Abstract
Summary. The problem studied here is a shortest path problem of the type encountered in sub-Riemannian geometry. It is distinguished by special structures related to its Lie group setting and the Z 2 graded structure on the relevant Lie algebra. In spite of the fact that the first order necessary conditions lead to differential equations that are integrable in terms of elementary functions, in this case there remain questions related to the existence of appropriate values for the parameters which appear. In this paper we treat the problem in some generality but establish the existence of suitable parameter values only in the case of the general linear group of dimension two.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Baillieul. Some Optimization Problems in Geometric Control Theory. PhD thesis, Harvard University, Cambridge MA, 1965.
C.S. Ballantine. Products of positive definite matrices. Linear Algebra and Appl., 3:79–114, 1970.
R.W. Brockett. Control theory and singular Riemannian geometry. In P. Hilton and G. Young, editors, New Directions in Applied Mathematics, pages 11–27. Springer-Verlag, New York, 1981.
R.W. Brockett. Explicitly solvable control problems with nonholonomic constraints. Proc. of the 38th IEEE Conf. on Decision and Contr., pages 13–16, 1999.
P. d’Alessandro, A. Isidori, and A. Ruberti. Structure analysis of linear and bilinear systems. In R. R. Mohler and A. Ruberti, editors, Theory and Analysis of Variable Structure Systems, pages 25–35. Academic Press, New York, 1972.
B. Gaveau. Principe de moindre action propagation de la chaleur et estimees sons elliptiques sur certains groupes nilpotents. Acta Mathematica, 139:95–153, 1977.
A. Isidori. Nonlinear Control Systems. Springer Verlag, London, UK, 3rd edition, 1995.
N. Khaneja, B. Luy, and S.J. Glaser. Boundary of quantum evolution under decoherence. Proceedings of National Academy of Sciences, 100(23):13162–13166, 2003.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Brockett, R. (2008). Nonholonomic Trajectory Optimization and the Existence of Twisted Matrix Logarithms. In: Astolfi, A., Marconi, L. (eds) Analysis and Design of Nonlinear Control Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74358-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-74358-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74357-6
Online ISBN: 978-3-540-74358-3
eBook Packages: EngineeringEngineering (R0)