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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

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, Vol 87, Control Theory and Optimization Vol II, Springer Verlag.

[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, Grenoble, 16 (1966), 319-361.

[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), 357-373. doi: 10.1007/s002200050140.

[4]

A. M. Bloch and P. E. Crouch, Optimal control and geodesic fows, Syst. & Control Lett., 28 (1996), 65-76. 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), 1273-1279. http://xxx.lanl.gov/abs/nlin.CD/0103042

[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), 2249-2254.

[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), 469-500. 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), 2247-2276. 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, algorithms and control, 69-76, Fields Inst. Commun., 3 (1994), Amer. Math. Soc., Providence, RI.

[11]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. 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), 344003, 18 pp.

[13]

C. J. Cotter and D. D. Holm, Continuous and discrete Clebsch variational principles, Found. Comput. Math., 9 (2009), 221-242. 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), 51-68. 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), 2671-2687.

[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., to appear.

[17]

F. Gay-Balmaz and T. S. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids, J. Symplectic Geom., 6 (2008), 189-237.

[18]

F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Adv. Appl. Math., 42 (2008), 176-275.

[19]

F. Gay-Balmaz and C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems, Physica D, 239 (2010), 1929-1947. 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, to appear.

[21]

D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids, In Geometry, Dynamics and Mechanics: 60th Birthday Volume for J. E. Marsden. P. Holmes, P. Newton and A. Weinstein, eds., Springer-Verlag (2002), 113-167.

[22]

D. D. Holm, Euler's fluid equations: Optimal control vs optimization, Phys. Lett. A, 373 (2009), 4354-4359. 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), 347-363. 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, A Festshrift for Alan Weinstein, 203-235, Progr. Math., 232, J. E. Marsden and T. S. Ratiu, Editors, Birkhäuse Boston, Boston, MA, (2004).

[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), 1-81. 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), 4173-4177. doi: 10.1103/PhysRevLett.80.4173.

[27]

M. Ise and M. Takeuchi, "Lie groups. I, II," Translations of Mathematical Monographs, 85, American Mathematical Society, Providence, 1991.

[28]

O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata, 81 (2000), 209-214. 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), 328-329. doi: 10.1007/BF01076037.

[30]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Second Edition, Springer, 1999.

[31]

J. E. Marsden, T. S. Ratiu and A. Weinstein, Semidirect product and reduction in mechanics, Trans. Amer. Math. Soc., 281 (1984), 147-177. 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), 305-323. 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., $4^e$ série, 18 (1985), 553-561.

[34]

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

[35]

A. S. Mishchenko and A. T. Fomenko, Integrability of Euler equations on semisimple Lie algebras, Sel. Math. Sov., 2 (1982), 207-291.

[36]

T. Ratiu, The motion of the free $n$-dimensional rigid body, Indiana U. Math J., 29 (1980), 609-629.

[37]

L. Younes, "Shapes and Diffeomorphisms," Applied Mathematical Sciences 171, Springer-Verlag New York, 2010.

show all references

References:
[1]

A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint," Encyclopaedia of Mathematical Sciences, Vol 87, Control Theory and Optimization Vol II, Springer Verlag.

[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, Grenoble, 16 (1966), 319-361.

[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), 357-373. doi: 10.1007/s002200050140.

[4]

A. M. Bloch and P. E. Crouch, Optimal control and geodesic fows, Syst. & Control Lett., 28 (1996), 65-76. 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), 1273-1279. http://xxx.lanl.gov/abs/nlin.CD/0103042

[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), 2249-2254.

[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), 469-500. 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), 2247-2276. 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, algorithms and control, 69-76, Fields Inst. Commun., 3 (1994), Amer. Math. Soc., Providence, RI.

[11]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. 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), 344003, 18 pp.

[13]

C. J. Cotter and D. D. Holm, Continuous and discrete Clebsch variational principles, Found. Comput. Math., 9 (2009), 221-242. 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), 51-68. 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), 2671-2687.

[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., to appear.

[17]

F. Gay-Balmaz and T. S. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids, J. Symplectic Geom., 6 (2008), 189-237.

[18]

F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Adv. Appl. Math., 42 (2008), 176-275.

[19]

F. Gay-Balmaz and C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems, Physica D, 239 (2010), 1929-1947. 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, to appear.

[21]

D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids, In Geometry, Dynamics and Mechanics: 60th Birthday Volume for J. E. Marsden. P. Holmes, P. Newton and A. Weinstein, eds., Springer-Verlag (2002), 113-167.

[22]

D. D. Holm, Euler's fluid equations: Optimal control vs optimization, Phys. Lett. A, 373 (2009), 4354-4359. 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), 347-363. 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, A Festshrift for Alan Weinstein, 203-235, Progr. Math., 232, J. E. Marsden and T. S. Ratiu, Editors, Birkhäuse Boston, Boston, MA, (2004).

[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), 1-81. 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), 4173-4177. doi: 10.1103/PhysRevLett.80.4173.

[27]

M. Ise and M. Takeuchi, "Lie groups. I, II," Translations of Mathematical Monographs, 85, American Mathematical Society, Providence, 1991.

[28]

O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata, 81 (2000), 209-214. 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), 328-329. doi: 10.1007/BF01076037.

[30]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Second Edition, Springer, 1999.

[31]

J. E. Marsden, T. S. Ratiu and A. Weinstein, Semidirect product and reduction in mechanics, Trans. Amer. Math. Soc., 281 (1984), 147-177. 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), 305-323. 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., $4^e$ série, 18 (1985), 553-561.

[34]

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

[35]

A. S. Mishchenko and A. T. Fomenko, Integrability of Euler equations on semisimple Lie algebras, Sel. Math. Sov., 2 (1982), 207-291.

[36]

T. Ratiu, The motion of the free $n$-dimensional rigid body, Indiana U. Math J., 29 (1980), 609-629.

[37]

L. Younes, "Shapes and Diffeomorphisms," Applied Mathematical Sciences 171, Springer-Verlag New York, 2010.

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