Geodesic
Projection Singularities of Extremals and Morse Property for Minimum Time
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 399-407

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For a generic minimum time problem on the plane, we study the projections of the support of extremals (regarded as a two-dimensional object, after normalization) from $\mathbb R^2\times S^1$ to $\mathbb R^2$. Moreover, we study the topology of the reachable set and we give a positive answer to a question of V. I. Arnold: Is the minimum time function, generically, a Morse function in topological sense? 
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     author = {U. Boscain},
     title = {Projection {Singularities} of {Extremals} and {Morse} {Property} for {Minimum} {Time}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     url = {http://geodesic.mathdoc.fr/item/TM_2002_236_a41/}
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U. Boscain. Projection Singularities of Extremals and Morse Property for Minimum Time. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 399-407. http://geodesic.mathdoc.fr/item/TM_2002_236_a41/