# American Institute of Mathematical Sciences

March  2013, 5(1): 1-38. doi: 10.3934/jgm.2013.5.1

## Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems

 1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States 2 Department of Electrical Engineering, University of Hawai‘i at Mānoa, Honolulu, HI 96822 3 Mathematics and Sciences Division, Leeward Community College, Pearl City, HI 96782, United States

Received  July 2012 Revised  February 2013 Published  April 2013

In this paper we define embedded optimal control problems" which prescribe parametrized families of well defined associated optimal control problems. We show that the extremal generating Hamiltonian equations for an embedded optimal control problem and any associated optimal control problem are simply related by a projection. Furthermore normal extremals project to normal extremals and similarly for abnormal extremals. An interesting class of embedded optimal control problems consists of Clebsch optimal control problems. We provide necessary conditions for a Clebsch optimal control problem to describe a variational problem and thereby a mechanical system. There may be many advantages to analyzing an embedded optimal control problem instead of a particular associated optimal control problem, for example the former being defined on a linear space and the latter on a nonlinear space. The continuous analysis is paralleled by a similar discrete analysis. We define a discrete embedded/Clebsch optimal control problem along with associated discrete optimal control problems and we show results that are analogous to the continuous results. We apply the theory, both in the continuous and the discrete setting, to two example systems: mechanical systems on matrix Lie groups and mechanical systems on $n$-spheres.
Citation: Anthony M. Bloch, Peter E. Crouch, Nikolaj Nordkvist. Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems. Journal of Geometric Mechanics, 2013, 5 (1) : 1-38. doi: 10.3934/jgm.2013.5.1
##### References:
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Mech. Engrg., 196 (2007), 2907.  doi: 10.1016/j.cma.2007.01.017.  Google Scholar [22] T. Lee, M. Leok and N. H. McClamroch, Lagrangian mechanics and variational integrators on two-spheres,, International Journal for Numerical Methods in Engineering, 79 (2009), 1147.  doi: 10.1002/nme.2603.  Google Scholar [23] 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.   Google Scholar [24] J. E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincare and Lie-Poisson equations,, Nonlinearity, 12 (1999), 1647.  doi: 10.1088/0951-7715/12/6/314.  Google Scholar [25] J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", $2^{nd}$ edition, 17 (1999).   Google Scholar [26] J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, 10 (2001), 357.  doi: 10.1017/S096249290100006X.  Google Scholar [27] A. S. Mischenko and A. T. Fomenko, On the integration of the Euler equations on semisimple Lie algebras,, Sov. Math. Dokl., 17 (1976), 1591.   Google Scholar [28] J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials,, Communications in Mathematical Physics, 139 (1991), 217.  doi: 10.1007/BF02352494.  Google Scholar [29] N. Nordkvist and A. K. Sanyal, A Lie group variational integrator for rigid body motion in $SE(3)$ with applications to underwater vehicles,, in, (2010), 5414.  doi: 10.1109/CDC.2010.5717622.  Google Scholar [30] J. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Birkhäuser Verlag, (2004).   Google Scholar [31] T. S. Ratiu, The motion of the free n-dimensional rigid body,, Indiana University Mathematics Journal, 29 (1980), 609.  doi: 10.1512/iumj.1980.29.29046.  Google Scholar

