01188nas a2200157 4500008004100000245005800041210005700099260003700156300001600193490000700209520065100216100002500867700002400892700002100916856009300937 2015 eng d00aGeodesics and horizontal-path spaces in Carnot groups0 aGeodesics and horizontalpath spaces in Carnot groups bMathematical Sciences Publishers a1569–16300 v193 a

We study properties of the space of horizontal paths joining the origin with a vertical point on a generic two-step Carnot group. The energy is a Morse-Bott functional on paths and its critical points (sub-Riemannian geodesics) appear in families (compact critical manifolds) with controlled topology. We study the asymptotic of the number of critical manifolds as the energy grows. The topology of the horizontal-path space is also investigated, and we find asymptotic results for the total Betti number of the sublevels of the energy as it goes to infinity. We interpret these results as local invariants of the sub-Riemannian structure.

1 aAgrachev, Andrei, A.1 aGentile, Alessandro1 aLerario, Antonio uhttps://www.math.sissa.it/publication/geodesics-and-horizontal-path-spaces-carnot-groups01122nas a2200145 4500008004100000245005400041210005100095260001300146520066000159653006000819100002500879700001600904700002000920856003600940 2014 en d00aOn conjugate times of LQ optimal control problems0 aconjugate times of LQ optimal control problems bSpringer3 aMotivated by the study of linear quadratic optimal control problems, we consider a dynamical system with a constant, quadratic Hamiltonian, and we characterize the number of conjugate times in terms of the spectrum of the Hamiltonian vector field $\vec{H}$. We prove the following dichotomy: the number of conjugate times is identically zero or grows to infinity. The latter case occurs if and only if $\vec{H}$ has at least one Jordan block of odd dimension corresponding to a purely imaginary eigenvalue. As a byproduct, we obtain bounds from below on the number of conjugate times contained in an interval in terms of the spectrum of $\vec{H}$.10aOptimal control, Lagrange Grassmannian, Conjugate point1 aAgrachev, Andrei, A.1 aRizzi, Luca1 aSilveira, Pavel uhttp://hdl.handle.net/1963/722700752nas a2200121 4500008004100000245004700041210004600088260001000134520040400144100002500548700002100573856003600594 2012 en d00aSub-Riemannian structures on 3D Lie groups0 aSubRiemannian structures on 3D Lie groups bSISSA3 a

We give a complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. As a corollary we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups $SL(2)$ and $A^+(\mathbb{R})\times S^1$, where $A^+(\mathbb{R})$ denotes the group of orientation preserving affine maps on the real line.

1 aAgrachev, Andrei, A.1 aBarilari, Davide uhttp://hdl.handle.net/1963/645301439nas a2200181 4500008004300000245007000043210006800113260001300181300001200194490000700206520090200213100002501115700001701140700002301157700002001180700002101200856003601221 2010 en_Ud 00aTwo-dimensional almost-Riemannian structures with tangency points0 aTwodimensional almostRiemannian structures with tangency points bElsevier a793-8070 v273 a

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including the presence of tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss-Bonnet formula for almost-Riemannian structures with tangency points.

1 aAgrachev, Andrei, A.1 aBoscain, Ugo1 aCharlot, Grégoire1 aGhezzi, Roberta1 aSigalotti, Mario uhttp://hdl.handle.net/1963/3870