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Asymptotics of accessibility sets along an abnormal trajectory. (English) Zbl 0996.93009
The author starts by giving a description of the geometry of the accessibility set in the neighborhood of an abnormal trajectory of corank 1 of a generic single-input smooth control system with bounded control. This is done by writing the system in an appropriate normal form in which the intrinsic second-order derivative along the abnormal trajectory can be represented by an explicit differential operator. This operator is extended into a larger space of controls, making possible the use of spectral analytic tools to infer the shape of the accessibility set.
This result is applied to the case of a generic sub-Riemannian geometry of rank 2. It is shown that sub-Riemannian spheres of small radius split in two sectors in the neighborhood of the end point of an abnormal geodesic. The points in one of these sectors are end points of geodesics that are close to the abnormal geodesic in the \(L^{\infty}\) topology, while the points in the other sector are end points of geodesics that lie close to the abnormal geodesic in the \(L^{2}\) topology but not in the \(L^{\infty}\) topology.
Some previously known conditions for optimality of abnormal trajectories are discussed in the light of the description of the accessibility set.

MSC:
93B03 Attainable sets, reachability
49K15 Optimality conditions for problems involving ordinary differential equations
93B10 Canonical structure
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