The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems

Chang, D., A. M. Bloch, N. Leonard, J. E. Marsden and C. Woolsey

Control and the Calculus of Variations (special issue dedicated to J. L. Lions) 8, 393-422.

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.