Advertisement

Vietnam Journal of Mathematics

, Volume 44, Issue 1, pp 181–202 | Cite as

Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems

  • Eduardo Casas
  • Fredi TröltzschEmail author
Article

Abstract

Second-order sufficient optimality conditions are considered for a simplified class of semilinear parabolic equations with quadratic objective functional including distributed and terminal observation. Main emphasis is laid on problems where the objective functional does not include a Tikhonov regularization term. Here, standard second-order conditions cannot be expected to hold. For this case, new second-order conditions are established that are based on different types of critical cones. Depending on the choice of this cones, the second-order conditions are sufficient for local minima that are weak or strong in the sense of calculus of variations.

Keywords

Optimal control Parabolic equation Semilinear equation Second-order optimality conditions Weak local minimum Strong local minimum 

Mathematics Subject Classification (2010)

49J20 49K20 

Notes

Acknowledgments

The first author was partially supported by Spanish Ministerio de Economía y Competitividad under projects MTM2011-22711 and MTM2014-57531-P. The second is supported by DFG in the framework of the Collaborative Research Center SFB 910, project B6.

References

  1. 1.
    Bayen, T., Bonnans, J., Silva, F.: Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations. Trans. Am. Math. Soc. 366, 2063–2087 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bayen, T., Silva, F.: Second order analysis for strong solutions in the optimal control of parabolic equations. Preprint Hal-01096149, https://hal.archives-ouvertes.fr/hal-01096149 (2014)
  3. 3.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2012)zbMATHGoogle Scholar
  4. 4.
    Buchholz, R., Engel, H., Kammann, E., Tröltzsch, F.: On the optimal control of the Schlögl-model. Comput. Optim. Appl. 56, 153–185 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Casas, E.: Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35, 1297–1327 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Casas, E.: Second order analysis for bang-bang control problems of PDEs. SIAM J. Control Optim. 50, 2355–2372 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Casas, E., Chrysafinos, K.: Error estimates for the approximation of the velocity tracking problem with bang-bang controls. Submitted (2015)Google Scholar
  8. 8.
    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, 795–820 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Casas, E., Ryll, C., Tröltzsch, F.: Sparse optimal control of the Schlögl and FitzHugh–Nagumo systems. Comput. Methods Appl. Math. 13, 415–442 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Casas, E., Ryll, C., Tröltzsch, F.: Second order and stability analysis for optimal sparse control of the FitzHugh–Nagumo equation. SIAM J. Control Optim. 53, 2168–2202 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Casas, E., Tröltzsch, F.: Second order analysis for optimal control problems: improving results expected from from abstract theory. SIAM J. Optim. 22, 261–279 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Casas, E., Tröltzsch, F.: Second order optimality conditions and their role in PDE control. Jahresber. Dtsch. Math.-Ver. 117, 3–44 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Casas, E., Tröltzsch, F., Unger, A.: Second order sufficient optimality conditions for a nonlinear elliptic control problem. Z. Anal. Anwend. (ZAA) 15, 687–707 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dunn, J.: On second order sufficient optimality conditions for structured nonlinear programs in infinite-dimensional function spaces. In: Fiacco, A. (ed.) Mathematical Programming with Data Perturbations, pp 83–107. Marcel Dekker (1998)Google Scholar
  15. 15.
    Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Co., Amsterdam (1979)zbMATHGoogle Scholar
  16. 16.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations. Theory, Methods and Applications, vol. 112. American Math. Society, Providence (2010)zbMATHGoogle Scholar
  17. 17.
    Wachsmuth, G., Wachsmuth, D.: Convergence and regularisation results for optimal control problems with sparsity functional. ESAIM Control Optim. Calc. Var. 17, 858–886 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications III. Springer, New York (1985). Variational Methods and Optimization, Translated from the German by Leo F. BoronCrossRefzbMATHGoogle Scholar
  19. 19.
    Zeidler, E.: Applied Functional Analysis and its Applications. Main Principles and their Applications. Springer, New York (1995)zbMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de TelecomunicaciónUniversidad de CantabriaSantanderSpain
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations