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The Equivalence of Controlled Lagrangianand Controlled HamiltonianSystems

Published online by Cambridge University Press:  15 August 2002

Dong Eui Chang ,
Anthony M. Bloch ,
Naomi E. Leonard ,
Jerrold E. Marsden  and
Craig A. Woolsey
Dong Eui Chang
Affiliation:
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.
Anthony M. Bloch
Affiliation:
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.
Naomi E. Leonard
Affiliation:
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.
Jerrold E. Marsden
Affiliation:
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.
Craig A. Woolsey
Affiliation:
Aerospace & Ocean Engineering, Virginia Tech., Blacksburg, VA 24061, USA; cwoolsey@vt.edu.
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Abstract

The purpose of this paper is to show that the method of controlledLagrangians and its Hamiltonian counterpart (based on the notionof passivity) are equivalent under rather general hypotheses. Westudy the particular case of simple mechanical control systems(where the underlying Lagrangian is kinetic minus potentialenergy) subject to controls and external forces in some detail.The equivalence makes use of almost Poisson structures (Poissonbrackets that may fail to satisfy the Jacobi identity) on theHamiltonian side, which is the Hamiltonian counterpart of a classof gyroscopic forces on the Lagrangian side.

Keywords

Controlled Lagrangiancontrolled HamiltonianenergyshapingLyapunov stabilitypassivityequivalence.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 393 - 422
Copyright
© EDP Sciences, SMAI, 2002

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