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.
Mots-clés : left-invariant control system, (detached) feedback equivalence, Lie − Poisson structure, energy-casimir method, Jacobi elliptic function
@article{COCV_2013__19_4_947_0,
author = {Adams, Ross M. and Biggs, Rory and Remsing, Claudiu C.},
title = {Two-input control systems on the euclidean group {~SE~(2)}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {947--975},
publisher = {EDP-Sciences},
volume = {19},
number = {4},
year = {2013},
doi = {10.1051/cocv/2012040},
mrnumber = {3182676},
zbl = {1283.49003},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2012040/}
}
TY - JOUR AU - Adams, Ross M. AU - Biggs, Rory AU - Remsing, Claudiu C. TI - Two-input control systems on the euclidean group SE (2) JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 947 EP - 975 VL - 19 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2012040/ DO - 10.1051/cocv/2012040 LA - en ID - COCV_2013__19_4_947_0 ER -
%0 Journal Article %A Adams, Ross M. %A Biggs, Rory %A Remsing, Claudiu C. %T Two-input control systems on the euclidean group SE (2) %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 947-975 %V 19 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2012040/ %R 10.1051/cocv/2012040 %G en %F COCV_2013__19_4_947_0
Adams, Ross M.; Biggs, Rory; Remsing, Claudiu C. Two-input control systems on the euclidean group SE (2). ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 947-975. doi : 10.1051/cocv/2012040. http://archive.numdam.org/articles/10.1051/cocv/2012040/
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