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arXiv · 1411.4974

Optimal control of elliptic PDEs on surfaces of codimension 1

C. Brett·A. S. Dedner·C. M. Elliott
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Abstract

We consider an elliptic optimal control problem where the objective functional contains an integral along a surface of codimension 1, also known as a hypersurface. In particular, we use a fidelity term that encourages the state to take certain values along a curve in 2D or a surface in 3D. In the discretisation of this problem, which uses piecewise linear finite elements, we allow the hypersurface to be approximated e.g. by a polyhedral hypersurface. This can lead to simpler numerical methods, however it complicates the numerical analysis. We prove a priori $L^2$ error estimates for the control and present numerical results that agree with these. A comparison is also made to point control problems.

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C. Brett, A. S. Dedner, C. M. Elliott. 2014-11-18. Optimal control of elliptic PDEs on surfaces of codimension 1. https://arxiv.org/abs/1411.4974

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