Skip to main content
SpringerLink
Log in

Error estimates for abstract linear–quadratic optimal control problems using proper orthogonal decomposition

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

Abstract

In this paper we investigate POD discretizations of abstract linear–quadratic optimal control problems with control constraints. We apply the discrete technique developed by Hinze (Comput. Optim. Appl. 30:45–61, 2005) and prove error estimates for the corresponding discrete controls, where we combine error estimates for the state and the adjoint system from Kunisch and Volkwein (Numer. Math. 90:117–148, 2001; SIAM J. Numer. Anal. 40:492–515, 2002). Finally, we present numerical examples that illustrate the theoretical results.

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

Similar content being viewed by others

References

  1. Afanasiev, K., Hinze, M.: Adaptive control of a wake flow using proper orthogonal decomposition. In: Lecture Notes in Pure and Applied Mathematics, vol. 216. pp. 317–332. Decker, New York (2001)

    Google Scholar 

  2. Arian, E., Fahl, M., Sachs, E.W.: Trust-region proper orthogonal decomposition for flow control. Technical report 2000-25, ICASE (2000)

  3. Atwell, J.A., Borggaard, J.T., King, B.B.: Reduced order controllers for Burgers’ equation with a nonlinear observer. Int. J. Appl. Math. Comput. Sci. 11, 1311–1330 (2001)

    MATH  MathSciNet  Google Scholar 

  4. Aubry, N., Holmes, P., Lumley, J.L., Stone, E.: The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  5. Banks, H.T., Joyner, M.L., Winchesky, B., Winfree, W.P.: Nondestructive evaluation using a reduced-order computational methodology. Inverse Problems 16, 1–17 (2000)

    Article  Google Scholar 

  6. Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5. Evolution Problems I. Springer, Berlin (1992)

    MATH  Google Scholar 

  7. Deckelnick, K., Hinze, M.: Error estimates in space and time for tracking-type control of the instationary Stokes system. In: International Series of Numerical Mathematics, vol. 143, pp. 87–103. Birkhäuser, Basel (2002)

    Google Scholar 

  8. Deckelnick, K., Hinze, M.: Semidiscretization and error estimates for distributed control of the instationary Navier–Stokes equations. Numer. Math. 97, 297–320 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  10. Fukunaga, K.: Introduction to Statistical Pattern Recognition. Academic, New York (1990)

    MATH  Google Scholar 

  11. Griesse, R., Volkwein, S.: A primal-dual active set strategy for optimal boundary control of a nonlinear reaction-diffusion system. SIAM J. Control Optim. 44, 467–494 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Henri, T., Yvon, M.: Convergence estimates of POD Galerkin methods for parabolic problems. Technical report No. 02-48, Institute of Mathematical Research of Rennes (2002)

  13. 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 

  14. Hinze, M., Kunisch, K.: Three control methods for time-dependent fluid flow. Flow Turbul. Combust. 65, 273–298 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hinze, M., Volkwein, S.: Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In: Benner, P., Mehrmann, V., Sorensen, D.C. (eds.) Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol. 45, pp. 261–306. Springer, Berlin (2005)

    Chapter  Google Scholar 

  16. Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  17. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)

    MATH  Google Scholar 

  18. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90, 117–148 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40, 492–515 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lall, S., Marsden, J.E., Glavaski, S.: A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int. J. Robust Nonlinear Control 12, 519–535 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    MATH  Google Scholar 

  22. Ly, H.V., Tran, H.T.: Modeling and control of physical processes using proper orthogonal decomposition. Math. Comput. Model. 33, 223–236 (2001)

    Article  MATH  Google Scholar 

  23. Noack, B., Afanasiev, K., Morzynski, M., Tadmor, G., Thiele, F.: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid. Mech. 497, 335–363 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Rathinam, M., Petzold, L.: Dynamic iteration using reduced order models: a method for simulation of large scale modular systems. SIAM J. Numer. Anal. 40, 1446–1474 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Ravindran, S.S.: Adaptive reduced order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J. Sci. Comput. 28, 1924–1942 (2002)

    Article  MathSciNet  Google Scholar 

  26. Rempfer, D., Fasel, H.F.: Dynamics of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid Mech. 275, 257–283 (1994)

    Article  Google Scholar 

  27. Rowley, C.W.: Model reduction for fluids using balanced proper orthogonal decomposition. Int. J. Bifurc. Chaos 15, 997–1013 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Shvartsman, S.Y., Kevrikidis, Y.: Nonlinear model reduction for control of distributed parameter systems: a computer-assisted study. AIChE J. 44, 1579–1595 (1998)

    Article  Google Scholar 

  29. Sirovich, L.: Turbulence and the dynamics of coherent structures parts I–III. Q. Appl. Math. XLV, 561–590 (1987)

    MathSciNet  Google Scholar 

  30. Tang, K.Y., Graham, W.R., Peraire, J.: Optimal control of vortex shedding using low-order models I: open loop model development. II. Model based control. Int. J. Numer. Methods Eng. 44, 945–990 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  31. Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40, 2323–2330 (2002)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department Mathematik, Schwerpunkt Approximation und Optimierung, Geomatikum, Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    M. Hinze

  2. Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010, Graz, Austria

    S. Volkwein

Authors
  1. M. Hinze
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Volkwein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Volkwein.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hinze, M., Volkwein, S. Error estimates for abstract linear–quadratic optimal control problems using proper orthogonal decomposition. Comput Optim Appl 39, 319–345 (2008). https://doi.org/10.1007/s10589-007-9058-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-007-9058-4

Keywords

Search

Navigation