We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term Elog E. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalized Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.

1 aBoscain, Ugo1 aPrandi, Dario1 aSeri, M. uhttps://doi.org/10.1080/03605302.2015.109576601202nas a2200133 4500008004100000245004900041210004800090300001200138490000700150520084200157100001300999700001801012856003801030 2015 eng d00aComplexity of Control-Affine Motion Planning0 aComplexity of ControlAffine Motion Planning a816-8440 v533 aIn this paper we study the complexity of the motion planning problem for control-affine systems. Such complexities are already defined and rather well understood in the particular case of nonholonomic (or sub-Riemannian) systems. Our aim is to generalize these notions and results to systems with a drift. Accordingly, we present various definitions of complexity, as functions of the curve that is approximated, and of the precision of the approximation. Due to the lack of time-rescaling invariance of these systems, we consider geometric and parametrized curves separately. Then, we give some asymptotic estimates for these quantities. As a byproduct, we are able to treat the long time local controllability problem, giving quantitative estimates on the cost of stabilizing the system near a nonequilibrium point of the drift.

1 aJean, F.1 aPrandi, Dario uhttps://doi.org/10.1137/13095079301517nas a2200121 4500008004100000245010200041210006900143260001000212520108600222653001901308100001801327856005001345 2014 en d00aGeometry and analysis of control-affine systems: motion planning, heat and Schrödinger evolution0 aGeometry and analysis of controlaffine systems motion planning h bSISSA3 aThis thesis is dedicated to two problems arising from geometric control theory, regarding control-affine systems $\dot q= f_0(q)+\sum_{j=1}^m u_j f_j(q)$, where $f_0$ is called the drift. In the first part we extend the concept of complexity of non-admissible trajectories, well understood for sub-Riemannian systems, to this more general case, and find asymptotic estimates. In order to do this, we also prove a result in the same spirit as the Ball-Box theorem for sub-Riemannian systems, in the context of control-affine systems equipped with the L1 cost. Then, in the second part of the thesis, we consider a family of 2-dimensional driftless control systems. For these, we study how the set where the control vector fields become collinear affects the diffusion dynamics. More precisely, we study whether solutions to the heat and Schrödinger equations associated with this Laplace-Beltrami operator are able to cross this singularity, and how its the presence affects the spectral properties of the operator, in particular under a magnetic Aharonov–Bohm-type perturbation.10acontrol theory1 aPrandi, Dario uhttp://urania.sissa.it/xmlui/handle/1963/747400468nas a2200121 4500008004100000245007300041210007000114260001700184300001600201490000700217100001800224856010400242 2014 eng d00aHölder equivalence of the value function for control-affine systems0 aHölder equivalence of the value function for controlaffine syste bEDP Sciences a1224–12480 v201 aPrandi, Dario uhttps://www.math.sissa.it/publication/h%C3%B6lder-equivalence-value-function-control-affine-systems00494nas a2200097 4500008004100000245012100041210006900162100001700231700001800248856013000266 2013 eng d00aSelf-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces0 aSelfadjoint extensions and stochastic completeness of the Laplac1 aBoscain, Ugo1 aPrandi, Dario uhttps://www.math.sissa.it/publication/self-adjoint-extensions-and-stochastic-completeness-laplace-beltrami-operator-conic-and