Geodesic
Codimension-two singularities in 3D affine control systems with a scalar control
Sbornik. Mathematics, Tome 199 (2008) no. 4, pp. 613-627

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Codimension-two singularities of the field of totally singular extremal trajectories in 3D affine control systems with scalar control are investigated. These singularities can be of two types: the first is related to singularities of the field of the Hamiltonian system of the maximum principle itself, while the second is related to the degenerate projection of the field of totally singular extremals onto the phase space. The fields of extremal trajectories occurring in these two cases have completely different normal forms and phase portraits. Bibliography: 7 titles. 
@article{SM_2008_199_4_a6,
     author = {A. O. Remizov},
     title = {Codimension-two singularities in {3D} affine control systems with a scalar control},
     journal = {Sbornik. Mathematics},
     pages = {613--627},
     publisher = {mathdoc},
     volume = {199},
     number = {4},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2008_199_4_a6/}
}
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A. O. Remizov. Codimension-two singularities in 3D affine control systems with a scalar control. Sbornik. Mathematics, Tome 199 (2008) no. 4, pp. 613-627. http://geodesic.mathdoc.fr/item/SM_2008_199_4_a6/