Skip to main content
Springer Nature Link
Log in

Discretization of Optimal Control Problems

  • Chapter
  • First Online:
Constrained Optimization and Optimal Control for Partial Differential Equations

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 160))

  • 3054 Accesses

Abstract

Solutions to optimization problems with pde constraints inherit special properties; the associated state solves the pde which in the optimization problem takes the role of a equality constraint, and this state together with the associated control solves an optimization problem, i.e., together with multipliers satisfies first- and second-order necessary optimality conditions. In this note we review the state of the art in designing discrete concepts for optimization problems with pde constraints with emphasis on structure conservation of solutions on the discrete level, and on error analysis for the discrete variables involved. As model problem for the state we consider an elliptic pde which is well understood from the analytical point of view. This allows to focus on structural aspects in discretization. We discuss the approaches First discretize, then optimize and First optimize, then discretize, and consider in detail two variants of the First discretize, then optimize approach, namely variational discretization, a discrete concept which avoids explicit discretization of the controls, and piecewise constant control approximations. We consider general constraints on the control, and also consider pointwise bounds on the state. We outline the basic ideas for providing optimal error analysis and accomplish our analytical findings with numerical examples which confirm our analytical results. Furthermore we present a brief review on recent literature which appeared in the field of discrete techniques for optimization problems with pde constraints.

Mathematics Subject Classification (2000). 49J20, 49K20, 35B37.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Comm. Pure Appl. Math., 12, 623–727 (1959).

    MathSciNet  MATH  Google Scholar 

  2. Apel, T., Rösch, A., Sirch, D.: 𝐿 ∞-Error Estimates on Graded Meshes with Application to Optimal Control. SIAM Journal Control and Optimization, 48(3):1771–1796, 2009.

    MathSciNet  MATH  Google Scholar 

  3. Apel, T., Rösch, A., Winkler, G: Optimal control in non-convex domains: a priori discretization error estimates. Calcolo, 44(3), 137–158 (2007).

    MathSciNet  MATH  Google Scholar 

  4. Arada, N., Casas, E., Tröltzsch, F.: Error estimates for the numerical approximation of a semilinear elliptic control problem. Computational Optimization and Applications 23, 201–229 (2002).

    MathSciNet  MATH  Google Scholar 

  5. Bergounioux, M., Ito, K., and Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control and Optimization, 37:1176–1194, 1999.

    MathSciNet  MATH  Google Scholar 

  6. Bergounioux, M. and Kunisch, K.: Primal-dual strategy for state-constrained optimal control problems. Computational Optimization and Applications, 22:193–224, 2002.

    MathSciNet  MATH  Google Scholar 

  7. Casas, E.: 𝐿 2 estimates for the finite element method for the Dirichlet problem with singular data. Numer. Math. 47, 627–632 (1985).

    MathSciNet  MATH  Google Scholar 

  8. Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Cont. Optim. 4, 1309–1322 (1986).

    MathSciNet  MATH  Google Scholar 

  9. Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Cont. Optim. 31, 993–1006 (1993).

    MathSciNet  MATH  Google Scholar 

  10. 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).

    MathSciNet  MATH  Google Scholar 

  11. Casas, E., Mateos, M.. Uniform convergence of the FEM. Applications to state constrained control problems. Comp. Appl. Math. 21, (2002).

    Google Scholar 

  12. Casas, E., Mateos, M.: Error Estimates for the Numerical Approximation of Neumann Control Problems. Comp. Appl. Math. 39(3):265–295, 2008.

    MathSciNet  MATH  Google Scholar 

  13. Casas, E., Mateos, M., Tröltzsch, F.: Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl. 31, 193–219 (2005).

    MathSciNet  MATH  Google Scholar 

  14. Casas, E., Raymond, J.P.: Error estimates for the numerical approximation of Dirichlet Boundary control for semilinear elliptic equations. SIAM J. Cont. Optim. 45, 1586–1611 (2006).

    MathSciNet  MATH  Google Scholar 

  15. Casas, E., Tröltzsch, F.: Error estimates for the finite element approximation of a semilinear elliptic control problems. Contr. Cybern. 31, 695–712 (2005).

    MathSciNet  MATH  Google Scholar 

  16. Casas, E., Mateos, M., and Tröltzsch, F.: Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl., 31(2):193–219, 2005.

