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.
Classification : 34D20, 70H03, 70H05, 93D15
Mots clés : controlled lagrangian, controlled hamiltonian, energy shaping, Lyapunov stability, passivity, equivalence
@article{COCV_2002__8__393_0,
author = {Chang, Dong Eui and Bloch, Anthony M. and Leonard, Naomi E. and Marsden, Jerrold E. and Woolsey, Craig A.},
title = {The equivalence of controlled lagrangian and controlled hamiltonian systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {393--422},
publisher = {EDP-Sciences},
volume = {8},
year = {2002},
doi = {10.1051/cocv:2002045},
zbl = {1070.70013},
mrnumber = {1932957},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2002045/}
}
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 PY - 2002 DA - 2002/// SP - 393 EP - 422 VL - 8 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2002045/ UR - https://zbmath.org/?q=an%3A1070.70013 UR - https://www.ams.org/mathscinet-getitem?mr=1932957 UR - https://doi.org/10.1051/cocv:2002045 DO - 10.1051/cocv:2002045 LA - en ID - COCV_2002__8__393_0 ER -
Chang, Dong Eui; Bloch, Anthony M.; Leonard, Naomi E.; Marsden, Jerrold E.; Woolsey, Craig A. The equivalence of controlled lagrangian and controlled hamiltonian systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 393-422. doi : 10.1051/cocv:2002045. http://www.numdam.org/articles/10.1051/cocv:2002045/
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