Clebsch optimal control formulation in mechanics

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March 2011, 3(1): 41-79. doi: 10.3934/jgm.2011.3.41

Clebsch optimal control formulation in mechanics

François Gay-Balmaz ^1, and Tudor S. Ratiu ^2,

1.	Centre National de la Recherche Scientifique (CNRS), Laboratoire de Météorologie Dynamique, École Normale Supérieure, Paris, France
2.	Section de Mathématiques and Bernoulli Center, Ecole Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland

Received September 2010 Revised April 2011 Published April 2011

Abstract
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This paper introduces and studies a class of optimal control problems based on the Clebsch approach to Euler-Poincaré dynamics. This approach unifies and generalizes a wide range of examples appearing in the literature: the symmetric formulation of $N$-dimensional rigid body and its generalization to other matrix groups; optimal control for ideal flow using the back-to-labels map; the double bracket equations associated to symmetric spaces. New examples are provided such as the optimal control formulation for the $N$-Camassa-Holm equation and a new geodesic interpretation of its singular solutions.

Keywords: momentum map, Clebsch variables, Euler equation, Optimal control, normal metric, geodesic., double bracket equation, Euler-Poincaré equation.

Mathematics Subject Classification: 49J15, 49J20, 37K05, 58E30, 70H3.

Citation: François Gay-Balmaz, Tudor S. Ratiu. Clebsch optimal control formulation in mechanics. Journal of Geometric Mechanics, 2011, 3 (1) : 41-79. doi: 10.3934/jgm.2011.3.41

References:

