The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems

¹ Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125, USA; dchang@cds.caltech.edu. Work partially supported by the California Institute of Technology and AFOSR grant ASOSR F49620-99-1-0190.
² Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA; abloch@math.lsa.umich.edu. Work partially supported by NSF grants DMS 981283 and 0103895 and AFOSR.
³ Mechanical & Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA; naomi@princeton.edu. Work partially supported by NSF grant CCR-9980058, ONR grant N00014-98-1-0649 and AFOSR grant F49620-01-1-0382.
⁴ Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125, USA; marsden@cds.caltech.edu. Work partially supported by the California Institute of Technology and AFOSR grant ASOSR F49620-99-1-0190.
⁵ Aerospace & Ocean Engineering, Virginia Tech., Blacksburg, VA 24061, USA; cwoolsey@vt.edu.

Received: 27 December 2001
Revised: 26 February 2002

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.