Geodesic
Geometry of neighborhoods of singular trajectories in problems with multidimensional control
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical control theory and differential equations, Tome 277 (2012), pp. 74-90

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It is shown that the order of a singular trajectory in problems with multidimensional control is described by a flag of linear subspaces in the control space. In terms of this flag, we construct necessary conditions for the junction of a nonsingular trajectory with a singular one in affine control systems. We also give examples of multidimensional problems in which the optimal control has the form of an irrational winding of a torus that is passed in finite time. 
@article{TM_2012_277_a5,
     author = {M. I. Zelikin and L. V. Lokutsievskiy and R. Hildebrand},
     title = {Geometry of neighborhoods of singular trajectories in problems with multidimensional control},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {74--90},
     publisher = {mathdoc},
     volume = {277},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2012_277_a5/}
}
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M. I. Zelikin; L. V. Lokutsievskiy; R. Hildebrand. Geometry of neighborhoods of singular trajectories in problems with multidimensional control. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical control theory and differential equations, Tome 277 (2012), pp. 74-90. http://geodesic.mathdoc.fr/item/TM_2012_277_a5/