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

Clebsch optimal control formulation in mechanics

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.
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]

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

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

[1]

David González-Sánchez, Onésimo Hernández-Lerma. On the Euler equation approach to discrete--time nonstationary optimal control problems. Journal of Dynamics & Games, 2014, 1 (1) : 57-78. doi: 10.3934/jdg.2014.1.57

[2]

Jeffrey K. Lawson, Tanya Schmah, Cristina Stoica. Euler-Poincaré reduction for systems with configuration space isotropy. Journal of Geometric Mechanics, 2011, 3 (2) : 261-275. doi: 10.3934/jgm.2011.3.261

[3]

Emanuel-Ciprian Cismas. Euler-Poincaré-Arnold equations on semi-direct products II. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5993-6022. doi: 10.3934/dcds.2016063

[4]

David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319

[5]

Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511

[6]

Terence Tao. On the universality of the incompressible Euler equation on compact manifolds. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1553-1565. doi: 10.3934/dcds.2018064

[7]

S. Huff, G. Olumolode, N. Pennington, A. Peterson. Oscillation of an Euler-Cauchy dynamic equation. Conference Publications, 2003, 2003 (Special) : 423-431. doi: 10.3934/proc.2003.2003.423

[8]

Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449

[9]

Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations & Control Theory, 2012, 1 (2) : 297-314. doi: 10.3934/eect.2012.1.297

[10]

M. Petcu. Euler equation in a channel in space dimension 2 and 3. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 755-778. doi: 10.3934/dcds.2005.13.755

[11]

Dongfen Bian, Huimin Liu, Xueke Pu. Modulation approximation for the quantum Euler-Poisson equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4375-4405. doi: 10.3934/dcdsb.2020292

[12]

In-Jee Jeong, Benoit Pausader. Discrete Schrödinger equation and ill-posedness for the Euler equation. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 281-293. doi: 10.3934/dcds.2017012

[13]

Juan Calvo. On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1341-1347. doi: 10.3934/cpaa.2013.12.1341

[14]

Adnan H. Sabuwala, Doreen De Leon. Particular solution to the Euler-Cauchy equation with polynomial non-homegeneities. Conference Publications, 2011, 2011 (Special) : 1271-1278. doi: 10.3934/proc.2011.2011.1271

[15]

Flavia Antonacci, Marco Degiovanni. On the Euler equation for minimal geodesics on Riemannian manifoldshaving discontinuous metrics. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 833-842. doi: 10.3934/dcds.2006.15.833

[16]

Thomas Y. Hou, Danping Yang, Hongyu Ran. Multiscale analysis in Lagrangian formulation for the 2-D incompressible Euler equation. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1153-1186. doi: 10.3934/dcds.2005.13.1153

[17]

Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013

[18]

Sergey A. Denisov. Infinite superlinear growth of the gradient for the two-dimensional Euler equation. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 755-764. doi: 10.3934/dcds.2009.23.755

[19]

Stephen C. Preston, Alejandro Sarria. One-parameter solutions of the Euler-Arnold equation on the contactomorphism group. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2123-2130. doi: 10.3934/dcds.2015.35.2123

[20]

Andrei Fursikov, Lyubov Shatina. Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1187-1242. doi: 10.3934/dcds.2018050

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (20)

Other articles
by authors

[Back to Top]