2015/02/26 by Francesco Boarotto, Boarotto, Francesco, Antonio Lerario +1
Mathematics · Medicine · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Growth Hormone and Insulin-like Growth Factors #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1502.07452
openalex publication_date 2015/02/26 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28
We discuss homotopy properties of endpoint maps for affine control systems.\nWe prove that these maps are Hurewicz fibrations with respect to some W1,p\ntopology on the space of trajectories, for a certain p>1. We study critical\npoints of geometric costs for these affine control systems, proving that if the\nbase manifold is compact then the number of their critical points is infinite\n(we use Lusternik-Schnirelmann category combined with the Hurewicz property).\nIn the special case where the control system is subriemannian this result can\nbe read as the corresponding version of Serre's theorem, on the existence of\ninfinitely many geodesics between two points on a compact riemannian manifold.\nIn the subriemannian case we show that the Hurewicz property holds for all\np\≥1 and the horizontal-loop space with the W1,2 topology has the\nhomotopy type of a CW-complex (as long as the endpoint map has at least one\nregular value); in particular the inclusion of the horizontal-loop space in the\nordinary one is a homotopy equivalence.\n