COCV 27
institution	Digital Phenikaa
collection	Digital Phenikaa
language	eng
topic	Optimal control non-smooth optimization second-order necessary optimality condition
spellingShingle	Optimal control non-smooth optimization second-order necessary optimality condition Christian Clason Vu Huu Nhu Arnd Rösch No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation
description	This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is finitely PC2 (continuous and C2 apart from finitely many points). We prove that the control-to-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish “no-gap” second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i.e., for which the sufficient condition only differs from the necessary one in the fact that the inequality is strict. A condition that is equivalent to the second-order sufficient optimality condition and could be useful for error estimates in, e.g., finite element discretizations is also provided.
format	Bài trích
author	Christian Clason Vu Huu Nhu Arnd Rösch
author_facet	Christian Clason Vu Huu Nhu Arnd Rösch
author_sort	Christian Clason
title	No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation
title_short	No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation
title_full	No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation
title_fullStr	No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation
title_full_unstemmed	No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation
title_sort	no-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation
publisher	ESAIM: COCV 27
publishDate	2021
url	https://www.esaim-cocv.org/articles/cocv/abs/2021/02/cocv200066/cocv200066.html https://dlib.phenikaa-uni.edu.vn/handle/PNK/2823 https://doi.org/10.1051/cocv/2020092
_version_	1751856302616739840
score	8.891695

No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation

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