Grégoire Charlot

Existence of Planar Curves Minimizing Length and Curvature

AbstractWe consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional

Optimal control of the Schrödinger equation with two or three levels

\smallint \sqrt {1 + K_\gamma ^2 ds}

Stability of Nonlinear Switched Systems on the Plane

, depending both on the length and curvature K. We fix starting and ending points as well as initial and final directions. For this functional we discuss the problem of existence of minimizers on various functional spaces. We find nonexistence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories may converge to curves with angles. We instead prove the existence of minimizers for the “time-reparametrized” functional

Resonance of minimizers for n-level quantum systems with an arbitrary cost

\smallint ||\dot \gamma (t)||\sqrt {1 + K_\gamma ^2 dt}

The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry

for all boundary conditions if the initial and final directions are considered regardless of orientation. In this case, minimizers may present cusps (at most two) but not angles.

Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces

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 Lagrangian and Legendrian singularities occur outside q0. As a consequence 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 certain way. This object is a three-dimensional surface that also has a natural structure of the wave front. The projection of the singular set of this “big wave front” on the 3-dimensional space is nothing else but the caustic. We show that in fact this big wave front is Legendre-stable at the origin.