Computational Optimization and Applications

, Volume 52, Issue 2, pp 559–581 | Cite as

Optimal control of Maxwell’s equations with regularized state constraints

  • Irwin YouseptEmail author


This paper is devoted to an optimal control problem of Maxwell’s equations in the presence of pointwise state constraints. The control is given by a divergence-free three-dimensional vector function representing an applied current density. To cope with the divergence-free constraint on the control, we consider a vector potential ansatz. Due to the lack of regularity of the control-to-state mapping, existence of Lagrange multipliers cannot be guaranteed. We regularize the optimal control problem by penalizing the pointwise state constraints. Optimality conditions for the regularized problem can be derived straightforwardly. It also turns out that the solution of the regularized problem enjoys higher regularity which then allows us to establish its convergence towards the solution of the unregularized problem. The second part of the paper focuses on the numerical analysis of the regularized optimal control problem. Here the state and the control are discretized by Nédélec’s curl-conforming edge elements. Employing the higher regularity property of the optimal control, we establish an a priori error estimate for the discretization error in the \(\boldsymbol{H}(\bold{curl})\)-norm. The paper ends by numerical results including a numerical verification of our theoretical results.


Optimal control with PDEs Maxwell’s equations Vector potential Pointwise state constraints Nédélec’s curl-conforming edge elements A priori error estimates 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alonso, A., Valli, A.: An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comput. 68(226), 607–631 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alonso, A., Valli, A.: Eddy Current Approximation of Maxwell Equations: Theory, Algorithms and Applications. Springer, Berlin (2010) zbMATHGoogle Scholar
  3. 3.
    Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Beck, R., Hiptmair, R., Hoppe, R.H.W., Wohlmuth, B.: Residual based a posteriori error estimators for eddy current computation. M2AN Math. Model. Numer. Anal. 34(1), 159–182 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bossavit, A.: Computational Electromagnetism. Academic Press, San Diego (1998) zbMATHGoogle Scholar
  6. 6.
    Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 4, 1309–1322 (1986) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ciarlet, P. Jr. , Zou, J.: Fully discrete finite element approaches for time-dependent Maxwell’s equations. Numer. Math. 82(2), 193–219 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151(3), 221–276 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Costabel, M., Dauge, M., Nicaise, S.: Singularities of Maxwell interface problems. M2AN Math. Model. Numer. Anal. 33(3), 627–649 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Girault, V., Raviart, P.: Finite Element Methods for Navier–Stokes Equations. Springer, Berlin (1986) zbMATHCrossRefGoogle Scholar
  11. 11.
    Hiptmair, R.: Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal. 36(1), 204–225 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11(1), 237–339 (2002) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hoppe, R.H.W.: Adaptive multigrid and domain decomposition methods in the computation of electromagnetic fields. J. Comput. Appl. Math. 168(1–2), 245–254 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hoppe, R.H.W., Schöberl, J.: Convergence of adaptive edge element methods for the 3D eddy currents equations. J. Comput. Math. 27, 657–676 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Ito, K., Kunisch, K.: Semi-smooth Newton methods for state-constrained optimal control problems. Syst. Control Lett. 50, 221–228 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. Society for Industrial and Applied Mathematics, Philadelphia (2008) zbMATHCrossRefGoogle Scholar
  17. 17.
    Landau, L.D., Lifshitz, E.M.: Electrodynamics of continuous media. In: Course of Theoretical Physics, vol. 8. Pergamon, Oxford (1960). Translated from the Russian by J.B. Sykes and J.S. Bell Google Scholar
  18. 18.
    Monk, P.: Analysis of a finite element method for Maxwell’s equations. SIAM J. Numer. Anal. 29, 714–729 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Clarendon, Oxford (2003) zbMATHCrossRefGoogle Scholar
  20. 20.
    Nédélec, J.C.: Mixed finite elements in ℝ3. Numer. Math. 35, 315–341 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Nédélec, J.C.: A new family of mixed finite elements in ℝ3. Numer. Math. 50, 57–81 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Schöberl, J.: NETGEN/NGSolve,
  23. 23.
    Schöberl, J.: A posteriori error estimates for Maxwell equations. Math. Comput. 77, 633–649 (2008) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations