Abstract
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.
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References
J.T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, SIAM, Philadelphia, 2001.
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.
A.E. Bryson and Y. Ho, Applied Optimal Control—Optimization, Estimation, and Control Hemisphere Publishing Corporation, New York, 1975.
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.
C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustandsbeschränkungen Dissertation, Westfälische Wilhelms-Universität Münster, 1998.
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.
J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations, John Wiley, New York, 1987.
J.-B. Caillau and J. Noailles, “Continuous optimal control sensitivity analysis with ad,” in [10], pp. 109–117.
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.
G.F. Corliss, C. Faure, A. Griewank, L. Hascoët, and U. Naumann (eds.), Automatic Differentiation: from Simulation to Optimization. Springer Verlag, New York, 2001.
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.
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.
Yu.G. Evtushenko, “Automatic differentiation viewed from optimal control theory,” in [17], pp. 25–30.
Yu.G. Evtushenko, “Computation of exact gradients in distributed dynamic systems,” Optim. Methods Softw. vol. 9, nos. 1–3, pp. 45–75, 1998.
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.
A. Griewank, Evaluating Derivatives, Principles and Techniques of Algorithmic Differentiation Frontiers in Appl. Math. 19, Phil., 2000.
A. Griewank and G. Corliss (eds.) Automatic Differentiation of Algorithms: Theory, Implementation, and Applications. SIAM, Philadelphia, Penn., 1991.
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.
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.
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.
W. Hager, “Runge-Kutta methods in optimal control and the transformed adjoint system,” Numer. Math. vol. 87, pp. 247–282, 2000.
P. Hiltmann, Numerische Lösung von Mehrpunkt-Randwertproblemen und Aufgaben Der Optimalen Steuerung mit Steuerfunktionen über endlichdimensionalen Räumen, Dissertation, TU München, Mathematisches Institut, Germany, 1989.
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.
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.
A. Quarteroni, R. Sacco, and F. Saleri, Numercial Mathematics Springer, New York, 2000.
K. Strehmel und R. Weiner, Numerik Gewöhnlicher Differentialgleichungen, Teubner Studienbücher: Mathematik. Teubner, Stuttgart, 1995.
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.
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Walther, A. Automatic differentiation of explicit Runge-Kutta methods for optimal control. Comput Optim Applic 36, 83–108 (2007). https://doi.org/10.1007/s10589-006-0397-3
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DOI: https://doi.org/10.1007/s10589-006-0397-3