[1]	A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint,", Encyclopaedia of Mathematical Sciences, 87 ().
[2]	V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimenson infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier, 16 (1966), 319.
[3]	A. M. Bloch, R. W. Brockett and P. E. Crouch, Double bracket equations and geodesic flows on symmetric spaces,, Commun. Math. Phys., 187 (1997), 357. doi: 10.1007/s002200050140.
[4]	A. M. Bloch and P. E. Crouch, Optimal control and geodesic fows,, Syst. & Control Lett., 28 (1996), 65. doi: 10.1016/0167-6911(96)00008-4.
[5]	A. M. Bloch, P. E. Crouch, D. D. Holm and J. E. Marsden, An optimal control formulation for inviscid incompressible ideal fluid flow,, Proc. CDC IEEE, 39 (2000), 1273.
[6]	A. M. Bloch, P. E. Crouch, J. E. Marsden and T. S. Ratiu, Discrete rigid body dynamics and optimal control,, Proc. IEEE Conf. on Decision and Control, 37 (1998), 2249.
[7]	A. M. Bloch, P. E. Crouch and N. Nordkvist, Clebsch optimal control representation of mechanical systems and corresponding discretization,, preprint., ().
[8]	A. M. Bloch, P. E. Crouch, J. E. Marsden and A. K Sanyal, Optimal control and geodesics on quadratic matrix Lie groups,, Found. Comput. Math., 8 (2008), 469. doi: 10.1007/s10208-008-9025-1.
[9]	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.
[10]	R. Brockett, The double bracket equation as the solution of a variational problem,, Hamiltonian and gradient flows, 3 (1994), 69.
[11]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661.
[12]	C. J. Cotter, The variational particle-mesh method for matching curves,, J. Phys. A: Math. Theor., 41 (2008).
[13]	C. J. Cotter and D. D. Holm, Continuous and discrete Clebsch variational principles,, Found. Comput. Math., 9 (2009), 221. doi: 10.1007/s10208-007-9022-9.
[14]	C. J. Cotter and D. D. Holm, Geodesic boundary value problems with symmetry,, J. Geom. Mech., 2 (2010), 51. doi: 10.3934/jgm.2010.2.51.
[15]	C. J. Cotter, D. D. Holm and P. E. Hydon, Multisymplectic formulation of fluid dynamics using the inverse map,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2671.
[16]	F. Gay-Balmaz, D. D. Holm, D. Meier, T. S. Ratiu and F.-X. Vialard [2011], Invariant higher-order variational problems,, Comm. Math. Phys., ().
[17]	F. Gay-Balmaz and T. S. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids,, J. Symplectic Geom., 6 (2008), 189.
[18]	F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids,, Adv. Appl. Math., 42 (2008), 176.
[19]	F. Gay-Balmaz and C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems,, Physica D, 239 (2010), 1929. doi: 10.1016/j.physd.2010.07.002.
[20]	F. Gay-Balmaz and C. Vizman [2011], Dual pairs in fluid dynamics,, Annals of Global Analysis and Geometry, ().
[21]	D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids,, In Geometry, Dynamics, and Mechanics: 60th Birthday Volume for J. E. Marsden, (2002), 113.
[22]	D. D. Holm, Euler's fluid equations: Optimal control vs optimization,, Phys. Lett. A, 373 (2009), 4354. doi: 10.1016/j.physleta.2009.09.061.
[23]	D. D. Holm and B. A. Kupershmidt, Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity,, Physica D, 6 (1983), 347. doi: 10.1016/0167-2789(83)90017-9.
[24]	D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation,, In The Breadth of Symplectic and Poisson Geometry, 232 (2004), 203.
[25]	D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. in Math., 137 (1998), 1. doi: 10.1006/aima.1998.1721.
[26]	D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion,, Phys. Rev. Lett., 349 (1998), 4173. doi: 10.1103/PhysRevLett.80.4173.
[27]	M. Ise and M. Takeuchi, "Lie groups. I, II,", Translations of Mathematical Monographs, 85 (1991).
[28]	O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds,, Geom. Dedicata, 81 (2000), 209. doi: 10.1023/A:1005287907806.
[29]	S. V. Manakov, A remark on the integration of Euler',s equations of the dynamics of an n-dimensional rigid body,, Funct. Anal. Appl., 10 (1976), 328. doi: 10.1007/BF01076037.
[30]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Second Edition, (1999).
[31]	J. E. Marsden, T. S. Ratiu and A. Weinstein, Semidirect product and reduction in mechanics,, Trans. Amer. Math. Soc., 281 (1984), 147. doi: 10.1090/S0002-9947-1984-0719663-1.
[32]	J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,, Physica D, 7 (1983), 305. doi: 10.1016/0167-2789(83)90134-3.
[33]	A. Medina and P. Revoy, Algèbres de Lie et produit scalaire invariant,, Ann. Scient. Ec. Norm Sup., 18 (1985), 553.
[34]	A. S. Mishchenko and A. T. Fomenko, On the integration of the Euler equations on semisimple Lie algebras,, Sov. Math. Dokl., 17 (1976), 1591.
[35]	A. S. Mishchenko and A. T. Fomenko, Integrability of Euler equations on semisimple Lie algebras,, Sel. Math. Sov., 2 (1982), 207.
[36]	T. Ratiu, The motion of the free $n$-dimensional rigid body,, Indiana U. Math J., 29 (1980), 609.
[37]	L. Younes, "Shapes and Diffeomorphisms,", Applied Mathematical Sciences, 171 (2010).

show all references

References:

