2018/10/30 by Andrei Agrachev, Agrachev, Andrei A., Francesco Boarotto +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1810.12662
openalex publication_date 2018/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a rank-two sub-Riemannian structure (M,Δ) and a point x0∈ M, a singular curve is a critical point of the endpoint map F:γ↦γ(1) defined on the space of horizontal curves starting at x0. The typical least degenerate singular curves of these structures are called regular singular curves; they are nice if their endpoint is not conjugate along γ. The main goal of this paper is to show that locally around a nice singular curve γ, once we choose a suitable topology on the control space we can find a normal form for the endpoint map, in which F writes as a sum of a linear map and a quadratic form. We also study the restriction of F to the level sets of the action functional and give a Morse-like formula for the inertia index of its Hessian at γ. This is a preparation for a forthcoming generalization of the Morse theory to rank-two sub-Riemannian structures.