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Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction

Koon, Wang-Sang and Marsden, Jerrold E. (1997) Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction. SIAM Journal on Control and Optimization, 35 (3). pp. 901-929. ISSN 0363-0129. doi:10.1137/S0363012995290367.

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In this paper we establish necessary conditions for optimal control using the ideas of Lagrangian reduction in the sense of reduction under a symmetry group. The techniques developed here are designed for Lagrangian mechanical control systems with symmetry. The benefit of such an approach is that it makes use of the special structure of the system, especially its symmetry structure and thus it leads rather directly to the desired conclusions for such systems. Lagrangian reduction can do in one step what one can alternatively do by applying the Pontryagin Maximum Principle followed by an application of Poisson reduction. The idea of using Lagrangian reduction in the sense of symmetry reduction was also obtained by Bloch and Crouch [1995a,b] in a somewhat different context and the general idea is closely related to those in Montgomery [1990] and Vershik and Gershkovich [1994]. Here we develop this idea further and apply it to some known examples, such as optimal control on Lie groups and principal bundles (such as the ball and plate problem) and reorientation examples with zero angular momentum (such as the satellite with moveable masses). However, one of our main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation in the context of the work of Bloch, Krishnaprasad, Marsden and Murray [1995]. The snakeboard is used to illustrate the method.

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Additional Information:© 1997, SIAM. Research partially supported by NSF and DOE. We thank Tony Bloch, P.S. Krishnaprasad, Naomi Leonard, Jim Ostrowski and Richard Montgomery for helpful comments on this paper.
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Department of EnergyUNSPECIFIED
Subject Keywords:constraints; Lagrangian reduction; mechanical systems with symmetry; nonholonomic; optimal control
Issue or Number:3
Record Number:CaltechAUTHORS:20100820-090419741
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19544
Deposited By: Ruth Sustaita
Deposited On:23 Aug 2010 21:38
Last Modified:08 Nov 2021 23:53

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