Advertisement

The Influence of the Tikhonov Term in Optimal Control of Partial Differential Equations

  • Eduardo CasasEmail author
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 17)

Abstract

In this paper, we analyze the importance of the presence of the Tikhonov term in an optimal control problem. The influence of this term in several aspects of control theory is analyzed: existence and regularity of a solution, convergence of the numerical approximations, and second order optimality conditions.

Keywords

Optimal control Bang-bang controls State constraints Second order optimality conditions 

Notes

Acknowledgements

The author was partially supported by Ministerio Español de Economía, Industria y Competitividad under the research projects MTM2014-57531-P and MTM2017- 83185-P.

References

  1. 1.
    Ali, A.A., Deckelnick, K., Hinze, M.: Global minima for semilinear optimal control problems. Comput. Optim. Appl. 65, 261–288 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bonnans, J.F.: Second-order analysis for control constrained optimal control problems of semilinear elliptic systems. Appl. Math. Optim. 38, 303–325 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bonnans, F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Berlin (2000)CrossRefGoogle Scholar
  4. 4.
    Casas, E.: Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35(4), 1297–1327 (1997)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Casas, E.: Second order analysis for bang-bang control problems of PDEs. SIAM J. Control Optim. 50(4), 2355–2372 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Casas, E., Chrysafinos, K.: Error estimates for the approximation of the velocity tracking problem with bang-bang controls. ESAIM Control Optim. Calc. Var. 23, 1267–1291 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Casas, E., Kunisch, K.: Parabolic control problems in space-time measure spaces. ESAIM Control Optim. Calc. Var. 22(2), 355–370 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Casas, E., Tröltzsch, F.: Second order analysis for optimal control problems: improving results expected from abstract theory. SIAM J. Optim. 22(1), 261–279 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Casas, E., Tröltzsch, F.: Second order optimality conditions and their role in PDE control. Jahresber. Dtsch. Math. Ver. 117(1), 3–44 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Casas, E., Tröltzsch, F.: Second order optimality conditions for weak and strong local solutions of parabolic optimal control problems. Vietnam J. Math. 44(1), 181–202 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Casas, E., Raymond, J.P., Zidani, H.: Pontryagin’s principle for local solutions of control problems with mixed control-state constraints. SIAM J. Control Optim. 39(4), 1182–1203 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Casas, E., Mateos, M., Rösch, A.: Finite element approximation of sparse parabolic control problems. Math. Control Relat. Fields 7(3), 393–417 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Casas, E., Wachsmuth, D., Wachsmuth, G.: Sufficient second-order conditions for bang-bang control problems. SIAM J. Control Optim. 55(5), 3066–3090 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ioffe, A.D.: Necessary and sufficient conditions for a local minimum. III. Second order conditions and augmented duality. SIAM J. Control Optim. 17(2), 266–288 (1979). MR 525027 (82j:49005c)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence (1988)zbMATHGoogle Scholar
  16. 16.
    Neitzel, I., Vexler, B.: A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numer. Math. 120, 345–386 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Raymond, J.P., Zidani, H.: Pontryagin’s principle for state-constrained control problems governed by parabolic equations with unbounded controls. SIAM J. Control Optim. 36(6), 1853–1879 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill, London (1970)zbMATHGoogle Scholar
  19. 19.
    Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs, vol. 49. American Mathematical Society, Providence (1997). MR 1422252 (98c:47076)Google Scholar
  20. 20.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Philadelphia (2010)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada y Ciencias de la ComputaciónUniversidad de CantabriaSantanderSpain

Personalised recommendations