Discrete variational integrators and optimal control theory

Abstract A geometric derivation of numerical integrators for optimal control problems is proposed. It is based in the classical technique of generating functions adapted to the special features of optimal control problems.

[1]  J. Marsden,et al.  Symplectic-energy-momentum preserving variational integrators , 1999 .

[2]  Anthony M. Bloch,et al.  Nonlinear Dynamical Control Systems (H. Nijmeijer and A. J. van der Schaft) , 1991, SIAM Review.

[3]  Frank L. Lewis,et al.  Optimal Control , 1986 .

[4]  C. Scovel,et al.  Symplectic integration of Hamiltonian systems , 1990 .

[5]  A. Bobenko,et al.  Discrete Time Lagrangian Mechanics on Lie Groups,¶with an Application to the Lagrange Top , 1999 .

[6]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

[7]  俊治 杉江 IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control , 2000 .

[8]  B. W. Jordan,et al.  Theory of a Class of Discrete Optimal Control Systems , 1964 .

[9]  E. Hairer,et al.  Structure-Preserving Algorithms for Ordinary Differential Equations , 2006 .

[10]  Mark J. Gotay,et al.  Presymplectic lagrangian systems. I : the constraint algorithm and the equivalence theorem , 1979 .

[11]  U. Nottingham,et al.  Principles of discrete time mechanics: II. Classical field theory , 1997, hep-th/9703080.

[12]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[13]  J. Marsden,et al.  Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs , 1998, math/9807080.

[14]  J. Marsden,et al.  Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems , 2000 .

[15]  David Martín de Diego,et al.  Geometric Numerical Integration of Nonholonomic Systems and Optimal Control Problems , 2004, Eur. J. Control.

[16]  M. de Leon,et al.  Variational integrators and time-dependent lagrangian systems , 2002 .

[17]  Symplectic Structure of Discrete Hamiltonian Systems , 2002 .

[18]  F. Pirani MATHEMATICAL METHODS OF CLASSICAL MECHANICS (Graduate Texts in Mathematics, 60) , 1982 .

[19]  M. de Leon,et al.  Geometric integrators and nonholonomic mechanics , 2002 .

[20]  A. Bobenko,et al.  Discrete Lagrangian Reduction, Discrete Euler–Poincaré Equations, and Semidirect Products , 1999, math/9906108.

[21]  L. Erbe,et al.  Disconjugacy for linear Hamiltonian difference systems , 1992 .

[22]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[23]  C. Scovel,et al.  Symplectic integration of Hamiltonian systems , 1990 .

[24]  D. D. Diego,et al.  TIME-DEPENDENT CONSTRAINED HAMILTONIAN SYSTEMS AND DIRAC BRACKETS , 1996 .

[25]  V. Arnold Mathematical Methods of Classical Mechanics , 1974 .

[26]  M. León,et al.  The constraint algorithm for time-dependent Lagrangians , 1994 .

[27]  Jorge Cortes,et al.  Non-holonomic integrators , 2001 .

[28]  James A. Cadzow,et al.  Discrete calculus of variations , 1970 .

[29]  J. Moser,et al.  Discrete versions of some classical integrable systems and factorization of matrix polynomials , 1991 .

[30]  C. Ahlbrandt Equivalence of Discrete Euler Equations and Discrete Hamiltonian Systems , 1993 .

[31]  T. D. Lee,et al.  Can time be a discrete dynamical variable , 1983 .

[32]  J. Marsden,et al.  Time‐discretized variational formulation of non‐smooth frictional contact , 2002 .

[33]  Jorge Cortes Geometric, Control and Numerical Aspects of Nonholonomic Systems , 2002 .

[34]  T. D. Lee,et al.  Difference equations and conservation laws , 1987 .

[35]  Principles of discrete time mechanics: III. Quantum field theory , 1997, hep-th/9707029.

[36]  G. Jaroszkiewicz,et al.  Principles of discrete time mechanics: I. Particle systems , 1997, hep-th/9703079.