Free Access
Volume 17, Number 2, April-June 2011
Page(s) 322 - 352
DOI https://doi.org/10.1051/cocv/2010012
Published online 31 March 2010
  1. U. Ascher, J. Christiansen and R.D. Russell, A collocation solver for mixed order systems of boundary value problems. Math. Comput. 33 (1979) 659–679. [Google Scholar]
  2. V. Bär, Ein Kollokationsverfahren zur numerischen Lösung allgemeiner Mehrpunktrandwertaufgaben mit Schalt- und Sprungbedingungen mit Anwendungen in der Optimalen Steuerung und der Parameteridentifizierung. Diploma Thesis, Bonn, Germany (1983). [Google Scholar]
  3. J.T. Betts, Survey of numerical methods for trajectory optimization. AIAA J. Guid. Control Dyn. 21 (1998) 193–207. [Google Scholar]
  4. J.T. Betts and W.P. Huffmann, Mesh refinement in direct transcription methods for optimal control. Optim. Control Appl. Meth. 19 (1998) 1–21. [CrossRef] [Google Scholar]
  5. A.I. Bobenko and Y.B. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations, and semidirect products. Lett. Math. Phys. 49 (1999) 79–93. [CrossRef] [MathSciNet] [Google Scholar]
  6. A.I. Bobenko and Y.B. Suris, Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top. Comm. Math. Phys. 204 (1999) 147–188. [CrossRef] [MathSciNet] [Google Scholar]
  7. H.G. Bock, Numerical solutions of nonlinear multipoint boundary value problems with applications to optimal control. Z. Angew. Math. Mech. 58 (1978) T407–T409. [CrossRef] [Google Scholar]
  8. H.G. Bock and K.J. Plitt, A multiple shooting algorithm for direct solution of optimal control problems, in 9th IFAC World Congress, Budapest, Hungary, Pergamon Press (1984) 242–247. [Google Scholar]
  9. J.F. Bonnans and J. Laurent-Varin, Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control. Numer. Math. 103 (2006) 1–10. [CrossRef] [MathSciNet] [Google Scholar]
  10. N. Bou-Rabee and H. Owhadi, Stochastic variational integrators. IMA J. Numer. Anal. 29 (2008) 421–443. [CrossRef] [Google Scholar]
  11. A.E. Bryson and Y.C. Ho, Applied Optimal Control. Hemisphere (1975). [Google Scholar]
  12. R. Bulirsch, Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung. Report of the Carl-Cranz-Gesellschaft e.V., DLR, Oberpfaffenhofen, Germany (1971). [Google Scholar]
  13. 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. 120 (2000) 85–108. [CrossRef] [MathSciNet] [Google Scholar]
  14. J.A. Cadzow, Discrete calculus of variations. Int. J. Control 11 (1970) 393–407. [Google Scholar]
  15. J.A. Cadzow, Discrete-Time Systems: An Introduction With Interdisciplinary Applications. Prentice-Hall (1973). [Google Scholar]
  16. A.L. Cauchy, Méthode générale pour la résolution des systèmes d'équations simultanées. C. R. Acad. Sci. 25 (1847) 536–538. [Google Scholar]
  17. F.L. Chernousko and A.A. Luybushin, Method of successive approximations for optimal control problems (survey paper). Opt. Control Appl. Meth. 3 (1982) 101–114. [CrossRef] [Google Scholar]
  18. P. Deuflhard, A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting. Numer. Math. 22 (1974) 289–315. [CrossRef] [MathSciNet] [Google Scholar]
  19. E.D. Dickmanns and K.H. Well, Approximate solution of optimal control problems using third order hermite polynomial functions. Lect. Notes Comput. Sci. 27 (1975) 158–166. [Google Scholar]
  20. A.L. Dontchev and W.W. Hager, The Euler Approximation in State Constrained Optimal Control, in Mathematics of Computation 70, American Mathematical Society, USA (2001) 173–203. [Google Scholar]
  21. A.L. Dontchev, W.W. Hager and V.M. Veliov, Second order Runge-Kutta approximations in control constrained optimal control. SIAM J. Numer. Anal. 38 (2000) 202–226. [CrossRef] [MathSciNet] [Google Scholar]
