9,228 research outputs found
Output Regulation for Systems on Matrix Lie-group
This paper deals with the problem of output regulation for systems defined on
matrix Lie-Groups. Reference trajectories to be tracked are supposed to be
generated by an exosystem, defined on the same Lie-Group of the controlled
system, and only partial relative error measurements are supposed to be
available. These measurements are assumed to be invariant and associated to a
group action on a homogeneous space of the state space. In the spirit of the
internal model principle the proposed control structure embeds a copy of the
exosystem kinematic. This control problem is motivated by many real
applications fields in aerospace, robotics, projective geometry, to name a few,
in which systems are defined on matrix Lie-groups and references in the
associated homogenous spaces
Flows on quaternionic-Kaehler and very special real manifolds
BPS solutions of 5-dimensional supergravity correspond to certain gradient
flows on the product M x N of a quaternionic-Kaehler manifold M of negative
scalar curvature and a very special real manifold N of dimension n >=0. Such
gradient flows are generated by the `energy function' f = P^2, where P is a
(bundle-valued) moment map associated to n+1 Killing vector fields on M. We
calculate the Hessian of f at critical points and derive some properties of its
spectrum for general quaternionic-Kaehler manifolds. For the homogeneous
quaternionic-Kaehler manifolds we prove more specific results depending on the
structure of the isotropy group. For example, we show that there always exists
a Killing vector field vanishing at a point p in M such that the Hessian of f
at p has split signature. This generalizes results obtained recently for the
complex hyperbolic plane (universal hypermultiplet) in the context of
5-dimensional supergravity. For symmetric quaternionic-Kaehler manifolds we
show the existence of non-degenerate local extrema of f, for appropriate
Killing vector fields. On the other hand, for the non-symmetric homogeneous
quaternionic-Kaehler manifolds we find degenerate local minima.Comment: 22 page
Maximal representations of complex hyperbolic lattices in SU(m,n)
Let denote a lattice in , with greater than 1. We show
that there exists no Zariski dense maximal representation with target
if . The proof is geometric and is based on the study of the rigidity
properties of the geometry whose points are isotropic -subspaces of a
complex vector space endowed with a Hermitian metric of signature
and whose lines correspond to the dimensional subspaces of on
which the restriction of has signature .Comment: 41 pages, 2 figures, accepted for pubblication in GAF
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