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–Carathéodory 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.

1 aBoscain, Ugo1 aCharlot, Grégoire1 aGhezzi, Roberta1 aSigalotti, Mario uhttps://doi.org/10.1007/s12220-011-9262-401439nas a2200181 4500008004300000245007000043210006800113260001300181300001200194490000700206520090200213100002501115700001701140700002301157700002001180700002101200856003601221 2010 en_Ud 00aTwo-dimensional almost-Riemannian structures with tangency points0 aTwodimensional almostRiemannian structures with tangency points bElsevier a793-8070 v273 aTwo-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.

1 aAgrachev, Andrei, A.1 aBoscain, Ugo1 aCharlot, Grégoire1 aGhezzi, Roberta1 aSigalotti, Mario uhttp://hdl.handle.net/1963/387001061nas a2200133 4500008004300000245009500043210006900138520060700207100002200814700001700836700002100853700001700874856003600891 2009 en_Ud 00aControllability of the discrete-spectrum Schrodinger equation driven by an external field0 aControllability of the discretespectrum Schrodinger equation dri3 aWe prove approximate controllability of the bilinear Schrodinger equation in the case in which the uncontrolled Hamiltonian has discrete nonresonant\\nspectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the\\nGalerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential controlled by suitable potentials.1 aChambrion, Thomas1 aMason, Paolo1 aSigalotti, Mario1 aBoscain, Ugo uhttp://hdl.handle.net/1963/254701551nas a2200121 4500008004300000245007900043210006900122520113900191100002501330700001701355700002101372856003601393 2008 en_Ud 00aA Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds0 aGaussBonnetlike formula on twodimensional almostRiemannian manif3 aWe consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and $Y$ are everywhere linearly independent, then they define a classical Riemannian metric on $M$ (the metric for which they are orthonormal) and they give to $M$ the structure of metric space. If $X$ and $Y$ become linearly dependent somewhere on $M$, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. They are special cases of rank-varying sub-Riemannian structures, which are naturally defined in terms of submodules of the space of smooth vector fields on $M$. Almost-Riemannian structures show interesting phenomena, in particular for what concerns the relation between curvature, presence of conjugate points, and topology of the manifold. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula.1 aAgrachev, Andrei, A.1 aBoscain, Ugo1 aSigalotti, Mario uhttp://hdl.handle.net/1963/186901180nas a2200109 4500008004300000245005200043210005000095520085100145100001700996700002101013856003601034 2007 en_Ud 00aHigh-order angles in almost-Riemannian geometry0 aHighorder angles in almostRiemannian geometry3 aLet X and Y be two smooth vector fields on a two-dimensional manifold M. If X and Y are everywhere linearly independent, then they define a Riemannian metric on M (the metric for which they are orthonormal) and they give to M the structure of metric space. If X and Y become linearly dependent somewhere on M, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula for domains with piecewise-C2 boundary. The main feature of such formula is the presence of terms that play the role of high-order angles at the intersection points with the set of singularities.1 aBoscain, Ugo1 aSigalotti, Mario uhttp://hdl.handle.net/1963/199500968nas a2200121 4500008004300000245005100043210005100094520060400145100001700749700002300766700002100789856003600810 2006 en_Ud 00aStability of planar nonlinear switched systems0 aStability of planar nonlinear switched systems3 aWe 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,∞) → {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.1 aBoscain, Ugo1 aCharlot, Grégoire1 aSigalotti, Mario uhttp://hdl.handle.net/1963/171001167nas a2200109 4500008004100000245010100041210006900142260001300211520077600224100002101000856003601021 2005 en d00aRegularity properties of optimal trajectories of single-input control systems in dimension three0 aRegularity properties of optimal trajectories of singleinput con bSpringer3 aLet q=f(q)+ug(q) be a smooth control system on a three-dimensional manifold. Given a point q 0 of the manifold at which the iterated Lie brackets of f and g satisfy some prescribed independence condition, we analyze the structure of a control function u(t) corresponding to a time-optimal trajectory lying in a neighborhood of q 0. The control turns out to be the concatenation of some bang-bang and some singular arcs. More general optimality criteria than time-optimality are considered. The paper is a step toward to the analysis of generic single-input systems affine in the control in dimension 3. The main techniques used are second-order optimality conditions and, in particular, the index of the second variation of the switching times for bang-bang trajectories.1 aSigalotti, Mario uhttp://hdl.handle.net/1963/479400867nas a2200121 4500008004100000245005700041210005000098260001800148520049700166100002500663700002100688856003600709 2003 en d00aOn the local structure of optimal trajectories in R30 alocal structure of optimal trajectories in R3 bSISSA Library3 aWe analyze the structure of a control function u(t) corresponding to an optimal trajectory for the system $\\\\dot q =f(q)+u\\\\, g(q)$ in a three-dimensional manifold, near a point where some nondegeneracy conditions are satisfied. The kind of optimality which is studied includes time-optimality. The control turns out to be the concatenation of some bang and some singular arcs. Studying the index of the second variation of the switching times, the number of such arcs is bounded by four.1 aAgrachev, Andrei, A.1 aSigalotti, Mario uhttp://hdl.handle.net/1963/161200397nas a2200097 4500008004100000245012200041210006900163260001000232100002100242856003600263 2003 en d00aSingle-Input Control Affine Systems: Local Regularity of Optimal Trajectories and a Geometric Controllability Problem0 aSingleInput Control Affine Systems Local Regularity of Optimal T bSISSA1 aSigalotti, Mario uhttp://hdl.handle.net/1963/534200481nas a2200121 4500008004300000245011600043210006900159260003000228100002000258700002400278700002100302856003600323 2002 en_Ud 00aThe passage from nonconvex discrete systems to variational problems in Sobolev spaces: the one-dimensional case0 apassage from nonconvex discrete systems to variational problems bMAIK Nauka/Interperiodica1 aBraides, Andrea1 aGelli, Maria Stella1 aSigalotti, Mario uhttp://hdl.handle.net/1963/3130