[1]	A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint,", Encyclopaedia of Mathematical Sciences, 87 ().
[2]	V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimenson infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier, 16 (1966), 319.
[3]	A. M. Bloch, R. W. Brockett and P. E. Crouch, Double bracket equations and geodesic flows on symmetric spaces,, Commun. Math. Phys., 187 (1997), 357. doi: 10.1007/s002200050140.
[4]	A. M. Bloch and P. E. Crouch, Optimal control and geodesic fows,, Syst. & Control Lett., 28 (1996), 65. doi: 10.1016/0167-6911(96)00008-4.
[5]	A. M. Bloch, P. E. Crouch, D. D. Holm and J. E. Marsden, An optimal control formulation for inviscid incompressible ideal fluid flow,, Proc. CDC IEEE, 39 (2000), 1273.
[6]	A. M. Bloch, P. E. Crouch, J. E. Marsden and T. S. Ratiu, Discrete rigid body dynamics and optimal control,, Proc. IEEE Conf. on Decision and Control, 37 (1998), 2249.
[7]	A. M. Bloch, P. E. Crouch and N. Nordkvist, Clebsch optimal control representation of mechanical systems and corresponding discretization,, preprint., ().
[8]	A. M. Bloch, P. E. Crouch, J. E. Marsden and A. K Sanyal, Optimal control and geodesics on quadratic matrix Lie groups,, Found. Comput. Math., 8 (2008), 469. doi: 10.1007/s10208-008-9025-1.
[9]	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.
[10]	R. Brockett, The double bracket equation as the solution of a variational problem,, Hamiltonian and gradient flows, 3 (1994), 69.
[11]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661.
[12]	C. J. Cotter, The variational particle-mesh method for matching curves,, J. Phys. A: Math. Theor., 41 (2008).
[13]	C. J. Cotter and D. D. Holm, Continuous and discrete Clebsch variational principles,, Found. Comput. Math., 9 (2009), 221. doi: 10.1007/s10208-007-9022-9.
[14]	C. J. Cotter and D. D. Holm, Geodesic boundary value problems with symmetry,, J. Geom. Mech., 2 (2010), 51. doi: 10.3934/jgm.2010.2.51.
[15]	C. J. Cotter, D. D. Holm and P. E. Hydon, Multisymplectic formulation of fluid dynamics using the inverse map,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2671.
[16]	F. Gay-Balmaz, D. D. Holm, D. Meier, T. S. Ratiu and F.-X. Vialard [2011], Invariant higher-order variational problems,, Comm. Math. Phys., ().
[17]	F. Gay-Balmaz and T. S. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids,, J. Symplectic Geom., 6 (2008), 189.
[18]	F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids,, Adv. Appl. Math., 42 (2008), 176.
[19]	F. Gay-Balmaz and C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems,, Physica D, 239 (2010), 1929. doi: 10.1016/j.physd.2010.07.002.
[20]	F. Gay-Balmaz and C. Vizman [2011], Dual pairs in fluid dynamics,, Annals of Global Analysis and Geometry, ().
[21]	D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids,, In Geometry, Dynamics, and Mechanics: 60th Birthday Volume for J. E. Marsden, (2002), 113.
[22]	D. D. Holm, Euler's fluid equations: Optimal control vs optimization,, Phys. Lett. A, 373 (2009), 4354. doi: 10.1016/j.physleta.2009.09.061.
[23]	D. D. Holm and B. A. Kupershmidt, Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity,, Physica D, 6 (1983), 347. doi: 10.1016/0167-2789(83)90017-9.
[24]	D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation,, In The Breadth of Symplectic and Poisson Geometry, 232 (2004), 203.
[25]	D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. in Math., 137 (1998), 1. doi: 10.1006/aima.1998.1721.
[26]	D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion,, Phys. Rev. Lett., 349 (1998), 4173. doi: 10.1103/PhysRevLett.80.4173.
[27]	M. Ise and M. Takeuchi, "Lie groups. I, II,", Translations of Mathematical Monographs, 85 (1991).
[28]	O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds,, Geom. Dedicata, 81 (2000), 209. doi: 10.1023/A:1005287907806.
[29]	S. V. Manakov, A remark on the integration of Euler',s equations of the dynamics of an n-dimensional rigid body,, Funct. Anal. Appl., 10 (1976), 328. doi: 10.1007/BF01076037.
[30]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Second Edition, (1999).
[31]	J. E. Marsden, T. S. Ratiu and A. Weinstein, Semidirect product and reduction in mechanics,, Trans. Amer. Math. Soc., 281 (1984), 147. doi: 10.1090/S0002-9947-1984-0719663-1.
[32]	J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,, Physica D, 7 (1983), 305. doi: 10.1016/0167-2789(83)90134-3.
[33]	A. Medina and P. Revoy, Algèbres de Lie et produit scalaire invariant,, Ann. Scient. Ec. Norm Sup., 18 (1985), 553.
[34]	A. S. Mishchenko and A. T. Fomenko, On the integration of the Euler equations on semisimple Lie algebras,, Sov. Math. Dokl., 17 (1976), 1591.
[35]	A. S. Mishchenko and A. T. Fomenko, Integrability of Euler equations on semisimple Lie algebras,, Sel. Math. Sov., 2 (1982), 207.
[36]	T. Ratiu, The motion of the free $n$-dimensional rigid body,, Indiana U. Math J., 29 (1980), 609.
[37]	L. Younes, "Shapes and Diffeomorphisms,", Applied Mathematical Sciences, 171 (2010).

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