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arXiv · 2003.11478

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

Christian Clason·Vu Huu Nhu·Arnd Rösch
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Abstract

This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is countably $PC^2$ (continuous and $C^2$ apart from countably 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.

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Christian Clason, Vu Huu Nhu, Arnd Rösch. 2020-09-09. No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation. https://doi.org/10.1051/cocv%2F2020092

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