  22. R.C. Fetecau, J.E. Marsden, M. Ortiz and M. West, Nonsmooth Lagrangian mechanics and variational collision integrators. SIAM J. Appl. Dyn. Syst. 2 (2003) 381–416. [Google Scholar]
  23. L. Flatto, Advanced calculus. Williams & Wilkins (1976). [Google Scholar]
  24. L. Fox, Some numerical experiments with eigenvalue problems in ordinary differential equations, in Boundary value problems in differential equations, R.E. Langer Ed. (1960). [Google Scholar]
  25. E. Frazzoli, M.A. Dahleh and E. Feron, Maneuver-based motion planning for nonlinear systems with symmetries. IEEE Trans. Robot. 21 (2005) 1077–1091. [CrossRef] [Google Scholar]
  26. A. Griewank, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM (2000). [Google Scholar]
  27. W.W. Hager, Convex control and dual approximations, in Constructive Approaches to Mathematical Models, Academic Press, New York, USA (1979) 189–202. [Google Scholar]
  28. W.W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87 (2000) 247–282. [Google Scholar]
  29. W.W. Hager, Numerical analysis in optimal control, in International Series of Numerical Mathematics 139, Birkhäuser Verlag, Basel, Switzerland (2001) 83–93. [Google Scholar]
  30. E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration. Springer (2002). [Google Scholar]
  31. S.P. Han, Superlinearly convergent variable-metric algorithms for general nonlinear programming problems. Math. Program. 11 (1976) 263–282. [CrossRef] [Google Scholar]
  32. P. Hiltmann, Numerische Lösung von Mehrpunkt-Randwertproblemen und Aufgaben der optimalen Steuerung über endlichdimensionalen Räumen. Ph.D. Thesis, Fakultät für Mathematik und Informatik, Technische Universität München, Germany (1990). [Google Scholar]
  33. C.L. Hwang and L.T. Fan, A discrete version of Pontryagin's maximum principle. Oper. Res. 15 (1967) 139–146. [CrossRef] [Google Scholar]
  34. B.W. Jordan and E. Polak, Theory of a class of discrete optimal control systems. J. Elec. Ctrl. 17 (1964) 697–711. [CrossRef] [Google Scholar]
  35. O. Junge and S. Ober-Blöbaum, Optimal reconfiguration of formation flying satellites, in IEEE Conference on Decision and Control and European Control Conference ECC, Seville, Spain (2005). [Google Scholar]
  36. O. Junge, J.E. Marsden and S. Ober-Blöbaum, Discrete mechanics and optimal control, in 16th IFAC World Congress, Prague, Czech Republic (2005). [Google Scholar]
  37. O. Junge, J.E. Marsden and S. Ober-Blöbaum, Optimal reconfiguration of formation flying spacecraft - a decentralized approach, in IEEE Conference on Decision and Control and European Control Conference ECC, San Diego, USA (2006) 5210–5215. [Google Scholar]
  38. C. Kane, J.E. Marsden and M. Ortiz, Symplectic energy-momentum integrators. Math. Phys. 40 (1999) 3353–3371. [CrossRef] [MathSciNet] [Google Scholar]
  39. C. Kane, J.E. Marsden, M. Ortiz and M. West, Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Meth. Eng. 49 (2000) 1295–1325. [Google Scholar]
  40. E. Kanso and J.E. Marsden, Optimal motion of an articulated body in a perfect fluid, in IEEE Conference on Decision and Control and European Control Conference ECC, Seville, Spain (2005). [Google Scholar]
  41. W. Karush, Minima of functions of several variables with inequalities as side constraints. Master's thesis, Department of Mathematics, University of Chicago, USA (1939). [Google Scholar]
  42. H.B. Keller, Numerical methods for two-point boundary value problems. Blaisdell, Waltham, USA (1968). [Google Scholar]
  43. H.J. Kelley, Gradient theory of optimal flight paths. Journal of the American Rocket Society 30 (1960) 947–953. [Google Scholar]
  44. L. Kharevych, P. Mullen, S. Leyendecker, Y. Tong, J.E. Marsden and M. Desbrun, Robust time-adaptive integrators for computer animation (in preparation). [Google Scholar]
