Browsing by Author "Agrachev, Andrei A."

Now showing 1 - 4 of 4
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    The curvature: a variational approach
    (SISSA, 2013-06-22) Agrachev, Andrei A.; Barilari, Davide; Rizzi, Luca; Mathematics
    The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.
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    Geodesics and admissible-path spaces in Carnot Groups
    (SISSA, 2013-11-26) Agrachev, Andrei A.; Gentile, A.; Lerario, Antonio; Mathematics
    We study the topology of admissible-loop spaces on a step-two Carnot group G. We use a Morse-Bott theory argument to study the structure and the number of geodesics on G connecting the origin with a 'vertical' point (geodesics are critical points of the 'Energy' functional, defined on the loop space). These geodesics typically appear in families (critical manifolds). Letting the energy grow, we obtain an upper bound on the number of critical manifolds with energy bounded by s: this upper bound is polynomial in s of degree l (the corank of the distribution). Despite this evidence, we show that Morse-Bott inequalities are far from sharp: the topology (i.e. the sum of the Betti numbers) of the loop space filtered by the energy grows at most as a polynomial in s of degree l-1. In the limit for s at infinity, all Betti numbers (except the zeroth) must actually vanish: the admissible-loop space is contractible. In the case the corank l=2 we compute exactly the leading coefficient of the sum of the Betti numbers of the admissible-loop space with energy less than s. This coefficient is expressed by an integral on the unit circle depending only on the coordinates of the final point and the structure constants of the Lie algebra of G.
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    Introduction to Riemannian and sub-Riemannian geometry
    (SISSA, 2012-04) Agrachev, Andrei A.; Barilari, Davide; Boscain, Ugo; Mathematics; Functional Analysis and Applications
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    Quadratic cohomology
    (SISSA, 2013-01-10) Agrachev, Andrei A.; Mathematics
    We study homological invariants of smooth families of real quadratic forms as a step towards a "Lagrange multipliers rule in the large" that intends to describe topology of smooth vector functions in terms of scalar Lagrange functions.