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 %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 %0 Journal Article %J Commun. Pure Appl. Anal. 7 (2008) 1-21 %D 2008 %T Stability of planar switched systems: the nondiagonalizable case %A Ugo Boscain %A Moussa Balde %B Commun. Pure Appl. Anal. 7 (2008) 1-21 %G en_US %U http://hdl.handle.net/1963/1857 %1 2361 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-09-28T09:15:19Z\\nNo. of bitstreams: 1\\n44-M.pdf: 3158552 bytes, checksum: c4fee14d81a9fcf026efd4448da88c75 (MD5) %R 10.3934/cpaa.2008.7.1 %0 Report %D 2006 %T Stability of planar nonlinear switched systems %A Ugo Boscain %A Grégoire Charlot %A Mario Sigalotti %X 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,∞) → {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. %B Discrete Contin. Dyn. Syst. 15 (2006) 415-432 %G en_US %U http://hdl.handle.net/1963/1710 %1 2441 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-01-18T08:35:31Z\\nNo. of bitstreams: 1\\nmath.OC0502361.pdf: 322404 bytes, checksum: e56f0d709d97e2e300e3cb9d4a629a1b (MD5) %0 Book Section %B Contrôle non linéaire et applications: Cours donnés à l\\\'école d\\\'été du Cimpa de l\\\'Université de Tlemcen / Sari Tewfit [ed.]. - Paris: Hermann, 2005 %D 2005 %T A short introduction to optimal control %A Ugo Boscain %A Benedetto Piccoli %B Contrôle non linéaire et applications: Cours donnés à l\\\'école d\\\'été du Cimpa de l\\\'Université de Tlemcen / Sari Tewfit [ed.]. - Paris: Hermann, 2005 %@ 2 7056 6511 0 %G en_US %U http://hdl.handle.net/1963/2257 %1 1990 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-18T11:30:24Z\\nNo. of bitstreams: 1\\nNotes-OptCont.pdf: 442775 bytes, checksum: 196e4d9cc52950dd18cf17feb0e89808 (MD5) %0 Journal Article %J SIAM J. Control Optim. 41 (2002), no. 1, 89-112 %D 2002 %T Stability of planar switched systems: the linear single input case %A Ugo Boscain %X We study the stability of the origin for the dynamical system $\\\\dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where A and B are two 2 × 2 real matrices with eigenvalues having strictly negative real part, $x\\\\in {\\\\mbox{{\\\\bf R}}}^2$, and $u(.):[0,\\\\infty[\\\\to[0,1]$ is a completely random measurable function. More precisely, we find a (coordinates invariant) necessary and sufficient condition on A and B for the origin to be asymptotically stable for each function u(.). The result is obtained without looking for a common Lyapunov function but studying the locus in which the two vector fields Ax and Bx are collinear. There are only three relevant parameters: the first depends only on the eigenvalues of A, the second depends only on the eigenvalues of B, and the third contains the interrelation among the two systems, and it is the cross ratio of the four eigenvectors of A and B in the projective line CP1. In the space of these parameters, the shape and the convexity of the region in which there is stability are studied. %B SIAM J. Control Optim. 41 (2002), no. 1, 89-112 %I SIAM %G en %U http://hdl.handle.net/1963/1529 %1 2634 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T13:03:40Z (GMT). No. of bitstreams: 0\\n Previous issue date: 2000 %R 10.1137/S0363012900382837