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Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

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Abstract

We discretize a directionally sparse parabolic control problem governed by a linear equation by means of control approximations that are piecewise constant in time and continuous piecewise linear in space. By discretizing the objective functional with the help of appropriate numerical quadrature formulas, we are able to show that the discrete optimal solution exhibits a directional sparse pattern alike the one enjoyed by the continuous solution. Error estimates are obtained and a comparison with the cases of having piecewise approximations of the control or a semilinear state equation are discussed. Numerical experiments that illustrate the theoretical results are included.

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References

  1. Boulanger, A.C., Trautmann, P.: Sparse optimal control of the KdV-Burgers equation on a bounded domain. SIAM J. Control Optim. 55(6), 3673–3706 (2017). https://doi.org/10.1137/15M1020745

    Article  MathSciNet  MATH  Google Scholar 

  2. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, vol. 15, 2nd edn. Springer, New York (2002). https://doi.org/10.1007/978-1-4757-3658-8

  3. Carstensen, C.: Quasi-interpolation and a posteriori error analysis in finite element methods. M2AN Math. Model. Numer. Anal. 33(6), 1187–1202 (1999). https://doi.org/10.1051/m2an:1999140

    Article  MathSciNet  MATH  Google Scholar 

  4. Casas, E., Clason, C., Kunisch, K.: Approximation of elliptic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 50(4), 1735–1752 (2012). https://doi.org/10.1137/110843216

    Article  MathSciNet  MATH  Google Scholar 

  5. Casas, E., Clason, C., Kunisch, K.: Parabolic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 51(1), 28–63 (2013). https://doi.org/10.1137/120872395

    Article  MathSciNet  MATH  Google Scholar 

  6. Casas, E., Herzog, R., Wachsmuth, G.: Approximation of sparse controls in semilinear equations by piecewise linear functions. Numer. Math. 122(4), 645–669 (2012). https://doi.org/10.1007/s00211-012-0475-7

    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(3), 795–820 (2012). https://doi.org/10.1137/110834366

    Article  MathSciNet  MATH  Google Scholar 

  8. Casas, E., Herzog, R., Wachsmuth, G.: Analysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations. ESAIM Control Optim. Calc. Var. 23(1), 263–295 (2017). https://doi.org/10.1051/cocv/2015048

    Article  MathSciNet  MATH  Google Scholar 

  9. Casas, E., Kunisch, K.: Parabolic control problems in space-time measure spaces. ESAIM Control Optim. Calc. Var. 22(2), 355–370 (2016). https://doi.org/10.1051/cocv/2015008

    Article  MathSciNet  MATH  Google Scholar 

  10. Casas, E., Mateos, M., Rösch, A.: Finite element approximation of sparse parabolic control problems. Math. Control Rel. Fields 7(3), 397–417 (2017). https://doi.org/10.3934/mcrf.2017014

    MathSciNet  MATH  Google Scholar 

  11. Casas, E., Tröltzsch, F.: A general theorem on error estimates with application to a quasilinear elliptic optimal control problem. Comput. Optim. Appl. 53(1), 173–206 (2012). https://doi.org/10.1007/s10589-011-9453-8

    Article  MathSciNet  MATH  Google Scholar 

  12. Casas, E., Zuazua, E.: Spike controls for elliptic and parabolic PDEs. Syst. Control Lett. 62(4), 311–318 (2013). https://doi.org/10.1016/j.sysconle.2013.01.001

    Article  MathSciNet  MATH  Google Scholar 

  13. Clason, C., Kunisch, K.: A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM Control Optim. Calc. Var. 17(1), 243–266 (2011). https://doi.org/10.1051/cocv/2010003

    Article  MathSciNet  MATH  Google Scholar 

  14. Clason, C., Kunisch, K.: A measure space approach to optimal source placement. Comput. Optim. Appl. 53(1), 155–171 (2012). https://doi.org/10.1007/s10589-011-9444-9

