Continuous and discrete embedded optimal control problemsand their application to the analysis of Clebsch optimal controlproblems and mechanical systems

Anthony M. Bloch; Peter E. Crouch; Nikolaj Nordkvist

doi:10.3934/jgm.2013.5.1

Article Contents

2013, Volume 5, Issue 1: 1-38. Doi: 10.3934/jgm.2013.5.1

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Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems

1.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
2.
Department of Electrical Engineering, University of Hawai‘i at Mānoa, Honolulu, HI 96822
3.
Mathematics and Sciences Division, Leeward Community College, Pearl City, HI 96782

Received: June 30, 2012

Revised: January 31, 2013

Published: February 2013

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Abstract

In this paper we define ``embedded optimal control problems" which prescribe parametrized families of well defined associated optimal control problems. We show that the extremal generating Hamiltonian equations for an embedded optimal control problem and any associated optimal control problem are simply related by a projection. Furthermore normal extremals project to normal extremals and similarly for abnormal extremals. An interesting class of embedded optimal control problems consists of Clebsch optimal control problems. We provide necessary conditions for a Clebsch optimal control problem to describe a variational problem and thereby a mechanical system. There may be many advantages to analyzing an embedded optimal control problem instead of a particular associated optimal control problem, for example the former being defined on a linear space and the latter on a nonlinear space. The continuous analysis is paralleled by a similar discrete analysis. We define a discrete embedded/Clebsch optimal control problem along with associated discrete optimal control problems and we show results that are analogous to the continuous results. We apply the theory, both in the continuous and the discrete setting, to two example systems: mechanical systems on matrix Lie groups and mechanical systems on $n$-spheres.

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Mathematics Subject Classification: Primary: 37J15, 49K15, 49Q99, 49S05, 58Z05, 65L99, 70H30; Secondary: 34G20.

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