Lorenzo D'Arca
(Sapienza University of Rome)
Title: Poincaré Inequalities and Hardy Improvements for Some Sub-Elliptic Operators
Abstract: We characterize the sharp constant and maximizing functions for weighted Poincaré inequalities. These results are used to derive \(L^p\) generalizations of the Brezis-Vazquez improvement of Hardy's inequality. Our techniques apply in Euclidean space, Heisenberg group, Carnot groups, and for operators such as the Baouendi-Grushin and Heisenberg-Greiner operators.
Marco Di Marco
(Università di Padova)
Title: Submanifolds with boundary and Stokes' Theorem in Heisenberg groups
Abstract: We present an analogue of Stokes’ Theorem in the sub-Riemannian Heisenberg setting. The main “ingredients” are H-regular submanifolds with boundary, that we introduce, and the Rumin’s complex of differential form in Heisenberg groups. The result is obtained by approximation with a sequence of C^1 submanifolds with boundary.
Joint work with Antoine Julia, Sebastiano Nicolussi Golo and Davide Vittone.
Zhengping Ji
(Laboratoire Jacques-Louis Lions, Sorbonne University, Paris)
Title: Continuation Method for Motion Planning of Control-Affine Systems via Regularization
Abstract: We investigate the motion planning problem of control-affine systems using a regularized homotopy continuation method. We prove that, when the intrinsic second order differential of the endpoint map is positive definite, the regularized solutions are well posed and converge to a well-posed solution to the motion planning problem. In particular, the endpoints of the trajectories derived from the regularized solutions converge to the desired target point. This provides a way to design the steering control in the presence of singular controls when the classical continuation method is not applicable. The effectiveness of the regularization is illustrated by numerical experiments on the rolling systems.
Michele Motta
(SISSA, Trieste)
Title: Asymptotics of motion planning complexity for control-affine systems
Abstract: We consider the complexity of motion planning for 3D control-affine systems satisfying the strong Hörmander condition (i.e., such that the underlying linear-control system is controllable). We provide precise asymptotics for this quantity as ε tends to 0, which encodes the cost required to track a
reference trajectory at a given precision ε. Our result extends to the control-affine case some well-known results for the control-linear case, showing precisely how the relative position of the reference trajectory and the trajectories of the drift appearing in the system yield different types of asymptotics.
Luca Nalon
(Université de Fribourg)
Title: Euclidean rectifiability of sub-Finsler spheres in free-Carnot groups of step 2
Abstract: We consider 2-step free-Carnot groups equipped with sub-Finsler distances. Without requiring any smoothness assumption on the norm, we prove that the boundaries of sub-Finsler balls are codimension-one rectifiable from the Euclidean viewpoint. The result is obtained by studying how the Lipschitz constant for the sub-Finsler distance function behaves near abnormal geodesics.
Alessandro Socionovo
(Laboratoire Jacques-Louis Lions, Sorbonne University, Paris)
Title: Non-optimality of non-smooth strictly abnormal extremals in sub-Riemannian geometry
Abstract: We study a class of examples of sub-Riemannian manifolds on $\mathbb{R}^3$ having non-smooth strictly abnormal extremals, and we prove that such abnormal curves cannot be length-minimizing if they are of class $C^1\setminus C^2$. In particular, our result prove that a minimality theorem providing the existence of a $C^2$ geodesic within the same class of examples is sharp.
Lucia Tessarolo
(Laboratoire Jacques-Louis Lions, Sorbonne University, Paris)
Title: On the Schrödinger Evolution on the Characteristic Foliation
Abstract: Let M be a 3D contact sub-Riemannian manifold and S a surface embedded in M. We study the Schrödinger evolution of a particle constrained on the characteristic foliation of S. Specifically, we define the Schrödinger operator on each leaf \ell as \Delta_\ell u=div_\mu \nabla_\ell u, where \nabla_\ell is the Euclidean gradient along the leaf and \mu is the surface measure inherited from the Popp volume using the sub-Riemannian normal to the surface. We analyze whether the operator \Delta_\ell at a characteristic point p is in the limit point or limit circle case, depending on a curvature-like invariant at p. Additionally, we study the self-adjoint extensions that allow communication between different leaves.