The equivalence of controlled lagrangian and controlled hamiltonian systems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8  (2002), p. 393-422

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.

DOI : https://doi.org/10.1051/cocv:2002045
Classification:  34D20,  70H03,  70H05,  93D15
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     pages = {393-422},
     doi = {10.1051/cocv:2002045},
     zbl = {1070.70013},
     mrnumber = {1932957},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2002__8__393_0}
}
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, Volume 8 (2002) , pp. 393-422. doi : 10.1051/cocv:2002045. http://www.numdam.org/item/COCV_2002__8__393_0/

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