Computational Optimization and Applications

, Volume 62, Issue 1, pp 157–180 | Cite as

Annular and sectorial sparsity in optimal control of elliptic equations

  • Roland Herzog
  • Johannes Obermeier
  • Gerd Wachsmuth
Article

Abstract

Optimal control problems are considered with linear elliptic equations in polar coordinates. The objective contains \(L^1\)-type norms, which promote sparse optimal controls. The particular iterated structure of these norms gives rise to either annular or sectorial sparsity patterns. Optimality conditions and numerical solution approaches are developed.

Keywords

Directional sparsity Polar coordinates Coordinate transformation 

Mathematics Subject Classification

49K20 65K10 49M15 

References

  1. 1.
    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). doi: 10.1007/s10589-007-9150-9 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Wachsmuth, G., Wachsmuth, D.: Convergence and regularization results for optimal control problems with sparsity functional. ESAIM 17(3), 858–886 (2011). doi:10.1051/cocv/2010027 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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). doi: 10.1137/110834366 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Casas, E., Herzog, R., Wachsmuth, G.: Approximation of sparse controls in semilinear equations by piecewise linear functions. Numer. Math. 122(4), 645–669 (2012). doi:10.1007/s00211-012-0475-7 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Clason, C., Kunisch, K.: A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM 17(1), 243–266 (2011). doi:10.1051/cocv/2010003 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Clason, C., Kunisch, K.: A measure space approach to optimal source placement. Comput. Optim. Appl. Int. J. 53(1), 155–171 (2012). doi:10.1007/s10589-011-9444-9 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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). doi:10.1137/110843216 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Herzog, R., Stadler, G., Wachsmuth, G.: Directional sparsity in optimal control of partial differential equations. SIAM J. Control Optim. 50(2), 943–963 (2012). doi:10.1137/100815037 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Casas, E., Clason, C., Kunisch, K.: Parabolic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 51(1), 28–63 (2013). doi:10.1137/120872395 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    van Niekerk, J., Tongue, B., Packard, A.: Active control of a circular plate to reduce transient noise transmission. J. Sound Vib. 183(4), 643–662 (1995). doi:10.1006/jsvi.1995.0277 CrossRefMATHGoogle Scholar
  11. 11.
    Coorpender, S., Finkel, D., Kyzar, J., Sims, R., Smirnova, A., Tawhid, M., Bouton, C., Smith, R.: Modeling and optimization issues concerning a circular piezoelectric actuator design. Tech. Rep. CRSC-TR99-22, North Carolina State University (1999). http://www.ncsu.edu/crsc/reports/reports99.html
  12. 12.
    Raghavan, A., Cesnik, C.E.S.: Modeling of piezoelectric-based lamb wave generation and sensing for structural health monitoring. In: Smart Structures and Materials 2004: Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems, vol. 5391, pp. 419–430 (2004). DOI: 10.1117/12.540269
  13. 13.
    Yeum, C.M., Sohn, H., Ihn, J.B.: Lamb wave mode decomposition using concentric ring and circular piezoelectric transducers. Wave Motion Int. J. Rep. Res. Wave Phenom. 48(4), 358–370 (2011). doi:10.1016/j.wavemoti.2011.01.001 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Li, X., Du, H., Xu, L., Hu, Y., Xu, L.: Optimization of a circular thin-film piezoelectric actuator lying on a clamped multilayered elastic plate. IEEE Trans. Ultrason. 56(7), 1469–1475 (2009)CrossRefGoogle Scholar
  15. 15.
    Fremlin, D.H.: Measure theory. Vol. 2. Torres Fremlin, Colchester. Broad foundations, Corrected second printing of the 2001 original (2003)Google Scholar
  16. 16.
    Diestel, J., Uhl, J.: Vector Measures. Mathematical Surveys and Monographs. American Mathematical Society, Providence (1977)Google Scholar
  17. 17.
    Ioffe, A.D., Tichomirov, V.M.: Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin (1979)MATHGoogle Scholar
  18. 18.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems, Classics in Applied Mathematics, vol. 28. SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  19. 19.
    Falk, R.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44, 28–47 (1973)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Casas, E., Mateos, M., Tröltzsch, F.: Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl. 31, 193–219 (2005)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Brandts, J., Hannukainen, A., Korotov, S., Křížek, M.: On angle conditions in the finite element method. SIAM J. 56, 81–95 (2011)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Roland Herzog
    • 1
  • Johannes Obermeier
    • 1
  • Gerd Wachsmuth
    • 1
  1. 1.Faculty of MathematicsTechnische Universität ChemnitzChemnitzGermany

Personalised recommendations