Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints

Article

Abstract

In this paper a class of semilinear elliptic optimal control problem with pointwise state and control constraints is studied. We show that sufficient second order optimality conditions for regularized problems with small regularization parameter can be obtained from a second order sufficient condition assumed for the unregularized problem. Moreover, error estimates with respect to the regularization parameter are derived.

Keywords

Optimal control Semilinear elliptic equation State constraints Regularization Virtual control Second order sufficient conditions 

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References

  1. 1.
    Arada, N., Raymond, J.P.: Optimal control problems with mixed control-state constraints. SIAM J. Control Optim. 39, 1391–1407 (2000) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37, 1176–1194 (1999) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000) MATHGoogle Scholar
  4. 4.
    Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 4, 1309–1322 (1986) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31, 993–1006 (1993) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Casas, E., de los Reyes, J.C., Tröltzsch, F.: Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19(2), 616–643 (2008) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Casas, E., Tröltzsch, F.: Error estimates for the finite-element approximation of a semilinear elliptic control problem. Control Cybern. 31, 695–712 (2002) MATHGoogle Scholar
  8. 8.
    Casas, E., Tröltzsch, F.: Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems. ESAIM: COCV 16(3), 581–600 (2010) MATHCrossRefGoogle Scholar
  9. 9.
    Cherednichenko, S., Krumbiegel, K., Rösch, A.: Error estimates for the Lavrentiev regularization of elliptic optimal control problems. Inverse Probl. 24(5), 055003 (2008) CrossRefGoogle Scholar
  10. 10.
    Cherednichenko, S., Rösch, A.: Error estimates for the regularization of optimal control problems with pointwise control and state constraints. ZAA 27(2), 195–212 (2008) MATHGoogle Scholar
  11. 11.
    Griesse, R., Metla, N., Rösch, A.: Convergence analysis of the SQP method for nonlinear mixed-constrained elliptic optimal control problems. ZAMM 88(10), 776–792 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985) MATHGoogle Scholar
  13. 13.
    Gröger, K.: A W 1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283, 679–687 (1989) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hager, W., Ianulescu, G.: Dual approximation in optimal control. SIAM J. Control Optim. 22, 423–466 (1984) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set method as a semi-smooth Newton method. SIAM J. Optim. 13, 865–888 (2003) MATHCrossRefGoogle Scholar
  16. 16.
    Hinze, M., Meyer, C.: Variational discretization of Lavrentiev-regularized state constrained elliptic optimal control problems. Comput. Optim. Appl. 46(3), 487–510 (2010) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Ito, K., Kunisch, K.: Augmented Lagrangian-SQP-methods for nonlinear optimal control problems of tracking type. SIAM J. Control Opt. 34(3), 874–891 (1996) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Ito, K., Kunisch, K.: Semi-smooth Newton methods for state-constrained optimal control problems. Syst. Control Lett. 50, 221–228 (2003) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Krumbiegel, K., Rösch, A.: A virtual control concept for state constrained optimal control problems. COAP 43(2), 213–233 (2009) MATHGoogle Scholar
  20. 20.
    Krumbiegel, K., Meyer, C., Rösch, A.: A priori error analysis for state constrained boundary control problems part I: control discretization (2009). Weierstrass Institute for Applied Analysis and Stochastics, WIAS Preprint 1393 Google Scholar
  21. 21.
    Krumbiegel, K., Rösch, A.: On the regularization error of state constrained Neumann control problems. Control Cybern. 37(2), 369–392 (2008) MATHGoogle Scholar
  22. 22.
    Kunisch, K., Rösch, A.: Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J. Optim. 13(2), 321–334 (2002) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33(2–3), 209–228 (2006) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Meyer, C., Yousept, I.: Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions. Comput. Optim. Appl. 44(2), 183–212 (2009) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Neitzel, I., Tröltzsch, F.: On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints. Control Cybern. 37(4), 1013–1043 (2008) MATHGoogle Scholar
  26. 26.
    Raymond, J.P., Tröltzsch, F.: Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dyn. Syst. 6, 431–450 (2000) MATHCrossRefGoogle Scholar
  27. 27.
    Rösch, A., Tröltzsch, F.: On regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints. SIAM J. Control Optim. 46(3), 1098–1115 (2007) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Schiela, A.: Barrier methods for optimal control problems with state constraints. SIAM J. Optim. 20(2), 1002–1031 (2009) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Tröltzsch, F.: An SQP method for the optimal control of a nonlinear heat equation. Control Cybern. 23(1/2), 267–288 (1994) MATHGoogle Scholar
  30. 30.
    Tröltzsch, F.: Regular Lagrange multipliers for problems with pointwise mixed control-state constraints. SIAM J. Optim. 15(2), 616–634 (2005) MATHCrossRefGoogle Scholar
  31. 31.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence (2010) MATHGoogle Scholar
  32. 32.
    Zanger, D.Z.: The inhomogeneous Neumann problem in Lipschitz domains. Commun. Partial Differ. Equ. 25, 1771–1808 (2000) MathSciNetMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Mathematics and Stochastics, Nonlinear Optimization and Inverse ProblemsBerlinGermany
  2. 2.Fakultät II–Mathematik und NaturwissenschaftenTechnische Universität BerlinBerlinGermany
  3. 3.Department of MathematicsUniversität Duisburg-EssenDuisburgGermany

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