Abstract
We study the sub-Riemannian structure determined by a left-invariant distribution of rank 2 on a step 3 Carnot group of dimension 5. We prove the conjectured cut times of Yu. Sachkov for the sub-Riemannian Cartan problem. Along the proof, we obtain a comparison with the known cut times in the sub-Riemannian Engel group, and a sufficient (generic) condition for the uniqueness of the length minimizer between two points. Hence we reduce the optimal synthesis to solving a certain system of equations in elliptic functions.
Highlights
The sub-Riemannian Cartan group C is the nilpotent model for all sub-Riemannian problems with growth vector (2, 3, 5)
A geometric description of the optimal control problem in the sub-Riemannian Cartan group can be given in terms of the generalized Dido problem: Given two points a, b ∈ R2 and a fixed “shoreline”, i.e., a curve γconnecting b and a, fix a desired oriented area S ∈ R and a desired center of mass c ∈ R2, see Figure 1
A closely related problem is that of optimal control in the sub-Riemannian Engel group E with the growth vector (2, 3, 4)
Summary
In both the Cartan and Engel groups, application of the PMP leads to a complete description of the geodesics. The normal extremal trajectories in both the Cartan and Engel cases project to Euler elasticae in the plane. Our main goal is to complete the second step and obtain the cut times in the Cartan case
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