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JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS, Vol. 6, No. 3, 2000, 365-395 ON SUB-RIEMANNIAN CAUSTICS AND WAVE FRONTS
 

Summary: JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS, Vol. 6, No. 3, 2000, 365-395
ON SUB-RIEMANNIAN CAUSTICS AND WAVE FRONTS
FOR CONTACT DISTRIBUTIONS IN THE THREE-SPACE
A. A. AGRACHEV, G. CHARLOT, J. P. A. GAUTHIER, and V. M. ZAKALYUKIN
Abstract. In a number of previous papers of the first and third
authors, caustics, cut-loci, spheres, and wave fronts of a system of
sub-Riemannian geodesics emanating from a point q0 were studied.
It turns out that only certain special arrangements of classical La-
grangian and Legendrian singularities occur outside q0. As a conse-
quence of this, for instance, the generic caustic is a globally stable
object outside the origin q0.
Here we solve two remaining stability problems.
The first part of the paper shows that in fact generic caustics have
moduli at the origin, and the first module that occurs has a simple
geometric interpretation.
On the contrary, the second part of the paper shows a stability
result at q0. We define the "big wave front": it is the graph of the
multivalued function arclength wave-front reparametrized in a cer-
tain way. This object is a three-dimensional surface that also has a
natural structure of the wave front. The projection of the singular

  

Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)
Charlot, Grégoire - Institut Fourier, Université Joseph Fourier Grenoble-I

 

Collections: Engineering; Mathematics