December  2014, 6(4): 451-478. doi: 10.3934/jgm.2014.6.451

Higher-order variational problems on lie groups and optimal control applications

1. 

Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, United States

2. 

Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Campus de Cantoblanco, UAM C/ Nicolas Cabrera, 15 - 28049 Madrid, Spain

Received  May 2014 Revised  August 2014 Published  December 2014

In this paper, we describe a geometric setting for higher-order La- grangian problems on Lie groups. Using left-trivialization of the higher-order tangent bundle of a Lie group and an adaptation of the classical Skinner-Rusk formalism, we deduce an intrinsic framework for this type of dynamical systems. Interesting applications as, for instance, a geometric derivation of the higher-order Euler-Poincaré equations, optimal control of underactuated control systems whose configuration space is a Lie group are shown, among others, along the paper.
Citation: Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451
References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition,, Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1029-0. Google Scholar

[2]

L. Abrunheiro, M. Camarinha, J. F. Cariñena, J. Clemente-Gallardo, E. Martínez and P. Santos, Some applications of quasi-velocities in optimal control,, Int. J. Geom. Methods Mod. Phys., 08 (2011), 835. doi: 10.1142/S0219887811005427. Google Scholar

[3]

M. Barbero-Liñán, A.Echeverría Enríquez, D. Martín de Diego, M.C Muñoz-Lecanda and N. Román-Roy, Skinner-Rusk unified formalism for optimal control systems and applications,, J. Phys. A: Math Theor., 40 (2007), 12071. doi: 10.1088/1751-8113/40/40/005. Google Scholar

[4]

L. Bates and R. Cushman, Global Aspect of Classical Integrable Systems,, Birkhäuser Verlag, (1997). doi: 10.1007/978-3-0348-8891-2. Google Scholar

[5]

R. Benito, Roberto and D. Martín de Diego, Hidden symplecticity in Hamilton's principle algorithms,, Differential geometry and its applications, (2005), 411. Google Scholar

[6]

A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, (2003). doi: 10.1007/b97376. Google Scholar

[7]

A. M. Bloch, I. I. Hussein, M. Leok and A. K. Sanyal, Geometric Structure-Preserving Optimal Control of the Rigid Body,, Journal of Dynamical and Control Systems, 15 (2009), 307. doi: 10.1007/s10883-009-9071-2. Google Scholar

[8]

C. Burnett, D. D. Holm and D. Meier, Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework,, Proc. R. Soc. A., 469 (2013). doi: 10.1098/rspa.2013.0249. Google Scholar

[9]

F. Bullo and A. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems,, Texts in Applied Mathematics, (2005). doi: 10.1007/978-1-4899-7276-7. Google Scholar

[10]

M. Crampin, W. Sarlet and F. Cantrijn, Higher order differential equations and higher order Lagrangian Mechanics,, Math. Proc. Camb. Phil. Soc., 99 (1986), 565. doi: 10.1017/S0305004100064501. Google Scholar

[11]

L. Colombo, F. Jimenez and D. Martín de Diego, Discrete Second-Order Euler-Poincaré Equations. An application to optimal control,, International Journal of Geometric Methods in Modern Physics, 9 (2012). doi: 10.1142/S0219887812500375. Google Scholar

[12]

L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometrical approach,, Journal Mathematical Physics, 51 (2010). doi: 10.1063/1.3456158. Google Scholar

[13]

L. Colombo and D. Martín de Diego, Quasivelocities and Optimal Control of Underactuated Mechanical Systems,, Geometry and Physics: XVIII Fall Workshop on Geometry and Physics. AIP Conference Proceedings, 1260 (2011), 133. Google Scholar

[14]

J. Cortés, M. de León, D. Martín de Diego and S. Martínez, Geometric description of vakonomic and nonholonomic dynamics,, SIAM J. Control Optim., 41 (2002), 1389. doi: 10.1137/S036301290036817X. Google Scholar

[15]

P. Crouch and F. Silva-Leite, Geometry and the dynamic interpolation problem,, American Control Conference, (1991), 1131. Google Scholar

[16]

