Coupling MPC and HJB for the Computation of POD-Based Feedback Laws

  • Giulia FabriniEmail author
  • Maurizio Falcone
  • Stefan Volkwein
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


In this paper we use a reference trajectory computed by a model predictive method to shrink the computational domain where we set the Hamilton-Jacobi Bellman (HJB) equation. Via a reduced-order approach based on proper orthogonal decomposition(POD), this procedure allows for an efficient computation of feedback laws for systems driven by parabolic equations. Some numerical examples illustrate the successful realization of the proposed strategy.



G. Fabrini gratefully acknowledges support by the German Science Fund DFG grant Reduced-Order Methods for Nonlinear Model Predictive Control.


  1. 1.
    A. Alla, G. Fabrini, M. Falcone, Coupling MPC and DP methods for an efficient solution of optimal control problems, in Conference Proceedings of IFIP (2015)Google Scholar
  2. 2.
    M. Bardi, I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (Birkhauser, Basel, 1997)CrossRefGoogle Scholar
  3. 3.
    R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5: Evolution Problems I (Springer, Berlin, 2000)Google Scholar
  4. 4.
    M. Falcone, R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations (SIAM, Philadelphia, 2013)CrossRefGoogle Scholar
  5. 5.
    L. Grüne, J. Pannek, Nonlinear Model Predictive Control (Springer, London, 2011)CrossRefGoogle Scholar
  6. 6.
    M. Gubisch, S. Volkwein, Proper orthogonal decomposition for linear-quadratic optimal control, in Model Reduction and Approximation: Theory and Algorithms, ed. by P. Benner, A. Cohen, M. Ohlberger, K. Willcox (SIAM, Philadelphia, 2017), pp. 5–66Google Scholar
  7. 7.
    P. Holmes, J.L. Lumley, G. Berkooz, C.W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics, 2nd edn. (Cambridge University Press, Cambridge, 2012)Google Scholar
  8. 8.
    J.B. Rawlings, D.Q. Mayne, Model Predictive Control: Theory and Design (Nob Hill Publishing, Madison, 2009)Google Scholar
  9. 9.
    F. Tröltzsch, S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44, 83–115 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Giulia Fabrini
    • 1
    Email author
  • Maurizio Falcone
    • 2
  • Stefan Volkwein
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of KonstanzKonstanzGermany
  2. 2.Dipartimento di MatematicaLa Sapienza Università di RomaRomaItaly

Personalised recommendations