01230nas a2200157   4500008004100000245009000041210006900131260002100200300001000221490000700231520073600238100001700974700001800991700001301009856005001022       2016                        eng d00aSpectral analysis and the Aharonov-Bohm effect on certain almost-Riemannian manifolds0 aSpectral analysis and the AharonovBohm effect on certain almostR  bTaylor & Francis  a32-500 v413 a<p>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.</p>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 a<p>In 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.</p>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/1963/747400464nas a2200121   4500008004100000245007300041210007000114260001700184300001600201490000700217100001800224856010000242       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://math.sissa.it/publication/h%C3%B6lder-equivalence-value-function-control-affine-systems00490nas a2200097   4500008004100000245012100041210006900162100001700231700001800248856012600266       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://math.sissa.it/publication/self-adjoint-extensions-and-stochastic-completeness-laplace-beltrami-operator-conic-and