M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of nonlinear systems,Introductory theory and examples,, International Journal of Control, 61 (1995), 1327. doi: 10.1080/00207179508921959. Google Scholar

[17]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems,, Communications in Mathematical Physics, 309 (2012), 413. doi: 10.1007/s00220-011-1313-y. Google Scholar

[18]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X Vialard, Invariant higher-order variational problems II,, Journal of Nonlinear Science, 22 (2012), 553. doi: 10.1007/s00332-012-9137-2. Google Scholar

[19]

F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Higher-Order Lagrange-Poincaré and Hamilton-Poincaré Reductions,, Bulletin of the Brazilian Math. Soc., 42 (2011), 579. doi: 10.1007/s00574-011-0030-7. Google Scholar

[20]

M. J. Gotay and J. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence theorem,, Ann. Inst. Henri Poincaré, 30 (1979), 129. Google Scholar

[21]

M. J. Gotay, J. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints,, J. Math. Phys., 19 (1978), 2388. doi: 10.1063/1.523597. Google Scholar

[22]

D. D. Holm, Geometric mechanics. Part I and II,, Imperial College Press, (2008). Google Scholar

[23]

D. D. Holm, J. E. Marsden and T. S. Ratiu, he Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1. doi: 10.1006/aima.1998.1721. Google Scholar

[24]

D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids,, Dyn. Syst., 23 (2008), 351. doi: 10.1080/14689360802294220. Google Scholar

[25]

T. Lee, M. Leok and N. H. McClamroch, Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on $SO(3)$,, Journal of Dynamical and Control Systems, 14 (2008), 465. doi: 10.1007/s10883-008-9047-7. Google Scholar

[26]

A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment,, Internat. J. Non-Linear Mech., 30 (1995), 793. doi: 10.1016/0020-7462(95)00024-0. Google Scholar

[27]

M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies 112, (1985). Google Scholar

[28]

J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry,, Second Edition, 17 (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar

[29]

D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications,, Ph.D thesis, (2013). Google Scholar

[30]

T. Mestdag and M. Crampin, Anholonomic frames in constrained dynamics,, Dynamical Systems. An International Journal, 25 (2010), 159. doi: 10.1080/14689360903360888. Google Scholar

[31]

M. van Nieuwstadt, M. Rathinam and R. M. Murray, Differential Flatness and Absolute Equivalence of Nonlinear Control Systems,, SIAM J. Control Optim., 36 (1998), 1225. doi: 10.1137/S0363012995274027. Google Scholar

[32]

H. Poincaré, Sur une forme nouvelle des équations de la mécanique,, C. R. Acad. Sci., 132 (1901), 369. Google Scholar

[33]

R. Skinner and R. Rusk, Generalized Hamiltonian dynamics I. Formulation on $T^{*}Q\oplus TQ$,, Journal of Mathematical Pyhsics, 24 (1983), 2589. Google Scholar

[34]

K. Spindler, Optimal attitude control of a rigid body,, Applied Mathematics& Optimization, 34 (1996), 79. doi: 10.1007/BF01182474. Google Scholar

show all references

References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition,, Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1029-0. Google Scholar

[2]

L. Abrunheiro, M. Camarinha, J. F. Cariñena, J. Clemente-Gallardo, E. Martínez and P. Santos, Some applications of quasi-velocities in optimal control,, Int. J. Geom. Methods Mod. Phys., 08 (2011), 835. doi: 10.1142/S0219887811005427. Google Scholar

[3]

M. Barbero-Liñán, A.Echeverría Enríquez, D. Martín de Diego, M.C Muñoz-Lecanda and N. Román-Roy, Skinner-Rusk unified formalism for optimal control systems and applications,, J. Phys. A: Math Theor., 40 (2007), 12071. doi: 10.1088/1751-8113/40/40/005. Google Scholar

[4]

L. Bates and R. Cushman, Global Aspect of Classical Integrable Systems,, Birkhäuser Verlag, (1997). doi: 10.1007/978-3-0348-8891-2. Google Scholar

[5]

R. Benito, Roberto and D. Martín de Diego, Hidden symplecticity in Hamilton's principle algorithms,, Differential geometry and its applications, (2005), 411. Google Scholar

