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{\textquoteright}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.

}, doi = {10.1080/03605302.2015.1095766}, url = {https://doi.org/10.1080/03605302.2015.1095766}, author = {Ugo Boscain and Dario Prandi and M. Seri} } @article {1305.5271, title = {Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces}, year = {2013}, doi = {10.1016/j.jde.2015.10.011}, author = {Ugo Boscain and Dario Prandi} } @article {2008, title = {Stability of planar switched systems: the nondiagonalizable case}, journal = {Commun. Pure Appl. Anal. 7 (2008) 1-21}, number = {SISSA;44/2006/M}, year = {2008}, doi = {10.3934/cpaa.2008.7.1}, url = {http://hdl.handle.net/1963/1857}, author = {Ugo Boscain and Moussa Balde} } @article {2006, title = {Stability of planar nonlinear switched systems}, number = {SISSA;04/2005/M}, year = {2006}, abstract = {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,$\infty$) {\textrightarrow} {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.}, url = {http://hdl.handle.net/1963/1710}, author = {Ugo Boscain and Gr{\'e}goire Charlot and Mario Sigalotti} } @inbook {2005, title = {A short introduction to optimal control}, booktitle = {Contr{\^o}le non lin{\'e}aire et applications: Cours donn{\'e}s {\`a} l\\\'{\'e}cole d\\\'{\'e}t{\'e} du Cimpa de l\\\'Universit{\'e} de Tlemcen / Sari Tewfit [ed.]. - Paris: Hermann, 2005}, number = {SISSA;10/2004/M}, year = {2005}, isbn = {2 7056 6511 0}, url = {http://hdl.handle.net/1963/2257}, author = {Ugo Boscain and Benedetto Piccoli} } @article {2002, title = {Stability of planar switched systems: the linear single input case}, journal = {SIAM J. Control Optim. 41 (2002), no. 1, 89-112}, number = {SISSA;72/00/M}, year = {2002}, publisher = {SIAM}, abstract = {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 {\texttimes} 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.}, doi = {10.1137/S0363012900382837}, url = {http://hdl.handle.net/1963/1529}, author = {Ugo Boscain} }