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Optimal Control of a Degenerate PDE for Surface Shape

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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.

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References

  1. Amster, P., Mariani, M.C.: The prescribed mean curvature equation with Dirichlet conditions. Nonlinear Anal. 44(1), 59–64 (2001)

    Article  MathSciNet  Google Scholar 

  2. Amster, P., Mariani, M.C.: The prescribed mean curvature equation for nonparametric surfaces. Nonlinear Anal. 52(4), 1069–1077 (2003)

    Article  MathSciNet  Google Scholar 

  3. Antil, H., Salgado, A.J.: Approximation of elliptic equations with BMO coefficients. IMA J. Numer. Anal. 36(1), 222–237 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Antil, H., Nochetto, R.H., Sodré, P.: Optimal control of a free boundary problem: analysis with second-order sufficient conditions. SIAM J. Control Optim. 52(5), 2771–2799 (2014)

    Article  MathSciNet  Google Scholar 

  5. Antil, H., Nochetto, R.H., Sodré, P.: Optimal control of a free boundary problem with surface tension effects: a priori error analysis. SIAM J. Numer. Anal. 53(5), 2279–2306 (2015)

    Article  MathSciNet  Google Scholar 

  6. Arada, N., Casas, E., Tröltzsch, F.: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23(2), 201–229 (2002)

    Article  MathSciNet  Google Scholar 

  7. Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization. Applications to PDEs and optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)

    Book  Google Scholar 

  8. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, 3rd edn. Springer, New York (2008)

    Book  Google Scholar 

  9. Casas, E., de los Reyes, J.C., Tröltzsch, F.: Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19(2), 616–643 (2008)

    Article  MathSciNet  Google Scholar 

  10. Casas, E., Tröltzsch, F.: Error estimates for the finite-element approximation of a semilinear elliptic control problem. Control Cybernet. 31(3), 695–712 (2002)

    MathSciNet  MATH  Google Scholar 

  11. Casas, E., Tröltzsch, F.: Second order analysis for optimal control problems: improving results expected from abstract theory. SIAM J. Optim. 22(1), 261–279 (2012)

    Article  MathSciNet  Google Scholar 

  12. Casas, E., Tröltzsch, F.: Second order optimality conditions and their role in PDE control. Jahresber. Dtsch. Math.-Ver. 117(1), 3–44 (2015)

    Article  MathSciNet  Google Scholar 

  13. Cavallaro, M., Botto, L., Lewandowski, E.P., Wang, M., Stebe, K.J.: Curvature-driven capillary migration and assembly of rod-like particles. Proc. Natl. Acad. Sci. 108(52), 20923–20928 (2011)

    Article  Google Scholar 

  14. Ciarlet, P.G.: The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]

  15. Diening, L., Kaplický, P., Schwarzacher, S.: BMO estimates for the \(p\)-Laplacian. Nonlinear Anal. 75(2), 637–650 (2012)

    Article  MathSciNet  Google Scholar 

  16. Ekeland, I., Témam, R.: Convex analysis and variational problems, Classics in Applied Mathematics vol. 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, english edition, 1999. Translated from the French

  17. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  18. Furst, E.M.: Directing colloidal assembly at fluid interfaces. Proc. Natl. Acad. Sci. 108(52), 20853–20854 (2011)

    Article  Google Scholar 

  19. Giaquinta, M.: On the Dirichlet problem for surfaces of prescribed mean curvature. Manuscripta Math. 12, 73–86 (1974)

    Article  MathSciNet  Google Scholar 

  20. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition

    MATH  Google Scholar 

  21. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications. Springer, New York (2009)

    MATH  Google Scholar 

  22. Irvine, W.T.M., Vitelli, V.: Geometric background charge: dislocations on capillary bridges. Soft Matter 8, 10123–10129 (2012)

    Article  Google Scholar 

  23. Irvine, W.T.M., Vitelli, V., Chaikin, P.M.: Pleats in crystals on curved surfaces. Nature 468(7326), 947–951 (2010)

    Article  Google Scholar 

  24. Kato, T.: Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer-Verlag New York, Inc., New York, 1966

  25. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1980)

    MATH  Google Scholar 

  26. Lipowsky, R., Döbereiner, H.-G., Hiergeist, C., Indrani, V.: Membrane curvature induced by polymers and colloids. Physica A 249(1–4), 536–543 (1998)

    Article  Google Scholar 

  27. Neitzel, I., Vexler, B.: A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numer. Math. 120(2), 345–386 (2012)

    Article  MathSciNet  Google Scholar 

  28. Nirenberg, L.: Topics in nonlinear functional analysis. Courant Institute of Mathematical Sciences New York University, New York, 1974. With a chapter by E. Zehnder, Notes by R. A. Artino, Lecture Notes, 1973–1974

  29. Nochetto, R.H.: Pointwise accuracy of a finite element method for nonlinear variational inequalities. Numer. Math. 54(6), 601–618 (1989)

    Article  MathSciNet  Google Scholar 

  30. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), 483–493 (1990)

    Article  MathSciNet  Google Scholar 

  31. Tröltzsch, F.: Optimal control of partial differential equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2010. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels

  32. Vitelli, V., Irvine, W.: The geometry and topology of soft materials. Soft Matter 9, 8086–8087 (2013)

    Article  Google Scholar 

  33. Walker, S.W.: FELICITY: Finite ELement Implementation and Computational Interface Tool for You. http://www.mathworks.com/matlabcentral/fileexchange/31141-felicity

  34. Wang, J.-S., Feng, X.-Q., Wang, G.-F., Yu, S.-W.: Twisting of nanowires induced by anisotropic surface stresses. Appl. Phys. Lett. 92(19), 191901 (2008)

    Article  Google Scholar 

  35. Wilson, R.M.: Custom shapes from swell gels. Phys. Today 65(5), 15 (2012)

    Article  Google Scholar 

  36. Zakhary, M.J., Sharma, P., Ward, A., Yardimici, S., Dogic, Z.: Geometrical edgeactants control interfacial bending rigidity of colloidal membranes. Soft Matter 9, 8306–8313 (2013)

    Article  Google Scholar 

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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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Authors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, 22030, USA

    Harbir Antil

  2. Department of Mathematics and Center for Computation and Technology (CCT), Louisiana State University, Baton Rouge, LA, 70803, USA

    Shawn W. Walker

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  1. Harbir Antil
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  2. Shawn W. Walker
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Correspondence to Harbir Antil.

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Antil, H., Walker, S.W. Optimal Control of a Degenerate PDE for Surface Shape. Appl Math Optim 78, 297–328 (2018). https://doi.org/10.1007/s00245-017-9407-3

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