My main academic focus is sub-Riemannian geometry and its related fields, such as control theory and contact geometry. I have grown fond of sub-Riemannian geometry because of its multidisciplinary aspects: it is an abstract subject, all the while arising from some very practical and physical problems. It is a prime example of how - through geometry - one can model physical phenomena with an intellectual and practical motivation.
During my PhD, I've been studying properties induced by sub-Riemannian 3-manifolds on embedded surfaces, with methods inspired by metric geometry and stochastic systems. Lately, I have been interested in the fundamental theory of control systems, particularly in approximate and local controllability.
Abstract. We are concerned with stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds. Employing the Riemannian approximations to the sub-Riemannian manifold which make use of the Reeb vector field, we obtain a second order partial differential operator on the surface arising as the limit of Laplace-Beltrami operators. The stochastic process associated with the limiting operator moves along the characteristic foliation induced on the surface by the contact distribution. We show that for this stochastic process elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices. We illustrate the results with examples and we identify canonical surfaces in the Heisenberg group, and in SU(2) and SL(2,ℝ)equipped with the standard sub-Riemannian contact structures as model cases for this setting. Our techniques further allow us to derive an expression for an intrinsic Gaussian curvature of a surface in a general three-dimensional contact sub-Riemannian manifold.
D. C. and M. Sigalotti Approximately controllable bilinear control systems are controllable, arXiv e-prints (April 2021); arXiv:2104.03375.
Abstract. We show that a bilinear control system is approximately controllable if and only if it is controllable in \(\mathbb{R}^n\setminus\{0\}\) We approach this problem by looking at the foliation made by the orbits of the system, and by showing that there does not exist a codimension-one foliation in \(\mathbb{R}^n\setminus\{0\}\) with dense leaves that are everywhere transversal to the radial direction. The proposed geometric approach allows to extend the results to homogeneous systems that are angularly controllable.
D. Barilari, U. Boscain and D. C. On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds, arXiv e-prints (September 2020); arXiv:2009.11748.
Abstract. Given a surface S in a 3D contact sub-Riemannian manifold M, we investigate the metric structure induced on S by M, in the sense of length spaces. First, we define a coefficient at characteristic points that determines locally the characteristic foliation of S. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.
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