The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems

Dong Eui Chang; Anthony M. Bloch; Naomi E. Leonard; Jerrold E. Marsden; Craig A. Woolsey

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 393-422
  • ISSN: 1292-8119

Abstract

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The purpose of this paper is to show that the method of controlled Lagrangians and its Hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying Lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity) on the Hamiltonian side, which is the Hamiltonian counterpart of a class of gyroscopic forces on the Lagrangian side.

How to cite

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Chang, Dong Eui, et al. "The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 393-422. <http://eudml.org/doc/90654>.

@article{Chang2010,
abstract = { The purpose of this paper is to show that the method of controlled Lagrangians and its Hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying Lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity) on the Hamiltonian side, which is the Hamiltonian counterpart of a class of gyroscopic forces on the Lagrangian side. },
author = {Chang, Dong Eui, Bloch, Anthony M., Leonard, Naomi E., Marsden, Jerrold E., Woolsey, Craig A.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Controlled Lagrangian; controlled Hamiltonian; energy shaping; Lyapunov stability; passivity; equivalence.; gyroscopic forces; equivalence; almost Poisson structures},
language = {eng},
month = {3},
pages = {393-422},
publisher = {EDP Sciences},
title = {The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems},
url = {http://eudml.org/doc/90654},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Chang, Dong Eui
AU - Bloch, Anthony M.
AU - Leonard, Naomi E.
AU - Marsden, Jerrold E.
AU - Woolsey, Craig A.
TI - The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 393
EP - 422
AB - The purpose of this paper is to show that the method of controlled Lagrangians and its Hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying Lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity) on the Hamiltonian side, which is the Hamiltonian counterpart of a class of gyroscopic forces on the Lagrangian side.
LA - eng
KW - Controlled Lagrangian; controlled Hamiltonian; energy shaping; Lyapunov stability; passivity; equivalence.; gyroscopic forces; equivalence; almost Poisson structures
UR - http://eudml.org/doc/90654
ER -

References

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