  45. M. Kobilarov, Discrete geometric motion control of autonomous vehicles. Ph.D. Thesis, University of Southern California, USA (2008). [Google Scholar]
  46. M. Kobilarov and G.S. Sukhatme, Optimal control using nonholonomic integrators, in IEEE International Conference on Robotics and Automation (ICRA), Rome, Italy (2007) 1832–1837. [Google Scholar]
  47. M. Kobilarov, M. Desbrun, J.E. Marsden and G.S. Sukhatme, A discrete geometric optimal control framework for systems with symmetries. Robotics: Science and Systems 3 (2007) 1–8. [Google Scholar]
  48. D. Kraft, On converting optimal control problems into nonlinear programming problems, in Computational Mathematical Programming F15 of NATO ASI series, K. Schittkowsky Ed., Springer (1985) 261–280. [Google Scholar]
  49. H.W. Kuhn and A.W. Tucker, Nonlinear programming, in Proceedings of the Second Berkeley Symposium on Mathematical Statisics and Probability, J. Neyman Ed., University of California Press, Berkeley, USA (1951). [Google Scholar]
  50. T.D. Lee, Can time be a discrete dynamical variable? Phys. Lett. B 121 (1983) 217–220. [Google Scholar]
  51. T.D. Lee, Difference equations and conservation laws. J. Stat. Phys. 46 (1987) 843–860. [Google Scholar]
  52. T. Lee, N.H. McClamroch and M. Leok, Attitude maneuvers of a rigid spacecraft in a circular orbit, in American Control Conference, Minneapolis, USA (2006) 1742–1747. [Google Scholar]
  53. T. Lee, N.H. McClamroch and M. Leok, Optimal control of a rigid body using geometrically exact computations on SE(3), in IEEE CDC and ECC, San Diego, USA (2006) 2710–2715. [Google Scholar]
  54. D.B. Leineweber, Efficient reduced SQP methods for the optimization of chemical processes described by large sparse DAE models, in Fortschr.-Bericht VDI Reihe 3, Verfahrenstechnik 613, VDI-Verlag (1999). [Google Scholar]
  55. A. Lew, J.E. Marsden, M. Ortiz and M. West, Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167 (2003) 85–146. [CrossRef] [MathSciNet] [Google Scholar]
  56. S. Leyendecker, S. Ober-Blöbaum, J.E. Marsden and M. Ortiz, Discrete mechanics and optimal control for constrained multibody dynamics, in 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, ASME International Design Engineering Technical Conferences, Las Vegas, USA (2007). [Google Scholar]
  57. S. Leyendecker, S. Ober-Blöbaum and J.E. Marsden, Discrete mechanics and optimal control for constrained systems. Optim. Contr. Appl. Meth. (2009) DOI: 10.1002/oca.912. [Google Scholar]
  58. J.D. Logan, First integrals in the discrete calculus of variation. Aequ. Math. 9 (1973) 210–220. [Google Scholar]
  59. R. MacKay, Some aspects of the dynamics of Hamiltonian systems, in The dynamics of numerics and the numerics of dynamics, D.S. Broomhead and A. Iserles Eds., Clarendon Press, Oxford, UK (1992) 137–193. [Google Scholar]
  60. S. Maeda, Canonical structure and symmetries for discrete systems. Math. Jap. 25 (1980) 405–420. [Google Scholar]
  61. S. Maeda, Extension of discrete Noether theorem. Math. Jap. 26 (1981) 85–90. [Google Scholar]
  62. S. Maeda, Lagrangian formulation of discrete systems and concept of difference space. Math. Jap. 27 (1981) 345–356. [Google Scholar]
  63. J.E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians, and water waves. Math. Proc. Camb. Phil. Soc. 125 (1999) 553–575. [CrossRef] [Google Scholar]
  64. J.E. Marsden and M. West, Discrete mechanics and variational integrators. Acta Numer. 10 (2001) 357–514. [CrossRef] [MathSciNet] [Google Scholar]
  65. J.E. Marsden, G.W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199 (1998) 351–395. [Google Scholar]
  66. J.E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincaré and Lie Poisson equations. Nonlinearity 12 (1999) 1647-1662. [CrossRef] [MathSciNet] [Google Scholar]
