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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, 87 ().   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[7]

A. M. Bloch, P. E. Crouch and N. Nordkvist, Clebsch optimal control representation of mechanical systems and corresponding discretization,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. Brockett, The double bracket equation as the solution of a variational problem,, Hamiltonian and gradient flows, 3 (1994), 69.   Google Scholar

[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.  Google Scholar

[12]

C. J. Cotter, The variational particle-mesh method for matching curves,, J. Phys. A: Math. Theor., 41 (2008).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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., ().   Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

F. Gay-Balmaz and C. Vizman [2011], Dual pairs in fluid dynamics,, Annals of Global Analysis and Geometry, ().   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. Ise and M. Takeuchi, "Lie groups. I, II,", Translations of Mathematical Monographs, 85 (1991).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Second Edition, (1999).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

A. Medina and P. Revoy, Algèbres de Lie et produit scalaire invariant,, Ann. Scient. Ec. Norm Sup., 18 (1985), 553.   Google Scholar

[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.   Google Scholar

[35]

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

[36]

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

[37]

L. Younes, "Shapes and Diffeomorphisms,", Applied Mathematical Sciences, 171 (2010).   Google Scholar

show all references

References:
[1]

A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint,", Encyclopaedia of Mathematical Sciences, 87 ().   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[7]

A. M. Bloch, P. E. Crouch and N. Nordkvist, Clebsch optimal control representation of mechanical systems and corresponding discretization,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. Brockett, The double bracket equation as the solution of a variational problem,, Hamiltonian and gradient flows, 3 (1994), 69.   Google Scholar

[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.  Google Scholar

[12]

C. J. Cotter, The variational particle-mesh method for matching curves,, J. Phys. A: Math. Theor., 41 (2008).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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., ().   Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

F. Gay-Balmaz and C. Vizman [2011], Dual pairs in fluid dynamics,, Annals of Global Analysis and Geometry, ().   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. Ise and M. Takeuchi, "Lie groups. I, II,", Translations of Mathematical Monographs, 85 (1991).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Second Edition, (1999).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

A. Medina and P. Revoy, Algèbres de Lie et produit scalaire invariant,, Ann. Scient. Ec. Norm Sup., 18 (1985), 553.   Google Scholar

[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.   Google Scholar

[35]

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

[36]

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

[37]

L. Younes, "Shapes and Diffeomorphisms,", Applied Mathematical Sciences, 171 (2010).   Google Scholar

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