TY - JOUR
T1 - Existence of planar curves minimizing length and curvature
JF - Proc. Steklov Inst. Math. 270 (2010) 43-56
Y1 - 2010
A1 - Ugo Boscain
A1 - GrĂ©goire Charlot
A1 - Francesco Rossi
AB - In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $\\\\int \\\\sqrt{1+K_\\\\gamma^2} ds$, depending both on length and curvature $K$. We fix starting and ending points as well as initial and final directions.\\nFor this functional we discuss the problem of existence of minimizers on various functional spaces. We find non-existence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories can converge to curves with angles.\\nWe instead prove existence of minimizers for the \\\"time-reparameterized\\\" functional $$\\\\int \\\\| \\\\dot\\\\gamma(t) \\\\|\\\\sqrt{1+K_\\\\ga^2} dt$$ for all boundary conditions if initial and final directions are considered regardless to orientation. In this case, minimizers can present cusps (at most two) but not angles.
PB - Springer
UR - http://hdl.handle.net/1963/4107
U1 - 297
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Projective Reeds-Shepp car on $S^2$ with quadratic cost
JF - ESAIM COCV 16 (2010) 275-297
Y1 - 2010
A1 - Ugo Boscain
A1 - Francesco Rossi
AB - Fix two points $x,\\\\bar{x}\\\\in S^2$ and two directions (without orientation) $\\\\eta,\\\\bar\\\\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $$ J[\\\\gamma]=\\\\int_0^T g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t))+\\nK^2_{\\\\gamma(t)}g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t)) ~dt$$ along all smooth curves starting from $x$ with direction $\\\\eta$ and ending in $\\\\bar{x}$ with direction $\\\\bar\\\\eta$. Here $g$ is the standard Riemannian metric on $S^2$ and $K_\\\\gamma$ is the corresponding geodesic curvature.\\nThe interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1).\\nWe compute the global solution for this problem: an interesting feature is that some optimal geodesics present cusps. The cut locus is a stratification with non trivial topology.
UR - http://hdl.handle.net/1963/2668
U1 - 1429
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups
JF - J. Funct. Anal. 256 (2009) 2621-2655
Y1 - 2009
A1 - Andrei A. Agrachev
A1 - Ugo Boscain
A1 - Jean-Paul Gauthier
A1 - Francesco Rossi
AB - We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp\\\'s volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares.\\nWe then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.\\nOur study is motivated by some recent results about the cut and conjugate loci on these sub-Riemannian manifolds. The perspective is to understand how singularities of the sub-Riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation.
UR - http://hdl.handle.net/1963/2669
U1 - 1428
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Invariant Carnot-Caratheodory metrics on S3, SO(3), SL(2) and Lens Spaces
JF - SIAM J. Control Optim. 47 (2008) 1851-1878
Y1 - 2008
A1 - Ugo Boscain
A1 - Francesco Rossi
AB - In this paper we study the invariant Carnot-Caratheodory metrics on SU(2) \\\' S3,\\nSO(3) and SL(2) induced by their Cartan decomposition. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary function. We then prove that the metric\\ngiven on SU(2) projects on the so called Lens Spaces L(p; q). Also for Lens Spaces, we compute\\nthe cut loci (globally).
UR - http://hdl.handle.net/1963/2144
U1 - 2099
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -