Necessary Conditions for Convergence Rates of Regularizations of Optimal Control Problems

  • Daniel Wachsmuth
  • Gerd Wachsmuth
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT, volume 391)

Abstract

We investigate the Tikhonov regularization of control constrained optimal control problems. We use a specialized source condition in combination with a condition on the active sets. In the case of high convergence rates, these conditions are necessary and sufficient.

Keywords

optimal control problem inequality constraints Tikhonov regularization source condition 

References

  1. 1.
    Casas, E., Herzog, R., Wachsmuth, G.: Optimality conditions and error analysis of semilinear elliptic control problems with L1 cost functional. SIAM J. Optim. (to appear, 2012)Google Scholar
  2. 2.
    Deckelnick, K., Hinze, M.: A note on the approximation of elliptic control problems with bang-bang controls. Comput. Optim. Appl. 51(2), 931–939 (2012)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of inverse problems. Mathematics and its Applications, vol. 375. Kluwer Academic Publishers Group, Dordrecht (1996)MATHCrossRefGoogle Scholar
  4. 4.
    Felgenhauer, U.: On stability of bang-bang type controls. SIAM J. Control Optim. 41(6), 1843–1867 (2003)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Neubauer, A.: Tikhonov-regularization of ill-posed linear operator equations on closed convex sets. J. Approx. Theory 53(3), 304–320 (1988)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations. Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence (2010); Theory, methods and applications, Translated from the 2005 German original by J. SprekelsMATHGoogle Scholar
  7. 7.
    Wachsmuth, D., Wachsmuth, G.: Convergence and regularization results for optimal control problems with sparsity functional. ESAIM Control Optim. Calc. Var. 17(3), 858–886 (2011)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Wachsmuth, D., Wachsmuth, G.: On the regularization of optimization problems with inequality constraints. Control and Cybernetics (2011) (to appear)Google Scholar

Copyright information

© IFIP International Federation for Information Processing 2013

Authors and Affiliations

  • Daniel Wachsmuth
    • 1
  • Gerd Wachsmuth
    • 2
  1. 1.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria
  2. 2.Department of MathematicsChemnitz University of TechnologyChemnitzGermany

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