    MathSciNet  MATH  Google Scholar 

  17. Cherednichenko, S., Krumbiegel, K., and Rösch, A.: Error estimates for the Lavrentiev regularization of elliptic optimal control problems. Inverse Problems, 24(5):055003, 2008.

    MathSciNet  MATH  Google Scholar 

  18. Cherednichenko, S. and Rösch, A.: Error estimates for the discretization of elliptic control problems with pointwise control and state constraints. Computational Optimization and Applications, 44:27–55, 2009.

    MathSciNet  MATH  Google Scholar 

  19. Deckelnick, K., Hinze, M.: Error estimates in space and time for tracking-type control of the instationary Stokes system. ISNM 143, 87–103 (2002)

    MATH  Google Scholar 

  20. Deckelnick, K., Hinze, M.: Semidiscretization And Error Estimates For Distributed Control Of The Instationary Navier-Stokes Equations. Numer. Math. 97, 297–320 (2004).

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Deckelnick, K., Hinze, M.: A finite element approximation to elliptic control problems in the presence of control and state constraints. Hamburger Beiträge zur Angewandten Mathematik HBAM2007-01 (2007).

    Google Scholar 

  23. Deckelnick, K., Hinze, M.: Numerical analysis of a control and state constrained elliptic control problem with piecewise constant control approximations. Proceedings of the ENUMATH (2007).

    Google Scholar 

  24. Deckelnick, K., Hinze, M.: Variational Discretization of Parabolic Control Problems in the Presence of Pointwise State Constraints. Journal of Computational Mathematics 29:1–16, 2011.

    MathSciNet  MATH  Google Scholar 

  25. Deckelnick, K., Hinze, M.: A note on the approximation of elliptic control problems with bang-bang controls. Comput. Optim. Appl. DOI: 10.1007/s10589-010-9365-z (2010).

    Google Scholar 

  26. Deckelnick, K., Günther, A., Hinze, M.: Finite element approximations of elliptic control problems with constraints on the gradient. Numer. Math. 111:335-350 (2009).

    MathSciNet  MATH  Google Scholar 

  27. Deckelnick, K., Günther, A, Hinze, M.: Finite Element Approximation of Dirichlet Boundary Control for Elliptic PDEs on Two- and Three-Dimensional Curved Domains. SIAM J. Cont. Optim. 48:2798–2819 (2009).

    MathSciNet  MATH  Google Scholar 

  28. Deckelnick, K., Hinze, M., Matthes, U., Schiela, A.: Approximation schemes for constrained optimal control with nonlinear pdes. In preparation (2010).

    Google Scholar 

  29. de los Reyes, J.C., Meyer, C., and Vexler, B.: Finite element error analysis for state-constrained optimal control of the Stokes equations. Control and Cybernetics, 37(2):251–284, 2008.

    MathSciNet  MATH  Google Scholar 

  30. Falk, R.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44, 28–47 (1997).

    MathSciNet  MATH  Google Scholar 

  31. Gastaldi, L., Nochetto, R.H.: On 𝐿 ∞-accuracy of mixed finite element methods for second-order elliptic problems. Mat. Apl. Comput. 7, 13–39 (1988).

    MathSciNet  MATH  Google Scholar 

  32. Geveci, T.: On the approximation of the solution of an optimal control problem governed by an elliptic equation. Math. Model. Numer. Anal. 13, 313–328 (1979).

    MathSciNet  MATH  Google Scholar 

  33. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order (2nd ed.). Springer (1983).

    Google Scholar 

  34. Giles, M., Ulbrich, S.: Convergence of linearised and adjoint approximations for discontinuous solutions of conservation laws. Part 1: Siam J. Numer. Anal. 48:882–904, Part 2: Siam J. Numer. Anal. 48:905–921 (2010).

    MATH  Google Scholar 

  35. Günther, A., Hinze, M.: Elliptic Control Problems with Gradient Constraints – Variational Discrete Versus Piecewise Constant Controls, Comput. Optim. Appl. 49:549–566, 2011.

    MathSciNet  MATH  Google Scholar 

  36. Gunzburger, M.D, Manservisi, S.: Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control. Siam J. Numer. Anal. 37, 1481–1512, 2000.

    MathSciNet  MATH  Google Scholar 

  37. Gunzburger, M.D., Manservisi, S.: The velocity tracking problem for Navier-Stokes flows with boundary controls. Siam J. Control and Optimization 39, 594–634 (2000).

