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Error estimates for integral constraint regularization of state-constrained elliptic control problems

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Abstract

In this paper, we study new aspects of the integral contraint regularization of state-constrained elliptic control problems (Jadamba et al. in Syst Control Lett 61(6):707–713, 2012). Besides giving new results on the regularity and the convergence of the regularized controls and associated Lagrange multipliers, the main objective of this paper is to give abstract error estimates for the regularization error. We also consider a discretization of the regularized problems and derive numerical estimates which are uniform with respect to the regularization parameter and the discretization parameter. As an application of these results, we prove that this discretization is indeed a full discretization of the original problem defined in terms of a problem with finitely many integral constraints. Detailed numerical results justifying the theoretical findings as well as a comparison of our work with the existing literature is also given.

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References

  1. Alibert, J.J., Raymond, J.P.: Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim. 18(3–4), 235–250 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Borwein, J.M., Vanderwerff, J.D.: Constructible convex sets. Set-Valued Anal. 12(1–2), 61–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24(6), 1309–1318 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Casas, E.: Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints. ESAIM Control Optim. Calc. Var. 8, 345–374 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Casas, E., Clason, C., Kunisch, K.: Approximation of elliptic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 50(4), 1735–1752 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Casas, E., Mateos, M.: Numerical approximation of elliptic control problems with finitely many pointwise constraints. Comput. Optim. Appl. 51(3), 1319–1343 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Casas Rentería, E., Mateos Alberdi, M., Vexler, B.: New regularity results and improved error estimates for optimal control problems with state constraints. ESAIM Control Optim. Calc. Var. 20(3), 803–822 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cherednichenko, S., Krumbiegel, K., Rösch, A.: Error estimates for the lavrentiev regularization of elliptic optimal control problems. Inverse Probl. 24(5), 055,003 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ciarlet, P.G.: The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002). doi:10.1137/1.9780898719208. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]

  10. Davydov, O.: Approximation by piecewise constants on convex partitions. J. Approx. Theory 164(2), 346–352 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Deckelnick, K., Hinze, M.: Convergence of a finite element approximation to a state-constrained elliptic control problem. SIAM J. Numer. Anal. 45(5), 1937–1953 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Deckelnick, K., Hinze, M.: A finite element approximation to elliptic control problems in the presence of control and state constraints. Fachbereich Mathematik, Univ.,[Department Mathematik] (2007)

  13. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  14. Gong, W., Yan, N.: A mixed finite element scheme for optimal control problems with pointwise state constraints. J. Sci. Comput. 46(2), 182–203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx (2014)

  16. Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston (1985)

    MATH  Google Scholar 

  17. Guzmán, J., Leykekhman, D., Rossmann, J., Schatz, A.H.: Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods. Numer. Math. 112(2), 221–243 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hecht, F.: New development in freefem++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Hintermüller, M., Hinze, M.: Moreau-Yosida regularization in state constrained elliptic control problems: error estimates and parameter adjustment. SIAM J. Numer. Anal. 47(3), 1666–1683 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hintermüller, M., Kunisch, K.: Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17(1), 159–187 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30(1), 45–61 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hinze, M., Meyer, C.: Variational discretization of Lavrentiev-regularized state constrained elliptic optimal control problems. Comput. Optim. Appl. 46(3), 487–510 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hinze, M., Schiela, A.: Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment. Comput. Optim. Appl. 48(3), 581–600 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jadamba, B., Khan, A.A., Sama, M.: Regularization for state constrained optimal control problems by half spaces based decoupling. Syst. Control Lett. 61(6), 707–713 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jiménez, B., Novo, V., Sama, M.: An extension of the basic constraint qualification to nonconvex vector optimization problems. J. Global Optim. 56(4), 1755–1771 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Khan, A.A., Sama, M.: A new conical regularization for some optimization and optimal control problems: convergence analysis and finite element discretization. Numer. Funct. Anal. Optim. 34(8), 861–895 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Krumbiegel, K., Neitzel, I., Rösch, A.: Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints. Comput. Optim. Appl. 52(1), 181–207 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Krumbiegel, K., Rösch, A.: A virtual control concept for state constrained optimal control problems. Comput. Optim. Appl. 43(2), 213–233 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Leykekhman, D., Meidner, D., Vexler, B.: Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints. Comput. Optim. Appl. 55(3), 769–802 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, W., Gong, W., Yan, N.: A new finite element approximation of a state-constrained optimal control problem. J. Comput. Math 27(1), 97–114 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Liu, W., Yang, D., Yuan, L., Ma, C.: Finite element approximations of an optimal control problem with integral state constraint. SIAM J. Numer. Anal. 48(3), 1163–1185 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Maz_i_a, V., rgen Rossmann, J.: Elliptic Equations in Polyhedral Domains, vol. 162. American Mathematical Society, Providence (2010)

    Google Scholar 

  33. Megginson, R.E.: An Introduction to Banach Space Theory. Graduate Texts in Mathematics, vol. 183. Springer, New York (1998)

    MATH  Google Scholar 

  34. Meyer, C.: Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybern. 37(1), 51–83 (2008)

    MathSciNet  MATH  Google Scholar 

  35. Meyer, C., Prüfert, U., Tröltzsch, F.: On two numerical methods for state-constrained elliptic control problems. Optim. Methods Softw. 22(6), 871–899 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33(2–3), 209–228 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rösch, A., Steinig, S.: A priori error estimates for a state-constrained elliptic optimal control problem. ESAIM Math. Model. Numer. Anal. 46(5), 1107–1120 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Schiela, A.: Barrier methods for optimal control problems with state constraints. SIAM J. Optim. 20(2), 1002–1031 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Baasansuren Jadamba’s work is supported by RIT’s COS FEAD Grant for 2016–2017. Akhtar Khan is support by a Grant from the Simons Foundation (#210443) and RIT’s COS FEAD Grant for 2015–2016. Miguel Sama’s work is partially supported by Ministerio de Economía y Competitividad (Spain), Project MTM2015-68103-P.

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Authors and Affiliations

  1. Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY, 14623, USA

    B. Jadamba & A. Khan

  2. Departamento de Matemática Aplicada, E.T.S.I.I. Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, 28040, Madrid, Spain

    M. Sama

Authors
  1. B. Jadamba
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  2. A. Khan
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  3. M. Sama
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Correspondence to M. Sama.

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Dedicated to Professor Zuhair Nashed on his 75th birthday.

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Jadamba, B., Khan, A. & Sama, M. Error estimates for integral constraint regularization of state-constrained elliptic control problems. Comput Optim Appl 67, 39–71 (2017). https://doi.org/10.1007/s10589-016-9885-2

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