[6]

A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, (2003). doi: 10.1007/b97376. Google Scholar

[7]

A. M. Bloch, I. I. Hussein, M. Leok and A. K. Sanyal, Geometric Structure-Preserving Optimal Control of the Rigid Body,, Journal of Dynamical and Control Systems, 15 (2009), 307. doi: 10.1007/s10883-009-9071-2. Google Scholar

[8]

C. Burnett, D. D. Holm and D. Meier, Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework,, Proc. R. Soc. A., 469 (2013). doi: 10.1098/rspa.2013.0249. Google Scholar

[9]

F. Bullo and A. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems,, Texts in Applied Mathematics, (2005). doi: 10.1007/978-1-4899-7276-7. Google Scholar

[10]

M. Crampin, W. Sarlet and F. Cantrijn, Higher order differential equations and higher order Lagrangian Mechanics,, Math. Proc. Camb. Phil. Soc., 99 (1986), 565. doi: 10.1017/S0305004100064501. Google Scholar

[11]

L. Colombo, F. Jimenez and D. Martín de Diego, Discrete Second-Order Euler-Poincaré Equations. An application to optimal control,, International Journal of Geometric Methods in Modern Physics, 9 (2012). doi: 10.1142/S0219887812500375. Google Scholar

[12]

L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometrical approach,, Journal Mathematical Physics, 51 (2010). doi: 10.1063/1.3456158. Google Scholar

[13]

L. Colombo and D. Martín de Diego, Quasivelocities and Optimal Control of Underactuated Mechanical Systems,, Geometry and Physics: XVIII Fall Workshop on Geometry and Physics. AIP Conference Proceedings, 1260 (2011), 133. Google Scholar

[14]

J. Cortés, M. de León, D. Martín de Diego and S. Martínez, Geometric description of vakonomic and nonholonomic dynamics,, SIAM J. Control Optim., 41 (2002), 1389. doi: 10.1137/S036301290036817X. Google Scholar

[15]

P. Crouch and F. Silva-Leite, Geometry and the dynamic interpolation problem,, American Control Conference, (1991), 1131. Google Scholar

[16]

M. Fliess, J. Lévine, P. Martin and P. Rouchon, Flatness and defect of nonlinear systems,Introductory theory and examples,, International Journal of Control, 61 (1995), 1327. doi: 10.1080/00207179508921959. Google Scholar

[17]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems,, Communications in Mathematical Physics, 309 (2012), 413. doi: 10.1007/s00220-011-1313-y. Google Scholar

[18]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X Vialard, Invariant higher-order variational problems II,, Journal of Nonlinear Science, 22 (2012), 553. doi: 10.1007/s00332-012-9137-2. Google Scholar

[19]

F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Higher-Order Lagrange-Poincaré and Hamilton-Poincaré Reductions,, Bulletin of the Brazilian Math. Soc., 42 (2011), 579. doi: 10.1007/s00574-011-0030-7. Google Scholar

[20]

M. J. Gotay and J. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence theorem,, Ann. Inst. Henri Poincaré, 30 (1979), 129. Google Scholar

[21]

M. J. Gotay, J. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints,, J. Math. Phys., 19 (1978), 2388. doi: 10.1063/1.523597. Google Scholar

[22]

D. D. Holm, Geometric mechanics. Part I and II,, Imperial College Press, (2008). Google Scholar

[23]

D. D. Holm, J. E. Marsden and T. S. Ratiu, he Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1. doi: 10.1006/aima.1998.1721. Google Scholar

[24]

D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids,, Dyn. Syst., 23 (2008), 351. doi: 10.1080/14689360802294220. Google Scholar

[25]

T. Lee, M. Leok and N. H. McClamroch, Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on $SO(3)$,, Journal of Dynamical and Control Systems, 14 (2008), 465. doi: 10.1007/s10883-008-9047-7. Google Scholar

[26]

A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment,, Internat. J. Non-Linear Mech., 30 (1995), 793. doi: 10.1016/0020-7462(95)00024-0. Google Scholar

[27]

M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Mathematical Studies 112, (1985). Google Scholar

[28]

J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry,, Second Edition, 17 (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar

