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 consider the Carnot{\textendash}Carath{\'e}odory distance canonically associated with an almost-Riemannian structure and study the problem of Lipschitz equivalence between two such distances on the same compact oriented surface. We analyze the generic case, allowing in particular for 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 characterization of the Lipschitz equivalence class of an almost-Riemannian distance in terms of a labeled graph associated with it.

}, issn = {1559-002X}, doi = {10.1007/s12220-011-9262-4}, url = {https://doi.org/10.1007/s12220-011-9262-4}, author = {Ugo Boscain and Gr{\'e}goire Charlot and Roberta Ghezzi and Mario Sigalotti} } @article {2011, title = {The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry}, journal = {Journal of Dynamical and Control Systems }, volume = {17 }, number = {arXiv:1009.2612;}, year = {2011}, pages = {141-161}, publisher = {Springer}, abstract = {We study the tangential case in 2-dimensional almost-Riemannian geometry. We\\r\\nanalyse the connection with the Martinet case in sub-Riemannian geometry. We\\r\\ncompute estimations of the exponential map which allow us to describe the\\r\\nconjugate locus and the cut locus at a tangency point. We prove that this last\\r\\none generically accumulates at the tangency point as an asymmetric cusp whose branches are separated by the singular set.

}, doi = {10.1007/s10883-011-9113-4}, url = {http://hdl.handle.net/1963/4914}, author = {Bernard Bonnard and Gr{\'e}goire Charlot and Roberta Ghezzi and Gabriel Janin} } @article {2010, title = {Existence of planar curves minimizing length and curvature}, journal = {Proc. Steklov Inst. Math. 270 (2010) 43-56}, number = {arXiv.org;0906.5290v2}, year = {2010}, publisher = {Springer}, abstract = {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.}, doi = {10.1134/S0081543810030041}, url = {http://hdl.handle.net/1963/4107}, author = {Ugo Boscain and Gr{\'e}goire Charlot and Francesco Rossi} } @article {boscain2010normal, title = {A normal form for generic 2-dimensional almost-Riemannian structures at a tangency point}, journal = {arXiv preprint arXiv:1008.5036}, year = {2010}, author = {Ugo Boscain and Gr{\'e}goire Charlot and Roberta Ghezzi} } @article {2010, title = {Two-dimensional almost-Riemannian structures with tangency points}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire }, volume = {27}, number = {arXiv.org;0908.2564v1}, year = {2010}, pages = {793-807}, publisher = {Elsevier}, abstract = {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.

}, doi = {10.1016/j.anihpc.2009.11.011}, url = {http://hdl.handle.net/1963/3870}, author = {Andrei A. Agrachev and Ugo Boscain and Gr{\'e}goire Charlot and Roberta Ghezzi and Mario Sigalotti} } @article {2006, title = {Stability of planar nonlinear switched systems}, number = {SISSA;04/2005/M}, year = {2006}, abstract = {We consider the time-dependent nonlinear system ˙ q(t) = u(t)X(q(t)) + (1 - u(t))Y (q(t)), where q ∈ R2, X and Y are two smooth vector fields, globally asymptotically stable at the origin and u : [0,$\infty$) {\textrightarrow} {0, 1} is an arbitrary measurable function. Analysing the topology of the set where X and Y are parallel, we give some sufficient and some necessary conditions for global asymptotic stability, uniform with respect to u(.). Such conditions can be verified without any integration or construction of a Lyapunov function, and they are robust under small perturbations of the vector fields.}, url = {http://hdl.handle.net/1963/1710}, author = {Ugo Boscain and Gr{\'e}goire Charlot and Mario Sigalotti} } @article {2005, title = {Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy}, journal = {Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 957-990}, number = {SISSA;56/2004/M}, year = {2005}, abstract = {We apply techniques of subriemannian geometry on Lie groups and of optimal synthesis on 2-D manifolds to the population transfer problem in a three-level quantum system driven by two laser pulses, of arbitrary shape and frequency. In the rotating wave approximation, we consider a nonisotropic model i.e. a model in which the two coupling constants of the lasers are different. The aim is to induce transitions from the first to the third level, minimizing 1) the time of the transition (with bounded laser amplitudes),\\n2) the energy of lasers (with fixed final time). After reducing the problem to real variables, for the purpose 1) we develop a theory of time optimal syntheses for distributional problem on 2-D-manifolds, while for the purpose 2) we use techniques of subriemannian geometry on 3-D Lie groups. The complete optimal syntheses are computed.}, url = {http://hdl.handle.net/1963/2259}, author = {Ugo Boscain and Thomas Chambrion and Gr{\'e}goire Charlot} } @article {2004, title = {Resonance of minimizers for n-level quantum systems with an arbitrary cost}, journal = {ESAIM COCV 10 (2004) 593-614}, number = {SISSA;58/2003/M}, year = {2004}, publisher = {EDP Sciences}, abstract = {We consider an optimal control problem describing a laser-induced population transfer on a $n$-level quantum system.\\nFor a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for $n=2$ and $n=3$): instead of looking for minimizers on the sphere $S^{2n-1}\\\\subset\\\\C^n$ one is reduced to look just for minimizers on the sphere $S^{n-1}\\\\subset \\\\R^n$. Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.}, doi = {10.1051/cocv:2004022}, url = {http://hdl.handle.net/1963/2910}, author = {Ugo Boscain and Gr{\'e}goire Charlot} }