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Embedded geodesic problems and optimal control for matrix Lie groups

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  • This paper is devoted to a detailed analysis of the geodesic problem on matrix Lie groups, with left invariant metric, by examining representations of embeddings of geodesic flows in suitable vector spaces. We show how these representations generate extremals for optimal control problems. In particular we discuss in detail the symmetric representation of the so-called $n$-dimensional rigid body equation and its relation to the more classical Euler description. We detail invariant manifolds of these flows on which we are able to define a strict notion of equivalence between representations, and identify naturally induced symplectic structures.
    Mathematics Subject Classification: Primary: 37J05, 49K15, 53C22, 70H30; Secondary: 34G20.


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  • [1]

    R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1978.


    V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," 2nd edition, Graduate Texts in Mathematics, 60, Springer Verlag, New York, 1989.


    A. M. Bloch, J. Ballieul, P. E. Crouch and J. E. Marsden, "Nonholonomic Mechanics and Control," number 24 in "Interdisciplinary Texts in Mathematics," Springer Verlag, New York, 2003.


    A. M. Bloch, P. E. Crouch, D. D. Holm and J. E. Marsden, An optimal control formulation for inviscid incompressible ideal fluid flow, In "Proc. IEEE Conf. on Decision and Control," Sydney, Australia, (December 2000), 1273-1279, arXiv:nlin/0103042.


    A. M. Bloch, P. E. Crouch, J. E. Marsden and T. S. Ratiu, The symmetric representation of the rigid body equations and their discretization, Nonlinearity, 15 (2002), 1309-1341.doi: 10.1088/0951-7715/15/4/316.


    A. M. Bloch, P. E. Crouch, J. E. Marsden and A. K. Sanyal, Optimal control and geodesics on quadratic matrix Lie groups, Foundations of Computational Mathematics, 8 (2008), 469-500.doi: 10.1007/s10208-008-9025-1.


    A. M. Bloch, P. E. Crouch and A. K. Sanyal, A variational problem on Stiefel manifolds, Nonlinearity, 19 (2006), 2247-2276.doi: 10.1088/0951-7715/19/10/002.


    Y. N. Federov and V. V. Kozlov, Various aspects of n-dimensional rigid body dynamics, in "Dynamical Systems in Classical Mechanics," American Mathematical Society Translations, 168, Amer. Math. Soc., Providence, RI, (1995), 141-171.


    F. Gay-Balmaz and T. S. RatiuClebsch optimal control formulation in mechanics, preprint.


    I. M. Gelfand and S. V. Fomin, "Calculus of Variations," Prentice-Hall, Inc., Englewood Cliffs, NJ, (reprinted by Dover, 2000), 1963.


    D. D. HolmRiemannian optimal control formulation of incompressible ideal fluid flow, preprint.


    D. E. Kirk, "Optimal Control Theory: An Introduction," Dover Publications, New York, 2004.


    S. V. Manakov, Note on the integration of Euler's equations of the dynamics of an n-dimensional rigid body, Functional Analysis and Its Applications, 10 (1976), 328-329.doi: 10.1007/BF01076037.


    J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," 2nd edition, Texts in Applied Mathematics, 17, Springer Verlag, New York, 1999.


    A. S. Mischenko and A. T. Fomenko, On the integration of the Euler equations on semisimple Lie algebras, Sov. Math. Dokl., 17 (1976), 1591-1593.


    T. S. Ratiu, The motion of the free n-dimensional rigid body, Indiana University Mathematics Journal, 29 (1980), 609-629.doi: 10.1512/iumj.1980.29.29046.

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