    MathSciNet  MATH  Google Scholar 

  38. Hintermüller, M. and Hinze, M.: Moreau-yosida regularization in state constrained elliptic control problems: error estimates and parameter adjustment. SIAM J. Numerical Analysis, 47:1666–1683, 2009.

    MathSciNet  MATH  Google Scholar 

  39. Hintermüller, M., Ito, K., and Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optimization, 13:865–888, 2003.

    MathSciNet  MATH  Google Scholar 

  40. Hintermüller, M. and Kunisch, K.: Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optimization, 17:159–187, 2006.

    MathSciNet  MATH  Google Scholar 

  41. Hintermüller, M., Kunisch, K.: Feasible and non-interior path following in constrained minimization with low multiplier regularity. Report, Universität Graz (2005).

    Google Scholar 

  42. Hintermüller, M., Ulbrich, M.: A mesh independence result for semismooth Newton methods. Mathematical Programming 101:151–184, 2004

    MathSciNet  MATH  Google Scholar 

  43. Hinze, M.: Optimal and instantaneous control of the instationary Navier-Stokes equations. Habilitationsschrift, Fachbereich Mathematik, Technische Universität Berlin (2000).

    Google Scholar 

  44. Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comp. Optim. Appl. 30:45–63, 2005.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  46. Hinze, M., Pinnau, R., Ulbrich, M., and Ulbrich, S.: Optimization with PDE constraints MMTA 23, Springer 2009.

    Google Scholar 

  47. Hinze, M. and Schiela, A.: Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment. Computational Optimization and Applications, DOI: 10.1007/s10589-009-9278-x, 2010.

    Google Scholar 

  48. Krumbiegel, K., Meyer, C., and Rösch, A.: A priori error analysis for linear quadratic elliptic Neumann boundary control problems with control and state constraints. SIAM Journal Control and Optimization 48(8):5108–5142, 2010.

    MathSciNet  MATH  Google Scholar 

  49. Krumbiegel, K., Neitzel, I., and Rösch, A.: Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints. Computational Optimization and Applications, DOI 10.1007/s10589-010-9357-z.

    Google Scholar 

  50. Krumbiegel, K. and Rösch, A.: On the regularization error of state constrained Neumann control problems. Control and Cybernetics, 37:369–392, 2008.

    MathSciNet  MATH  Google Scholar 

  51. Krumbiegel, K. and Rösch, A.: A virtual control concept for state constrained optimal control problems. Computational Optimization and Applications, 43:213–233, 2009.

    MathSciNet  MATH  Google Scholar 

  52. Kunisch, K. and Rösch, A.: Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J. Optimization, 13(2):321–334, 2002.

    MathSciNet  MATH  Google Scholar 

  53. Malanowski, K.: Convergence of approximations vs. regularity of solutions for convex, control constrained optimal-control problems. Appl. Math. Optim. 8 69–95 (1981).

    MathSciNet  MATH  Google Scholar 

  54. Mateos, M. and Rösch, A. On saturation effects in the Neumann boundary control of elliptic optimal control problems. Computational Optimization and Applications, 49:359–378, 2011.

    MathSciNet  MATH  Google Scholar 

  55. May, S., Rannacher, R., Vexler, B.: A priori error analysis for the finite element approximation of elliptic Dirichlet boundary control problems. Proceedings of ENUMATH 2007, Graz (2008).

    Google Scholar 

  56. Meidner, D., Vexler, B.: A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems. Part I: Problems without Control Constraints. SIAM Journal on Control and Optimization 47, 1150–1177 (2008).

    MathSciNet  MATH  Google Scholar 

  57. Meidner, D., Vexler, B.: A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems. Part II: Problems with Control Constraints. SIAM Journal on Control and Optimization 47, 1301–1329 (2008).

    MathSciNet  MATH  Google Scholar 

  58. Meyer, C.: Error estimates for the finite element approximation of an elliptic control problem with pointwise constraints on the state and the control, WIAS Preprint 1159 (2006).

    Google Scholar 

  59. Meyer, C., Prüfert, U., and Tröltzsch, F.: On two numerical methods for stateconstrained elliptic control problems. Optimization Methods and Software, 22:871–899, 2007.

    MathSciNet  MATH  Google Scholar 

  60. Meyer, C., Rösch, A.: Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43, 970–985 (2004).

    MathSciNet  MATH  Google Scholar 

  61. Meyer, C., Rösch, A.: 𝐿 ∞-estimates for approximated optimal control problems. SIAM J. Control Optim. 44, 1636–1649 (2005).

