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Invariant Control Systems on Lie Groups

  • Rory BiggsEmail author
  • Claudiu C. Remsing
Chapter
Part of the UNIPA Springer Series book series (USS)

Abstract

This is a survey of our research (conducted over the last few years) on invariant control systems, the associated optimal control problems, and the associated Hamilton–Poisson systems. The focus is on equivalence and classification. State space and detached feedback equivalence of control systems are characterized in simple algebraic terms; several classes of systems (in three dimensions, on the Heisenberg groups, and on the six-dimensional orthogonal group) are classified. Equivalence of cost-extended systems is shown to imply equivalence of the associated Hamilton–Poisson systems. Cost-extended systems of a certain kind are reinterpreted as invariant sub-Riemannian structures. A classification of quadratic Hamilton–Poisson systems in three dimensions is presented. As an illustrative example, the stability and integration of a typical system is investigated.

Notes

Acknowledgements

The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007–2013) under grant agreement no. 317721. Also, Rory Biggs would like to acknowledge the financial assistance of the Claude Leon Foundation.

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Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of PretoriaPretoriaSouth Africa
  2. 2.Department of MathematicsRhodes UniversityGrahamstownSouth Africa

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