Skip to main content
SpringerLink
Log in

Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We analyze a combined regularization–discretization approach for a class of linear-quadratic optimal control problems. By choosing the regularization parameter \(\alpha \) with respect to the mesh size \(h\) of the discretization we approximate the optimal bang–bang control. Under weaker assumptions on the structure of the switching function we generalize existing convergence results and prove error estimates of order \({\mathcal {O}}(h^{1/(k+1)})\) with respect to the controllability index \(k\).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Agrachev, A., Stefani, G., Zezza, P.L.: Strong optimality for a bang–bang trajectory. SIAM J. Control Optim. 41, 991–1014 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alt, W.: Discretization and mesh-independence of Newton’s method for generalized equations. In: Fiacco, A.V. (ed.) Mathematical Programming with Data Perturbations V”. Lecture Notes in Pure and Applied Mathematics, vol. 195, pp. 1–30. Marcel Dekker (1997)

  3. Alt, W., Bräutigam, N.: Finite-difference discretizations of quadratic control problems governed by ordinary elliptic differential equations. Comput. Optim. Appl. 43, 133–150 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Alt, W., Seydenschwanz, M.: Regularization and discretization of linear-quadratic control problems. Control Cybern. 40(4), 903–921 (2011)

    MathSciNet  Google Scholar 

  5. Alt, W., Seydenschwanz, M.: An implicit discretization scheme for linear-quadratic control problems with bang-bang solutions. Optim. Methods Softw. 29(3), 535–560 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  6. Alt, W., Baier, R., Gerdts, M., Lempio, F.: Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions. Numer. Algebr. Control Optim. 2, 547–570 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Alt, W., Baier, R., Gerdts, M., Lempio, F.: Approximations of linear control problems with bang-bang solutions. Optimization 62, 9–32 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Baier, R., Chahma, I.A., Lempio, F.: Stability and convergence of Euler’s method for state-constrained differential inclusions. SIAM J. Optim. 18, 1004–1026 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dontchev, A.L., Hager, W.W.: Lipschitzian stability in nonlinear control and optimization. SIAM J. Control Optim. 31, 569–603 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dontchev, A.L., Hager, W.W., Malanowski, K.: Error bounds for Euler approximation of a state and control constrained optimal control problem. Numer. Funct. Anal. Optim. 21, 653–682 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dontchev, A.L., Hager, W.W., Veliov, V.M.: Second-order Runge-Kutta approximations in control constrained optimal control. SIAM J. Numer. Anal. 38, 202–226 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dontchev, A.L., Hager, W.W.: The Euler approximation in state constrained optimal control. Math. Comput. 70, 173–203 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North Holland, Amsterdam-Oxford (1976)

    MATH  Google Scholar 

  14. Felgenhauer, U.: On stability of bang-bang type controls. SIAM J. Control Optim. 41, 1843–1867 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Felgenhauer, U.: Optimality properties of controls with bang-bang components in problems with semilinear state equation. Control Cybernet. 34, 763–785 (2005)

    MATH  MathSciNet  Google Scholar 

  16. Felgenhauer, U.: The shooting approach in analyzing bang-bang extremals with simultaneous control switches. Control Cybernet. 37, 307–327 (2008)

    MATH  MathSciNet  Google Scholar 

  17. Felgenhauer, U., Poggiolini, L., Stefani, G.: Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour. Control Cybernet. 38, 1305–1325 (2009)

    MATH  MathSciNet  Google Scholar 

  18. Hager, W.W.: Multiplier methods for nonlinear optimal control. SIAM J. Numer. Anal. 27(4), 1061–1080 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. Haunschmied, J., Pietrus, A., Veliov, V.M.: The Euler method for linear control systems revisited. In: Lirkov, I., Margenov, S., Wasniewski, J. (eds.) Large-Scale Scientific Computing—-LSSC 2013. Lecture Notes in Computer Science, vol. 8353, pp. 90–97 (2014)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Lorenz, D.A., Rösch, A.: Error estimates for joint Tikhonov- and Lavrentiev-regularization of constrained control problems. Appl. Anal. 89(11), 1679–1691 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. Malanowski, K., Büskens, C., Maurer, H.: Convergence of approximations to nonlinear optimal control problems. In: Fiacco, A.V. (ed.) Mathematical Programming with Data Perturbations V”. Lecture Notes in Pure and Applied Mathematics, vol. 195, pp. 253–284. Marcel Dekker (1997)

  24. Maurer, H., Osmolowski, N.P.: Second order sufficient conditions for time optimal bang-bang control. SIAM J. Control Optim. 42, 2239–2263 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Maurer, H., Büskens, C., Kim, J.-H.R., Kaya, C.Y.: Optimization methods for the verification of second-order sufficient conditions for bang-bang controls. Optimal Control Appl. Methods 26, 129–156 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. Neitzel, I., Tröltzsch, F.: On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints. Control Cybern. 37, 568–579 (2008)

    Google Scholar 

  29. Osmolowski, N.P., Lempio, F.: Transformation of quadratic forms to perfect squares for broken extremals. Set-Valued Anal. 10, 209–232 (2002)

    Article  MathSciNet  Google Scholar 

  30. Osmolowski, N.P., Maurer, H.: Equivalence of second order optimality conditions for bang-bang control problems. Part 1: main results. Control Cybernet. 34, 927–950 (2005)

    MathSciNet  Google Scholar 

  31. Osmolowski, N.P., Maurer, H.: Equivalence of second order optimality conditions for bang-bang control problems. Part 2: Proofs, variational derivatives and representations. Control Cybernet. 36, 5–45 (2007)

    MathSciNet  Google Scholar 

  32. Quincampoix, M., Veliov, V.M.: Metric regularity and stability of optimal control problems for linear systems. SIAM J. Control Optim. 51(5), 4118–4137 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  33. Seydenschwanz, M.: Diskretisierung und Regularisierung linear-quadratischer Steuerungsprobleme mit Bang-Bang Lösungen. Dissertation, Mathematisches Institut, Friedrich-Schiller-Universität Jena (2014)

  34. Tröltzsch, F., Yousept, I.: A regularization method for the numerical solution of elliptic boundary control problems with pointwise constraints. Comput. Optim. Appl. 42, 43–66 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  35. Veliov, V.M.: On the time-discretization of control systems. SIAM J. Control Optim. 35, 1470–1486 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  36. Veliov, V.M.: Error analysis of discrete approximations to bang-bang optimal control problems: the linear case. Control Cybernet. 34, 967–982 (2005)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author would like to thank the anonymous reviewers for their valuable comments.

Author information

Authors and Affiliations

  1. Siemens AG, Corporate Technology – Research and Technology Center, 81739, Munich, Germany

    Martin Seydenschwanz

Authors
  1. Martin Seydenschwanz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Seydenschwanz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Seydenschwanz, M. Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions. Comput Optim Appl 61, 731–760 (2015). https://doi.org/10.1007/s10589-015-9730-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-015-9730-z

Keywords

Mathematics Subject Classification

Search

Navigation