show all references

##### References:
 [1] A. M. Bloch, P. E. Crouch, D. D. Holm and J. E. Marsden, An optimal control formulation for inviscid incompressible ideal fluid flow,, in, (2000), 1273.  doi: 10.1109/CDC.2000.912030.  Google Scholar [2] 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.  doi: 10.1088/0951-7715/15/4/316.  Google Scholar [3] 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.  doi: 10.1007/s10208-008-9025-1.  Google Scholar [4] A. M. Bloch, P. E. Crouch, N. Nordkvist and A. K. Sanyal, Embedded geodesic problems and optimal control for matrix Lie groups,, Journal of Geometric Mechanics, 3 (2011), 197.  doi: 10.3934/jgm.2011.3.197.  Google Scholar [5] A. M. Bloch, P. E. Crouch and A. K. Sanyal, A variational problem on Stiefel manifolds,, Nonlinearity, 19 (2006), 2247.  doi: 10.1088/0951-7715/19/10/002.  Google Scholar [6] A. I Bobenko and Y. B. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations, and semidirect products,, Letters in Mathematical Physics, 49 (1999), 79.  doi: 10.1023/A:1007654605901.  Google Scholar [7] V. G. Boltyanskii, "Optimal Control of Discrete Systems,", John Wiley, (1978).   Google Scholar [8] N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups part I: Introduction and structure-preserving properties,, Foundations of Computational Mathematics, 9 (2008), 197.  doi: 10.1007/s10208-008-9030-4.  Google Scholar [9] J. R. Cardoso and F. Silva Leite, The Moser-Veselov equation,, Linear Algebra and its Applications, 360 (2003), 237.  doi: 10.1016/S0024-3795(02)00450-0.  Google Scholar [10] C. J. Cotter and D. D. Holm, Continuous and discrete Clebsch variational principles,, Foundations of Computational Mathematics, 9 (2009), 221.  doi: 10.1007/s10208-007-9022-9.  Google Scholar [11] P. E. Crouch and R. Grossman, Numerical integration of ordinary differential equations on manifolds,, Journal of Nonlinear Science, 3 (1993), 1.  doi: 10.1007/BF02429858.  Google Scholar [12] P. E. Crouch, N. Nordkvist and A. K. Sanyal, Optimal control and geodesics on matrix Lie groups,, in, (2010).   Google Scholar [13] M. de León, D. Martín de Diego and A. Santamaría-Merino, Discrete variational integrators and optimal control theory,, Advances in Computational Mathematics, 26 (2007), 251.  doi: 10.1007/s10444-004-4093-5.  Google Scholar [14] Y. N. Federov and V. V. Kozlov, Various aspects of n-dimensional rigid body dynamics,, in, 168 (1995), 141.   Google Scholar [15] F. Gay-Balmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics,, Journal of Geometric Mechanics, 3 (2011), 41.  doi: 10.3934/jgm.2011.3.41.  Google Scholar [16] E. Hairer, C. Lubich and G. Wanner., "Geometric Numerical Integration,", Springer Verlag, (2002).   Google Scholar [17] D. D. Holm, Riemannian optimal control formulation of incompressible ideal fluid flow,, preprint., ().   Google Scholar [18] V. Jurdjevic, "Geometric Control Theory,", Cambridge University Press, (1997).   Google Scholar [19] M. Kobilarov and J. E. Marsden, Discrete geometric optimal control on Lie groups,, IEEE Transactions on Robotics, 27 (2011), 641.  doi: 10.1007/s10208-011-9089-1.  Google Scholar [20] T. Lee, M. Leok and N. H. McClamroch, A Lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum,, in, (2005), 962.   Google Scholar [21] T. Lee, M. Leok and N. H. McClamroch, Lie group variational integrators for the full body problem,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2907.  doi: 10.1016/j.cma.2007.01.017.  Google Scholar [22] T. Lee, M. Leok and N. H. McClamroch, Lagrangian mechanics and variational integrators on two-spheres,, International Journal for Numerical Methods in Engineering, 79 (2009), 1147.  doi: 10.1002/nme.2603.  Google Scholar [23] 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.   Google Scholar [24] J. E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincare and Lie-Poisson equations,, Nonlinearity, 12 (1999), 1647.  doi: 10.1088/0951-7715/12/6/314.  Google Scholar [25] J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", $2^{nd}$ edition, 17 (1999).   Google Scholar [26] J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, 10 (2001), 357.  doi: 10.1017/S096249290100006X.  Google Scholar [27] A. S. Mischenko and A. T. Fomenko, On the integration of the Euler equations on semisimple Lie algebras,, Sov. Math. Dokl., 17 (1976), 1591.   Google Scholar [28] J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials,, Communications in Mathematical Physics, 139 (1991), 217.  doi: 10.1007/BF02352494.  Google Scholar [29] N. Nordkvist and A. K. Sanyal, A Lie group variational integrator for rigid body motion in $SE(3)$ with applications to underwater vehicles,, in, (2010), 5414.  doi: 10.1109/CDC.2010.5717622.  Google Scholar [30] J. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Birkhäuser Verlag, (2004).   Google Scholar [31] T. S. Ratiu, The motion of the free n-dimensional rigid body,, Indiana University Mathematics Journal, 29 (1980), 609.  doi: 10.1512/iumj.1980.29.29046.  Google Scholar
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