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[3] L. Abrunheiro, M. Camarinha, and J. Clemente-Gallardo. Minimum force optimal control of the spherical free rigid body. In Proc. Control’2010 9th Portuguese Conference on Automatic Control, 8-10 September 2010, Coimbra, pages 333-338. Portugal, 2010.
[4] L. Abrunheiro, M. Camarinha, and J. Clemente-Gallardo. Cubic polynomials on Lie groups: reduction of the Hamiltonian system. J. Phys. A: Math and Theor., 44, 355203, 2011.
[5] V. I. Arnold. Sur la g´eom´etrie di↵´erentielle des groupes de Lie de dimension infinie et ses applications `a l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier, 16:319–361, 1966.
[6] M. Barbero-Lin˜a´n, A. Echeverr´ıa-Enr´ıquez, D. Mart´ın de Diego, M. C. Mun˜oz-Lecanda, and N. Roma´n-Roy. Skinner-Rusk unified formalism for optimal control systems and applications. J. Phys. A: Math. Theor., 40:12071–12093, 2007.
[7] L.M. Bates and R. H. Cushman Global aspects of classical integrable systems. Birkha¨user Verlag, 1997.
[8] R. Benedito and D. Mart´ın de Diego. Hidden symplecticity in Hamilton’s principle algorithms. In Proc. 9th Int. Conf. on Di↵erential Geom. Appl., Prague, 30 August 3 September 2004, Charles University, Prague, pages 411–419. Czech Republic, 2005.
[9] A. Bloch and P. Crouch. Nonholonomic and Vakonomic Control Systems on Riemannian Manifolds. Fields Inst. Commun., 1:25–52, 1993.
[10] A. Bloch and P. Crouch. On the equivalence of higher order variational problems and optimal control problems. In Proc. 35th IEEE Conference on Decision and Control, December 1996, Kobe, volume 2, pages 1648–1653. Japan, 1996.
[11] M. Camarinha, P. Crouch, and F. Silva Leite. Hamiltonian structure of generalized cubic poly- nomials. In Proc. IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, 16-18 March 2000, Princeton University, pages 13–18. USA, 2000.
[12] P. Crouch and F. Silva Leite. The dynamic interpolation problem: on Riemannian manifolds, Lie groups and symmetric spaces. J. Dynam. Control Systems, 1(2):177–202, 1995.
[13] M. Delgado-T´ellez and A. Ibort. A Panorama of Geometrical Optimal Control Theory. Ex- tracta Math., 18(2):129–151, 2003.
[14] M. J. Gotay and J. M. Nester. Presymplectic Lagrangian systems. I: the constraint algorithm and the equivalence theorem. Ann. Inst. Henri Poincar´e, Section A, 30(2):129–142, 1979.
[15] M. J. Gotay and J. M. Nester. Presymplectic Lagrangian systems. II: the second-order equa- tion problem. Ann. Inst. Henri Poincar´e, Section A, 32(1):1–13, 1980.
[16] M. de Leo´n, J. Cort´es, D. Mart´ın de Diego, and S. Mart´ınez. General Symmetries in Optimal Control. Rep. Math. Phys., 53(1):55–78, 2004.
[17] J. E. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry. Rep. Math. Phys., 5(1):121–130, 1974.
[18] J.M. Maruskin and A.M. Bloch. The Boltzmann-Hamel equations for the optimal control of mechanical systems with nonholonomic constraints. Int. J. Robust. Nonlinear Control, 21 (4):373-386, 2011.
[19] L. Noakes. Null Cubics and Lie Quadratics. J. Math. Phys., (3) 44:1436–1448, 2003.
[20] L. Noakes. Duality and Riemmanian cubics. Adv. Comput. Math., 25:195–209, 2006.
[21] L. Noakes, G. Heinzinger, and B. Paden. Cubic splines on curved spaces. IMA J. Math.
Control Inform., 6:465–473, 1989.
[22] J. P. Ortega and T. S. Ratiu. Momentum Maps and Hamiltonian Reduction. Progress in
Mathematics, Vol. 222, Birkha¨user, 2004.
[23] T. Popiel. Higher order geodesics in Lie groups. Math. Control Signals Systems, 19:235–253,
2007.