  67. J.E. Marsden, S. Pekarsky and S. Shkoller, Symmetry reduction of discrete Lagrangian mechanics on Lie groups. Geometry and Physics 36 (1999) 140–151. [CrossRef] [Google Scholar]
  68. J. Martin, Discrete mechanics and optimal control. Master's Thesis, Department of Control and Dynamical Systems, California Institute of Technology, USA (2006). [Google Scholar]
  69. R.I. McLachlan and S. Marsland, Discrete mechanics and optimal control for image registration, in Computational Techniques and Applications Conference (CTAC) (2006). [Google Scholar]
  70. A. Miele, Gradient algorithms for the optimization of dynamic systems, in Control and Dynamic Systems 60, C.T. Leondes Ed. (1980) 1–52. [Google Scholar]
  71. S. Ober-Blöbaum, Discrete mechanics and optimal control. Ph.D. Thesis, University of Paderborn, Germany (2008). [Google Scholar]
  72. G.W. Patrick and C. Cuell, Error analysis of variational integrators of unconstrained lagrangian systems. Numer. Math. 113 (2009) 243–264. [CrossRef] [MathSciNet] [Google Scholar]
  73. D. Pekarek, A.D. Ames and J.E. Marsden, Discrete mechanics and optimal control applied to the compass gait biped, in IEEE Conference on Decision and Control and European Control Conference ECC, New Orleans, USA (2007). [Google Scholar]
  74. L.S. Pontryagin, V.G. Boltyanski, R.V. Gamkrelidze and E.F. Miscenko, The mathematical theory of optimal processes. John Wiley & Sons (1962). [Google Scholar]
  75. M.J.D. Powell, A fast algorithm for nonlinearly constrained optimization calculations, in Numerical Analysis Lecture Notes in Mathematics 630, G.A. Watson Ed., Springer (1978) 261–280. [Google Scholar]
  76. R. Pytlak, Numerical methods for optimal control problems with state constraints. Springer (1999). [Google Scholar]
  77. L.B. Rall, Automatic Differentiation: Techniques and Applications, Lect. Notes Comput. Sci. 120. Springer Verlag, Berlin, Germany (1981). [Google Scholar]
  78. S.D. Ross, Optimal flapping strokes for self-propulsion in a perfect fluid, in American Control Conference, Minneapolis, USA (2006) 4118–4122. [Google Scholar]
  79. B. Sendov and V.A. Popov, The averaged moduli of smoothness. John Wiley (1988). [Google Scholar]
  80. Y.B. Suris, Hamiltonian methods of Runge-Kutta type and their variational interpretation. Math. Model. 2 (1990) 78–87. [Google Scholar]
  81. H. Tolle, Optimization methods. Springer (1975). [Google Scholar]
  82. O. von Stryk, Numerical solution of optimal control problems by direct collocation, in Optimal Control - Calculus of Variation, Optimal Control Theory and Numerical Methods, R. Bulirsch, A. Miele, J. Stoer and K.H. Well Eds., International Series of Numerical Mathematics 111, Birkhäuser (1993) 129–143. [Google Scholar]
  83. O. von Stryk, Numerical hybrid optimal control and related topics. Habilitation Thesis, TU München, Germany (2000). [Google Scholar]
  84. A. Walther, A. Kowarz and A. Griewank, ADOL-C: a package for the automatic differentiation of algorithms written in C/C++. ACM TOMS 22 (1996) 131–167. [CrossRef] [Google Scholar]
  85. J.M. Wendlandt and J.E. Marsden, Mechanical integrators derived from a discrete variational principle. Physica D 106 (1997) 223–246. [CrossRef] [MathSciNet] [Google Scholar]
  86. J.M. Wendlandt and J.E. Marsden, Mechanical systems with symmetry, variational principles and integration algorithms, in Current and Future Directions in Applied Mathematics, M. Alber, B. Hu and J. Rosenthal Eds., Birkhäuser (1997) 219–261. [Google Scholar]
  87. R.E. Wengert, A simple automatic derivative evaluation program. Commun. ACM 7 (1964) 463–464. [CrossRef] [Google Scholar]

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