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Two-input control systems on the Euclidean group SE (2)

Published online by Cambridge University Press:  04 July 2013

Ross M. Adams,
Rory Biggs and
Claudiu C. Remsing
Ross M. Adams
Affiliation:
Department of Mathematics (Pure and Applied), Rhodes University, Grahamstown, South Africa. dros@webmail.co.za; rorybiggs@gmail.com; c.c.remsing@ru.ac.za
Rory Biggs
Affiliation:
Department of Mathematics (Pure and Applied), Rhodes University, Grahamstown, South Africa. dros@webmail.co.za; rorybiggs@gmail.com; c.c.remsing@ru.ac.za
Claudiu C. Remsing
Affiliation:
Department of Mathematics (Pure and Applied), Rhodes University, Grahamstown, South Africa. dros@webmail.co.za; rorybiggs@gmail.com; c.c.remsing@ru.ac.za

Abstract

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Any two-input left-invariant control affine system of full rank, evolving on theEuclidean group SE (2), is (detached) feedback equivalent to one ofthree typical cases. In each case, we consider an optimal control problem which is thenlifted, via the Pontryagin Maximum Principle, to a Hamiltonian system onthe dual space 𝔰𝔢 (2)*. These reduced Hamilton − Poisson systems are the maintopic of this paper. A qualitative analysis of each reduced system is performed. Thisanalysis includes a study of the stability nature of all equilibrium states, as well asqualitative descriptions of all integral curves. Finally, the reduced Hamilton equationsare explicitly integrated by Jacobi elliptic functions. Parametrisations for all integralcurves are exhibited.

Keywords

Left-invariant control system(detached) feedback equivalenceLie − Poisson structureenergy-Casimir methodJacobi elliptic function

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013, pp. 947 - 975
Copyright
© EDP Sciences, SMAI, 2013

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