The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems
Dong Eui Chang, Anthony M. Bloch, Naomi Ehrich Leonard, Jerrold E. Marsden and Craig A. Woolsey
To appear ESAIM: Control, Optimisation, and Calculus of Variations,
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.