    MathSciNet  MATH  Google Scholar 

  62. Meyer, C., Rösch, A., and Tröltzsch, F.: Optimal control of PDEs with regularized pointwise state constraints. Computational Optimization and Applications, 33(2-3):209–228, 2006.

    MathSciNet  MATH  Google Scholar 

  63. Neitzel, I. and Tröltzsch, F.: On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints. Control and Cybernetics, 37:1013–1043, 2008.

    MathSciNet  MATH  Google Scholar 

  64. Nochetto, R.H., Verdi, C.: Convergence past singularities for a fully discrete approximation of curvature-driven interfaces, SIAM J. Numer. Anal. 34, 490–512 (1997).

    MathSciNet  MATH  Google Scholar 

  65. Ortner, C., Wollner, W.: A priori error estimates for optimal control problems with pointwise constraints on the gradient of the state. DFG Schwerpunktprogramm 1253, preprint No. SPP1253-071, (2009).

    Google Scholar 

  66. Rösch, A.: Error estimates for parabolic optimal control problems with control constraints, Zeitschrift für Analysis und ihre Anwendungen ZAA 23, 353–376 (2004).

    MathSciNet  MATH  Google Scholar 

  67. Rösch, A. and Wachsmuth, D.: How to checknum erically the sufficient optimality conditions for infinite-dimensional optimization problems, in Optimal control of coupled systems of partial differential equations, eds. Kunisch, K., Leugering, G., Sprekels, J., and Tröltzsch, F., pages 297–318, Birkhäuser 2009.

    Google Scholar 

  68. Rösch, A. and Wachsmuth, D.: Numerical verification of optimality conditions, SIAM Journal Control and Optimization, 47, 5 (2008), 2557–2581.

    MathSciNet  MATH  Google Scholar 

  69. Schatz, A.H.: Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I: Global estimates. Math. Comput. 67(223), 877–899 (1998).

    MathSciNet  MATH  Google Scholar 

  70. Schiela, A.: Barrier methods for optimal control problems with state constraints. SIAM J. Optimization, 20:1002–1031, 2009.

    MathSciNet  MATH  Google Scholar 

  71. Tröltzsch, F.: Optimale Steuerung mit partiellen Differentialgleichungen, (2005).

    Google Scholar 

  72. Tröltzsch, F. and Yousept, I.: A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints. Computational Optimization and Applications, 42:43–66, 2009.

    MathSciNet  MATH  Google Scholar 

  73. Tröltzsch, F., and Yousept, I.: Source representation strategy for optimal boundary control problems with state constraints. Journal of Analysis and its Applications, 28:189–203, 2009.

    MathSciNet  MATH  Google Scholar 

  74. Vexler, B.: Finite element approximation of elliptic Dirichlet optimal control problems. Numer. Funct. Anal. Optim. 28, 957–975 (2007).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Hamburg, Bundesstrasse 55, D-20146, Hamburg, Germany

    Michael Hinze

  2. Fakultät für Mathematik, Universität Duisburg-Essen, Forsthausweg 2, D-47057, Duisburg, Germany

    Arnd Rösch

Authors
  1. Michael Hinze
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arnd Rösch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Hinze .

Editor information

Editors and Affiliations

  1. , Department Mathematik, Universität Erlangen-Nürnberg, Martensstr. 3, Erlangen, 91058, Germany

    Günter Leugering

  2. , Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Geschossbau 2, Dortmund, 44221, Germany

    Sebastian Engell

  3. , Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Berlin, 12489, Germany

    Andreas Griewank

  4. , Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Michael Hinze

  5. , Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 293, Heidelberg, 69120, Germany

    Rolf Rannacher

  6. , FB 4 - Department Mathematik, Universität Trier, Trier, 54286, Germany

    Volker Schulz

  7. Fakultät für Mathematik, M1, Lehrstuhl für Mathematische Optimierung, Technische Universität München, Boltzmannstr. 3, Garching bei München, 85748, Germany

    Michael Ulbrich

  8. , Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostrasse 15, Darmstadt, 64293, Germany

    Stefan Ulbrich

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel AG

About this chapter

Cite this chapter

Hinze, M., Rösch, A. (2012). Discretization of Optimal Control Problems. In: Leugering, G., et al. Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol 160. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0133-1_21

Download citation

Publish with us

Policies and ethics