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On the transversality of functions at the core of the transverse function approach to control

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Abstract

The transverse function approach to control, introduced by Morin and Samson in the early 2000s, is based on functions that are transverse to a set of vector fields in a sense formally similar to, although strictly speaking different from, the classical notion of transversality in differential topology. In this paper, a precise link is established between transversality and the functions used in the transverse function approach. It is first shown that a smooth function \({f : M \longrightarrow Q}\) is transverse to a set of vector fields which locally span a distribution D on Q if, and only if, its tangent mapping T f is transverse to D, where D is regarded as a submanifold of the tangent bundle T Q. It is further shown that each of these two conditions is equivalent to transversality of T f to D along the zero section of T M. These results are then used to rigorously state and prove that if M is compact and D is a distribution on Q, then the set of mappings of M into Q that are transverse to D is open in the strong (or “Whitney C ∞-”) topology on C ∞(M, Q).

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  1. División de Matemáticas Aplicadas, IPICYT, Camino a la Presa San José 2055, Col. Lomas 4a sección, CP 78216, San Luis Potosí, SLP, Mexico

    David A. Lizárraga

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Lizárraga, D.A. On the transversality of functions at the core of the transverse function approach to control. Math. Control Signals Syst. 23, 177–197 (2011). https://doi.org/10.1007/s00498-011-0067-6

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