NCSU Differential Equations/Nonlinear Analysis Seminars

Location and Time: SAS 4201, Wednesday 15:00 -16:00

Organizers: Lorena Bociu, Patrick Combettes, Ryan Murray, and Khai T. Nguyen

NEXT TALK

Tuesday, April 28, 15:00-16:00, SAS 4201

Speaker: Franco Rampazzo, University of Padova, Italy
Title: Vector fields with low regularity: From commutativity to controllability
Abstract: Commutativity of the integral trajectories of two vector fields f, g is a trivial property for constant vector fields.However, it is well-known that such a commutativity is verified also for non-constant vector fields as soon as their Lie bracket [f,g](x) := ∇g(x) · f(x) − ∇f(x) · g(x) vanishes identically. On the other hand, local controllability for a non-linear control system (linear in the controls) consists in the fact that, for every (sufficiently small) t > 0, the points reached at a time s ≤ t by the trajectories of the system starting from a point x∗ form a full neighborhood of x∗. Rashevskii-Chow’s theorem states that the so-called rank-condition, namely the fact that iterated Lie brackets of vector fields of the system generate the whole tangent space, is sufficient for local controllability. These results are classically obtained under strong hypotheses of regularity. In this talk I will illustrate how these theorems (and other similar ones) can be extended to a non-smooth setting by means of a generalized notion of (set-valued) Lie bracket. Besides being desirable for technological applications, such extension might constitute a starting point for a non-smooth version of sub-Riemannian geometry.

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