    Article  MathSciNet  MATH  Google Scholar 

  15. de los Reyes, J.C., Meyer, C., Vexler, B.: Finite element error analysis for state-constrained optimal control of the Stokes equations. Control Cybernet. 37(2), 251–284 (2008). http://control.ibspan.waw.pl:3000/contents/export?filename=2008-2-01_reyes_et_al.pdf

  16. Herzog, R., Stadler, G., Wachsmuth, G.: Directional sparsity in optimal control of partial differential equations. SIAM J. Control Optim. 50(2), 943–963 (2012). https://doi.org/10.1137/100815037

    Article  MathSciNet  MATH  Google Scholar 

  17. Kunisch, K., Pieper, K., Vexler, B.: Measure valued directional sparsity for parabolic optimal control problems. SIAM J. Control Optim. 52(5), 3078–3108 (2014). https://doi.org/10.1137/140959055

    Article  MathSciNet  MATH  Google Scholar 

  18. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170. Springer, Berlin (1971)

  19. Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems. I. Problems without control constraints. SIAM J. Control Optim. 47(3), 1150–1177 (2008). https://doi.org/10.1137/070694016

    Article  MathSciNet  MATH  Google Scholar 

  20. Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems. II. Problems with control constraints. SIAM J. Control Optim. 47(3), 1301–1329 (2008). https://doi.org/10.1137/070694028

    Article  MathSciNet  MATH  Google Scholar 

  21. Meidner D., Vexler B.: Optimal error estimates for fully discrete galerkin approximations of semilinear parabolic equations (2017). arXiv:1707.07889v1

  22. 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). https://doi.org/10.1007/s00211-011-0409-9

    Article  MathSciNet  MATH  Google Scholar 

  23. Pieper, K.: Finite element discretization and efficient numerical solution of elliptic and parabolic sparse control problems. Ph.D. thesis, Technische Universität München (2015). http://mediatum.ub.tum.de/node?id=1241413

  24. Pieper, K., Vexler, B.: A priori error analysis for discretization of sparse elliptic optimal control problems in measure space. SIAM J. Control Optim. 51(4), 2788–2808 (2013). https://doi.org/10.1137/120889137

    Article  MathSciNet  MATH  Google Scholar 

  25. Raviart, P.A., Thomas, J.M.: Introduction à l’analyse numérique des équations aux dérivées partielles. Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris (1983)

  26. Stadler, G.: Elliptic optimal control problems with \(L^1\)-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44(2), 159–181 (2009). https://doi.org/10.1007/s10589-007-9150-9

    Article  MathSciNet  MATH  Google Scholar 

  27. Trautmann, P.: Measure valued optimal control problems governed by wave-like equations. Ph.D. thesis, University of Graz (2015)

  28. Tröltzsch, F.: Optimal control of partial differential equations, Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence, RI (2010). https://doi.org/10.1090/gsm/112. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels

  29. Wachsmuth, G., Wachsmuth, D.: Convergence and regularization results for optimal control problems with sparsity functional. ESAIM Control Optim. Calc. Var. 17(3), 858–886 (2011). https://doi.org/10.1051/cocv/2010027

    Article  MathSciNet  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. Departamento de Matemáticas, Campus de Gijón, Universidad de Oviedo, 33203, Gijón, Spain

    Mariano Mateos

  3. Fakultät für Mathematik, Universtät Duisburg-Essen, 45127, Essen, Germany

    Arnd Rösch

Authors
  1. Eduardo Casas
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  2. Mariano Mateos
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  3. Arnd Rösch
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Corresponding author

Correspondence to Mariano Mateos.

Additional information

Eduardo Casas and Mariano Mateos were partially supported by the Spanish Ministerio de Economía y Competitividad under Projects MTM2014-57531-P and MTM2017-83185-P.

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Casas, E., Mateos, M. & Rösch, A. Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity. Comput Optim Appl 70, 239–266 (2018). https://doi.org/10.1007/s10589-018-9979-0

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  • DOI: https://doi.org/10.1007/s10589-018-9979-0

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