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

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Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2671-2687.  Google Scholar

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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

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

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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).  Google Scholar

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Physica D, 7 (1983), 305-323. doi: 10.1016/0167-2789(83)90134-3.  Google Scholar

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Ann. Scient. Ec. Norm Sup., $4^e$ série, 18 (1985), 553-561.  Google Scholar

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show all references

References:
[1]

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

[2]

Ann. Inst. Fourier, Grenoble, 16 (1966), 319-361.  Google Scholar

[3]

Commun. Math. Phys., 187 (1997), 357-373. doi: 10.1007/s002200050140.  Google Scholar

[4]

Syst. & Control Lett., 28 (1996), 65-76. doi: 10.1016/0167-6911(96)00008-4.  Google Scholar

[5]

Proc. CDC IEEE, 39 (2000), 1273-1279. http://xxx.lanl.gov/abs/nlin.CD/0103042 Google Scholar

[6]

Proc. IEEE Conf. on Decision and Control, 37 (1998), 2249-2254. 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]

Found. Comput. Math., 8 (2008), 469-500. doi: 10.1007/s10208-008-9025-1.  Google Scholar

[9]

Nonlinearity, 19 (2006), 2247-2276. doi: 10.1088/0951-7715/19/10/002.  Google Scholar

[10]

Hamiltonian and gradient flows, algorithms and control, 69-76, Fields Inst. Commun., 3 (1994), Amer. Math. Soc., Providence, RI.  Google Scholar

[11]

Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[12]

J. Phys. A: Math. Theor., 41 (2008), 344003, 18 pp.  Google Scholar

[13]

Found. Comput. Math., 9 (2009), 221-242. doi: 10.1007/s10208-007-9022-9.  Google Scholar

[14]

J. Geom. Mech., 2 (2010), 51-68. doi: 10.3934/jgm.2010.2.51.  Google Scholar

[15]

Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2671-2687.  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]

J. Symplectic Geom., 6 (2008), 189-237.  Google Scholar

[18]

Adv. Appl. Math., 42 (2008), 176-275.  Google Scholar

[19]

Physica D, 239 (2010), 1929-1947. 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]

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.  Google Scholar

[22]

Phys. Lett. A, 373 (2009), 4354-4359. doi: 10.1016/j.physleta.2009.09.061.  Google Scholar

[23]

Physica D, 6 (1983), 347-363. doi: 10.1016/0167-2789(83)90017-9.  Google Scholar

[24]

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).  Google Scholar

[25]

Adv. in Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.  Google Scholar

[26]

Phys. Rev. Lett., 349 (1998), 4173-4177. doi: 10.1103/PhysRevLett.80.4173.  Google Scholar

[27]

Translations of Mathematical Monographs, 85, American Mathematical Society, Providence, 1991.  Google Scholar

[28]

Geom. Dedicata, 81 (2000), 209-214. doi: 10.1023/A:1005287907806.  Google Scholar

[29]

Funct. Anal. Appl., 10 (1976), 328-329. doi: 10.1007/BF01076037.  Google Scholar

[30]

Second Edition, Springer, 1999.  Google Scholar

[31]

Trans. Amer. Math. Soc., 281 (1984), 147-177. doi: 10.1090/S0002-9947-1984-0719663-1.  Google Scholar

[32]

Physica D, 7 (1983), 305-323. doi: 10.1016/0167-2789(83)90134-3.  Google Scholar

[33]

Ann. Scient. Ec. Norm Sup., $4^e$ série, 18 (1985), 553-561.  Google Scholar

[34]

Sov. Math. Dokl., 17 (1976), 1591-1593. Google Scholar

[35]

Sel. Math. Sov., 2 (1982), 207-291.  Google Scholar

[36]

Indiana U. Math J., 29 (1980), 609-629.  Google Scholar

[37]

Applied Mathematical Sciences 171, Springer-Verlag New York, 2010.  Google Scholar

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