Computational Optimization and Applications

, Volume 36, Issue 1, pp 83–108 | Cite as

Automatic differentiation of explicit Runge-Kutta methods for optimal control

  • Andrea WaltherEmail author


This paper considers the numerical solution of optimal control problems based on ODEs. We assume that an explicit Runge-Kutta method is applied to integrate the state equation in the context of a recursive discretization approach. To compute the gradient of the cost function, one may employ Automatic Differentiation (AD). This paper presents the integration schemes that are automatically generated when differentiating the discretization of the state equation using AD. We show that they can be seen as discretization methods for the sensitivity and adjoint differential equation of the underlying control problem. Furthermore, we prove that the convergence rate of the scheme automatically derived for the sensitivity equation coincides with the convergence rate of the integration scheme for the state equation. Under mild additional assumptions on the coefficients of the integration scheme for the state equation, we show a similar result for the scheme automatically derived for the adjoint equation. Numerical results illustrate the presented theoretical results.


Optimal control Automatic differentiation Sensitivity equation Adjoint equation 


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  1. 1.
    J.T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, SIAM, Philadelphia, 2001.zbMATHGoogle Scholar
  2. 2.
    H.G. Bock and K.-J. Plitt, “A multiple shooting algorithm for direct solution of optimal control problems,” in Proceedings of the 9th IFAC World Congress, Budapest, Pergamon Press, 1984.Google Scholar
  3. 3.
    A.E. Bryson and Y. Ho, Applied Optimal Control—Optimization, Estimation, and Control Hemisphere Publishing Corporation, New York, 1975.Google Scholar
  4. 4.
    R. Bulirsch, E. Nerz, H.J. Pesch, and O. von Stryk, “Combining direct and indirect methods in nonlinear optimal control: range maximization of a hang glider,” in R. Bulirsch, A. Miele, J. Stoer, K.H. Well, (eds.), in Optimal Control, Calculus of Variations, Optimal Control Theory and Numerical Methods, Birkhäuser, Basel, 1993, pp. 273–288.Google Scholar
  5. 5.
    C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustandsbeschränkungen Dissertation, Westfälische Wilhelms-Universität Münster, 1998.Google Scholar
  6. 6.
    C. Büskens and H. Maurer “SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control. J Comput Appl Math, vol. 120, pp. 85–108, 2000.CrossRefMathSciNetGoogle Scholar
  7. 7.
    J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations, John Wiley, New York, 1987.zbMATHGoogle Scholar
  8. 8.
    J.-B. Caillau and J. Noailles, “Continuous optimal control sensitivity analysis with ad,” in [10], pp. 109–117.Google Scholar
  9. 9.
    D. Casanova, R.S. Sharp, M. Final, B. Christianson, and P. Symonds, “Application of automatic differentiation to race car performance optimisation,” in [10], pp. 117–124.Google Scholar
  10. 10.
    G.F. Corliss, C. Faure, A. Griewank, L. Hascoët, and U. Naumann (eds.), Automatic Differentiation: from Simulation to Optimization. Springer Verlag, New York, 2001.Google Scholar
  11. 11.
    A.L. Dontchev, W. Hager, and V. Veliov, “Second-order runge-kutta approximations in control constrained optimal control,” SIAM J. Numer. Anal. vol. 38, pp. 202–226, 2000.CrossRefMathSciNetGoogle Scholar
  12. 12.
    M. Hinze and T. Slawig, “Adjoint gradients compared to gradients from algorithmic differentiation in instantaneous control of the navier–stokes equations. Optim. Methods Softw. vol. 18, no. 3, 299–315, 2003.CrossRefMathSciNetGoogle Scholar
  13. 13.
    Yu.G. Evtushenko, “Automatic differentiation viewed from optimal control theory,” in [17], pp. 25–30.Google Scholar
  14. 14.
    Yu.G. Evtushenko, “Computation of exact gradients in distributed dynamic systems,” Optim. Methods Softw. vol. 9, nos. 1–3, pp. 45–75, 1998.MathSciNetGoogle Scholar
  15. 15.
    R. Griesse and A. Walther, “Evaluating gradients in optimal control—Continuous adjoints versus automatic differentiation,” Journal of Optimization Theory and Applications (JOTA), vol. 122, no. 1, pp. 63–86, 2004.CrossRefMathSciNetGoogle Scholar
  16. 16.
    A. Griewank, Evaluating Derivatives, Principles and Techniques of Algorithmic Differentiation Frontiers in Appl. Math. 19, Phil., 2000.Google Scholar
  17. 17.
    A. Griewank and G. Corliss (eds.) Automatic Differentiation of Algorithms: Theory, Implementation, and Applications. SIAM, Philadelphia, Penn., 1991.Google Scholar
  18. 18.
    A. Griewank, D. Juedes, and J. Utke, “ADOL-C: A package for the automatic differentiation of algorithms written in C/C++,” TOMS vol. 22, pp. 131–167, 1996.Google Scholar
  19. 19.
    A. Griewank and A. Walther, “Applying the checkpointing routine treeverse to discretizations of burgers’ equation, Lect. Notes Comput. Sci.and Engin., 8,” in High Performance Scientific and Engineering Computing, H.-J. Bungartz, F. Durst, and C. Zenger (eds.), Springer Berlin Heidelberg, 1999.Google Scholar
  20. 20.
    A. Griewank and A. Walther, “Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation,” ACM Trans. Math. Software, vol. 26, pp. 19–45, 2000.CrossRefGoogle Scholar
  21. 21.
    W. Hager, “Runge-Kutta methods in optimal control and the transformed adjoint system,” Numer. Math. vol. 87, pp. 247–282, 2000.CrossRefMathSciNetGoogle Scholar
  22. 22.
    P. Hiltmann, Numerische Lösung von Mehrpunkt-Randwertproblemen und Aufgaben Der Optimalen Steuerung mit Steuerfunktionen über endlichdimensionalen Räumen, Dissertation, TU München, Mathematisches Institut, Germany, 1989.Google Scholar
  23. 23.
    H. Oberle and W. Grimm, BNDSCO—A program for the numerical solution of optimal control problems, Report No. 515, Institute for Flight System Dynamics, Oberpfaffenhofen, German Aerospace Research Establishment DLR, 1989.Google Scholar
  24. 24.
    H.J. Pesch, “Offline and online computation of optimal trajectories in the aerospace field,” in Applied Mathematics in Aerospace Science and Engineering, A. Miele. A. Salvetti (eds.), Plenum Press, New York, Mathematical Concepts and Methods in Science and Engineering, 1994 vol. 44, pp. 165–220.Google Scholar
  25. 25.
    A. Quarteroni, R. Sacco, and F. Saleri, Numercial Mathematics Springer, New York, 2000.Google Scholar
  26. 26.
    K. Strehmel und R. Weiner, Numerik Gewöhnlicher Differentialgleichungen, Teubner Studienbücher: Mathematik. Teubner, Stuttgart, 1995.Google Scholar
  27. 27.
    O. von Stryk, User’s Guide for DIRCOL (Version 2.1): A Direct Collocation Method for the Numerical Solution of Optimal Control Problems. Fachgebiet Simulation und Systemoptimierung (SIM), Technische Universität Darmstadt, 2000.Google Scholar

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© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Institute of Scientific ComputingTechnische Universität DresdenGermany

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