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Applied Mathematics & Optimization

, Volume 78, Issue 2, pp 297–328 | Cite as

Optimal Control of a Degenerate PDE for Surface Shape

  • Harbir Antil
  • Shawn W. Walker
Article

Abstract

Controlling the shapes of surfaces provides a novel way to direct self-assembly of colloidal particles on those surfaces and may be useful for material design. This motivates the investigation of an optimal control problem for surface shape in this paper. Specifically, we consider an objective (tracking) functional for surface shape with the prescribed mean curvature equation in graph form as a state constraint. The control variable is the prescribed curvature. We prove existence of an optimal control, and using improved regularity estimates, we show sufficient differentiability to make sense of the first order optimality conditions. This allows us to rigorously compute the gradient of the objective functional for both the continuous and discrete (finite element) formulations of the problem. Numerical results are shown to illustrate the minimizers and optimal controls on different domains.

Keywords

Locally elliptic nonlinear PDE \(L^{p}-L^{2}\) norm discrepancy Mean curvature 

Mathematics Subject Classification

49J20 35Q35 35R35 65N30 

Notes

Acknowledgements

H.A. acknowledges the support by the NSF-DMS-1521590 and S.W.W. acknowledges the support by the NSF-DMS-1418994.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesGeorge Mason UniversityFairfaxUSA
  2. 2.Department of Mathematics and Center for Computation and Technology (CCT)Louisiana State UniversityBaton RougeUSA

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