Non-minimality of corners in subriemannian geometry

Abstract

We give a short solution to one of the main open problems in subriemannian geometry. Namely, we prove that length minimizers do not have corner-type singularities. With this result we solve Problem II of Agrachev's list, and provide the first general result toward the 30-year-old open problem of regularity of subriemannian geodesics.Comment: 11 pages, final versio

Optimal Strokes at Low Reynolds Number: A Geometric and Numerical Study of Copepod and Purcell Swimmers

“…Applying twice the normal stroke, we obtain the same displacementx than with the abnormal stroke but with a length strictly lower thanl. The contribution [18] proves that such abnormal trajectory with a corner is not C 0 −optimal in the unconstrained case. We were not able to prove the same result for our abnormal stroke taking into account the triangle constraint.…”

“…Notice that it provides in the (θ 1 , θ 2 )−plane the boundary of the physical domain for the Copepod model. A recent contribution [18], applying SR-geometry arguments, proves that such an abnormal curve with corners cannot be optimal (in [18] the authors consider problems in absence of state constraints). Introducing the efficiency term (see Def.…”

On Tangent Cones to Length Minimizers in Carnot--Carathéodory Spaces

“…We give a detailed proof of some facts about the blow-up of horizontal curves in Carnot-Carathéodory spaces. These results are crucially used in [6,7,10]. The proof of a fraction of these results was already sketched, in a special case, in [13,Section 3.2].…”

A numerical approach to the optimal control and efficiency of the copepod swimmer

“…Remark 2: A recent contribution [12] proves that a trajectory with a corner of this type cannot be optimal (not taking into account the state constraints). To analyze the first situation of Fig.2, the mechanical energy has to be used in relation with SR-geometry.…”