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.

%B Communications in Partial Differential Equations %I Taylor & Francis %V 41 %P 32-50 %G eng %U https://doi.org/10.1080/03605302.2015.1095766 %R 10.1080/03605302.2015.1095766 %0 Journal Article %J SIAM Journal on Control and Optimization %D 2015 %T Complexity of Control-Affine Motion Planning %A Jean, F. %A Dario Prandi %XIn 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.

%B SIAM Journal on Control and Optimization %V 53 %P 816-844 %G eng %U https://doi.org/10.1137/130950793 %R 10.1137/130950793 %0 Thesis %D 2014 %T Geometry and analysis of control-affine systems: motion planning, heat and Schrödinger evolution %A Dario Prandi %K control theory %X This 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. %I SISSA %G en %U http://urania.sissa.it/xmlui/handle/1963/7474 %1 7576 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by dprandi@sissa.it (dprandi@sissa.it) on 2014-10-28T17:03:22Z No. of bitstreams: 1 thesis_SISSA.pdf: 2610057 bytes, checksum: b1f6802988c3d34412407d628c7c1963 (MD5) %0 Journal Article %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %T Hölder equivalence of the value function for control-affine systems %A Dario Prandi %B ESAIM: Control, Optimisation and Calculus of Variations %I EDP Sciences %V 20 %P 1224–1248 %G eng %R 10.1051/cocv/2014014 %0 Journal Article %D 2013 %T Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces %A Ugo Boscain %A Dario Prandi %G eng %R 10.1016/j.jde.2015.10.011