Volume 19, Number 4, October-December 2013
|Page(s)||947 - 975|
|Published online||04 July 2013|
Two-input control systems on the Euclidean group SE (2)
Revised: 4 June 2012
Any two-input left-invariant control affine system of full rank, evolving on the Euclidean group SE (2), is (detached) feedback equivalent to one of three typical cases. In each case, we consider an optimal control problem which is then lifted, via the Pontryagin Maximum Principle, to a Hamiltonian system on the dual space 𝔰𝔢 (2)*. These reduced Hamilton − Poisson systems are the main topic of this paper. A qualitative analysis of each reduced system is performed. This analysis includes a study of the stability nature of all equilibrium states, as well as qualitative descriptions of all integral curves. Finally, the reduced Hamilton equations are explicitly integrated by Jacobi elliptic functions. Parametrisations for all integral curves are exhibited.
Mathematics Subject Classification: 49J15 / 93D05 / 22E60 / 53D17
Key words: Left-invariant control system / (detached) feedback equivalence / Lie − Poisson structure / energy-Casimir method / Jacobi elliptic function
© EDP Sciences, SMAI, 2013
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.