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Analysis of Optimal Control Problems of Semilinear Elliptic Equations by BV-Functions

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Abstract

Optimal control problems for semilinear elliptic equations with control costs in the space of bounded variations are analysed. BV-based optimal controls favor piecewise constant, and hence ’simple’ controls, with few jumps. Existence of optimal controls, necessary and sufficient optimality conditions of first and second order are analysed. Special attention is paid on the effect of the choice of the vector norm in the definition of the BV-seminorm for the optimal primal and adjoined variables.

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References

  1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2000)

    MATH  Google Scholar 

  2. Bredies, K, Holler, M.: A pointwise characterization of the subdifferential of the total variation functional, tech. report

  3. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

    MATH  Google Scholar 

  4. Casas, E.: Second order analysis for bang-bang control problems of PDEs. SIAM J. Control Optim. 50, 2355–2372 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Casas, E., Clason, C., Kunisch, K.: Approximation of elliptic control problems in measure spaces with sparse solutions. SIAM J. Control. Optim. 50, 1735–1752 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Casas, E., Clason, C., Kunisch, K.: Parabolic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 51, 28–63 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Casas, E., Herzog, R., Wachsmuth, G.: Optimality conditions and error analysis of semilinear elliptic control problems with l 1 cost functional. SIAM J. Optim. 22, 795–820 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Casas, E., Kogut, P., Leugering, G.: Approximation of optimal control problems in the coefficient for the p-Laplace equation. I. Convergence result. SIAM J. Control Optim. 54, 1406–1422 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Casas, E., Kruse, F., Kunisch, K.: Optimal control of semilinear parabolic equations by BV-functions. SIAM J. Control Optim. 55, 1752–1788 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Casas, E., Kunisch, K.: Optimal control of semilinear elliptic equations in measure spaces. SIAM J. Control Optim. 52, 339–364 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Casas, E., Tröltzsch, F.: Second order optimality conditions for weak and strong local solutions of parabolic optimal control problems. Vietnam J. Math. 44, 181–202 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Casas, E., Kunisch, K., Pola, C.: Regularization by functions of bounded variation and applications to image enhancement. App. Math. Optim. 40, 229–257 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chambolle, A, Duval, V, Peyré, G, Poon, C: Geometric properties of solutions to the total variation denoising problem. Inverse Problems 33, 015002, 44 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Clason, C., Kunisch, K.: A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM: Control, Optimisation and Calculus of Variations 17, 243–266 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland-Elsevier, New York (1976)

    MATH  Google Scholar 

  17. Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  18. Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston (1984)

    Book  MATH  Google Scholar 

  19. Hafemayer, D.: Regularization and Discretization of a BV-Controlled, Elliptic Problem: A Completely Adaptive Approach. master’s thesis, TU-Munich, Germany (2017)

    Google Scholar 

  20. Jalalzai, K.: Some remarks on the staircasing phenomenon in total variation-based image denoising. J. Math. Imaging Vision 54, 256–268 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ring, W.: Structural properties of solutions to total variation regularization problems, M2AN. Math. Model. Numer. Anal. 34, 799–810 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rudin, L. I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1992). Experimental mathematics: computational issues in nonlinear science (Los Alamos, NM, 1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rudin, W.: Real and Complex Analysis. McGraw-Hill Book Co., London (1970)

    MATH  Google Scholar 

  24. Rudin, W.: Functional Analysis. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg (1973)

    MATH  Google Scholar 

  25. Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15, 189–258 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Philadelphia (2010)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005, Santander, Spain

    Eduardo Casas

  2. Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, A-8010, Graz, Austria

    Karl Kunisch

Authors
  1. Eduardo Casas
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  2. Karl Kunisch
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Correspondence to Eduardo Casas.

Additional information

Eduardo Casas was partially supported by Spanish Ministerio de Economía, Industria y Competitividad under research projects MTM2014-57531-P and MTM2017-83185-P. Karl Kunisch was partially supported by the ERC advanced grant 668998 (OCLOC) under the EU’s H2020 research program.

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Casas, E., Kunisch, K. Analysis of Optimal Control Problems of Semilinear Elliptic Equations by BV-Functions. Set-Valued Var. Anal 27, 355–379 (2019). https://doi.org/10.1007/s11228-018-0482-7

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