Geodesic
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

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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.

DOI : 10.1051/cocv/2012040
Classification : 49J15, 93D05, 22E60, 53D17
Keywords: 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://geodesic.mathdoc.fr/articles/10.1051/cocv/2012040/}
}
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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

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