Advances in Computational Mathematics

, Volume 41, Issue 5, pp 1073–1102 | Cite as

Asymptotic stability of POD based model predictive control for a semilinear parabolic PDE

  • Alessandro Alla
  • Stefan VolkweinEmail author


In this article a stabilizing feedback control is computed for a semilinear parabolic partial differential equation utilizing a nonlinear model predictive (NMPC) method. In each level of the NMPC algorithm the finite time horizon open loop problem is solved by a reduced-order strategy based on proper orthogonal decomposition (POD). A stability analysis is derived for the combined POD-NMPC algorithm so that the lengths of the finite time horizons are chosen in order to ensure the asymptotic stability of the computed feedback controls. The proposed method is successfully tested by numerical examples.


Dynamic programming Nonlinear model predictive control Asymptotic stability Suboptimal control Proper orthogonal decomposition 

Mathematics Subject Classifications (2010)

35K58 49L20 65K10 90C30. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alla, A., Falcone, M.: An adaptive POD approximation method for the control of advection-diffusion equations. In: Kunisch, K., Bredies, K., Clason, C., von Winckel, G. (eds.) Control and Optimization with PDE Constraints. International Series of Numerical Mathematics, vol. 164, pp. 1–17. Birkhäuser, Basel (2013)CrossRefGoogle Scholar
  2. 2.
    Alla, A., Falcone, M.: A time-adaptive pod method for optimal control problems. In: Proceedings of the 1st IFAC Workshop on Control of Systems Modeled by Partial Differential Equations, pp 245–250 (2013)Google Scholar
  3. 3.
    Allgöwer, F., Findeisen, R., Nagy, Z.K.: Nonlinear model predictive control: from theory to application. J. Chin. Inst. Chem. Engrs. 35, 299–315 (2004)Google Scholar
  4. 4.
    Allgöwer, F., Chen, H.: A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34, 1205–1217 (1998)zbMATHCrossRefGoogle Scholar
  5. 5.
    Altmüller, N., Grüne, L.: A comparative stability analysis of Neumann and Dirichlet boundary MPC for the heat equation. In: Proceedings of the 1st IFAC Workshop on Control of Systems Modeled by Partial Differential Equations, pp. 1161–1166 (2013)Google Scholar
  6. 6.
    Altmüller, N., Grüne, L., Worthmann, K.: Receding horizon optimal control for the wave equation. In: Proceedings of the 49th IEEE Conference on Decision and Control, pp. 3427–3432, Atlanta (2010)Google Scholar
  7. 7.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Basel (1997)zbMATHCrossRefGoogle Scholar
  8. 8.
    Cazenave, T., Haraux, A.: An Introduction to Semilinear Evolution Equation. Oxford Science Publications (1998)Google Scholar
  9. 9.
    Chaturantabut, S., Sorensen, D.C.: Discrete Empirical Interpolation for NonLinear Model Reduction. SIAM J. Sci. Comput. 32, 2737–2764 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5: Evolution Problems I. Springer, Berlin (2000)CrossRefGoogle Scholar
  11. 11.
    Evans, L.C.: Partial differential equations. American Math. Society, Providence, Rhode Island (2008)Google Scholar
  12. 12.
    Findeisen, R., Allgöwer, F.: An introduction to nonlinear model predictive control. In: Scherer, C.W., Schumacher, J.M. (eds.) Summerschool on The Impact of Optimization in Control. Dutch Institute of Systems and Control, DISC (2001)Google Scholar
  13. 13.
    Findeisen, R., Allgöwer, F.: The quasi-infinte horizon approach to nonlinear model predictive control. In: Zinober, A., Owens, D. (eds.) Nonlinear and Adaptive Control, Lecture Notes in Control and Information Sciences, pp. 89–105. Springer-Verlag, Berlin (2002)Google Scholar
  14. 14.
    Ghiglieri, J., Ulbrich, S.: Optimal Flow Control Based on POD and MPC and an Application to the Cancellation of Tollmien-Schlichting Waves. Submitted (2012)Google Scholar
  15. 15.
    Grüne, L., Pannek, J.: Nonlinear Model Predictive Control. Springer, London (2011)zbMATHCrossRefGoogle Scholar
  16. 16.
    Grüne, L., Panneck, J., Seehafer, M., Worthmann, K.: Analysis of unconstrained nonlinear MPC schemes with time varying control horizon. SIAM J. Control. Optim. 48, 4938–4962 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Gubisch, M., Volkwein, S.: Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control. Submitted (2013).
  18. 18.
    Holmes, P., Lumley, J.L., Berkooz, G., Romley, C.W.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics, 2nd edn. Cambridge University Press (2012)Google Scholar
  19. 19.
    Ito, K., Kunisch, K.: Receding horizon control for infinite dimensional systems. ESAIM, Control, Optim. Calc. Var. 8, 741–760 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90, 117–148 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40, 492–515 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Kunisch, K., Volkwein, S., Xie, L.: HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst. 3, 701–722 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Pannocchia, G., Rawlings, J.B., Wright, S.J.: Conditions under which suboptimal nonlinear MPC is inherently robust. In: 18th IFAC World Congress, Milan (2011)Google Scholar
  24. 24.
    Rawlings, J.B., Mayne, D.Q.: Model Predictive Control: Theory and Design. Nob Hill Publishing, LLC (2009)Google Scholar
  25. 25.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York (1980)zbMATHGoogle Scholar
  26. 26.
    Sachs, E.W., Schu, M.: A-priori error estimates for reduced order models in finance. ESAIM: Math. Model. Numer. Anal. 47, 449–469 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Singler, J.R.: New POD expressions, error bounds, and asymptotic results for reduced order models of parabolic PDEs. Submitted (2013)Google Scholar
  28. 28.
    Sirovich, L.: Turbulence and the dynamics of coherent structures. Parts I–II. Q. Appl. Math. XVL, 561–590 (1987)MathSciNetGoogle Scholar
  29. 29.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and applications. Graduate Studies in Mathematics, Vol. 112, American Mathematical Society (2010)Google Scholar
  30. 30.
    Volkwein, S.: Lagrange-SQP techniques for the control constrained optimal boundary control for the Burgers equation. Comput. Optim. Appl. 26(253), 284 (2003)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HamburgHamburgGermany
  2. 2.Department of Mathematics and StatisticsUniversity of KonstanzKonstanzGermany

Personalised recommendations