[29]

D. Meier, Invariant Higher-Order Variational Problems: Reduction, Geometry and Applications,, Ph.D thesis, (2013). Google Scholar

[30]

T. Mestdag and M. Crampin, Anholonomic frames in constrained dynamics,, Dynamical Systems. An International Journal, 25 (2010), 159. doi: 10.1080/14689360903360888. Google Scholar

[31]

M. van Nieuwstadt, M. Rathinam and R. M. Murray, Differential Flatness and Absolute Equivalence of Nonlinear Control Systems,, SIAM J. Control Optim., 36 (1998), 1225. doi: 10.1137/S0363012995274027. Google Scholar

[32]

H. Poincaré, Sur une forme nouvelle des équations de la mécanique,, C. R. Acad. Sci., 132 (1901), 369. Google Scholar

[33]

R. Skinner and R. Rusk, Generalized Hamiltonian dynamics I. Formulation on $T^{*}Q\oplus TQ$,, Journal of Mathematical Pyhsics, 24 (1983), 2589. Google Scholar

[34]

K. Spindler, Optimal attitude control of a rigid body,, Applied Mathematics& Optimization, 34 (1996), 79. doi: 10.1007/BF01182474. Google Scholar

[1]

Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81

[2]

Michał Jóźwikowski, Mikołaj Rotkiewicz. Bundle-theoretic methods for higher-order variational calculus. Journal of Geometric Mechanics, 2014, 6 (1) : 99-120. doi: 10.3934/jgm.2014.6.99

[3]

Cédric M. Campos, Sina Ober-Blöbaum, Emmanuel Trélat. High order variational integrators in the optimal control of mechanical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4193-4223. doi: 10.3934/dcds.2015.35.4193

[4]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990

[5]

Leonardo Colombo, David Martín de Diego. Optimal control of underactuated mechanical systems with symmetries. Conference Publications, 2013, 2013 (special) : 149-158. doi: 10.3934/proc.2013.2013.149

[6]

Anthony M. Bloch, Peter E. Crouch, Nikolaj Nordkvist, Amit K. Sanyal. Embedded geodesic problems and optimal control for matrix Lie groups. Journal of Geometric Mechanics, 2011, 3 (2) : 197-223. doi: 10.3934/jgm.2011.3.197

[7]

Weisheng Niu, Yao Xu. Convergence rates in homogenization of higher-order parabolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4203-4229. doi: 10.3934/dcds.2018183

[8]

Clesh Deseskel Elion Ekohela, Daniel Moukoko. On higher-order anisotropic perturbed Caginalp phase field systems. Electronic Research Announcements, 2019, 26: 36-53. doi: 10.3934/era.2019.26.004

[9]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

[10]

Anthony M. Bloch, Rohit Gupta, Ilya V. Kolmanovsky. Neighboring extremal optimal control for mechanical systems on Riemannian manifolds. Journal of Geometric Mechanics, 2016, 8 (3) : 257-272. doi: 10.3934/jgm.2016007

[11]

Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247

[12]

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064

[13]

Anthony M. Bloch, Peter E. Crouch, Nikolaj Nordkvist. Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems. Journal of Geometric Mechanics, 2013, 5 (1) : 1-38. doi: 10.3934/jgm.2013.5.1

[14]

Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004

[15]

Firas Hindeleh, Gerard Thompson. Killing's equations for invariant metrics on Lie groups. Journal of Geometric Mechanics, 2011, 3 (3) : 323-335. doi: 10.3934/jgm.2011.3.323

[16]

Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387

[17]

Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012

[18]

Juan Carlos Marrero. Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes. Journal of Geometric Mechanics, 2010, 2 (3) : 243-263. doi: 10.3934/jgm.2010.2.243

[19]

Adriano Da Silva, Alexandre J. Santana, Simão N. Stelmastchuk. Topological conjugacy of linear systems on Lie groups. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3411-3421. doi: 10.3934/dcds.2017144

[20]

Benjamin Couéraud, François Gay-Balmaz. Variational discretization of thermodynamical simple systems on Lie groups. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-28. doi: 10.3934/dcdss.2020064

2018 Impact Factor: 0.525

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]