Optimality of singular curves in the problem of a car with \(n\) trailers (Q2519227)

The mathematical model of a car with \(n\) trailers is one of the first nontrivial examples of a nonholonomic systems in robot technology. Initially, the stabilization problem of this control system was mainly considered. Later on, it was shown that the system is described by a Goursat 2-distribution on the manifold \(\mathbb R^2 \times (S^1)^{(n+1)}\) and that the local classification of this 2-distribution and an arbitrary Goursat distribution of corank \((n+1)\) is the same problem. In particular, every singularity of an arbitrary Goursat distribution of corank \((n+1)\) can be realized as a certain singular configuration of a car with \(n\) trailers. The singular configurations of the car with \(n\) trailers are described by the following simple geometric condition: at least one of the trailers (except for the last one) is located at a right angle to the previous. The curves in \(\mathbb R^2 \times (S^1)^{(n+1)}\) defined by this condition are called \textit{singular curves}. M. I. Zelikin conjectured that singular curves can be described as singular trajectories yielding the minimum in the time-optimal control problem with constant linear and bounded angular velocities of the car. By explicitly constructing the field of extremals in a neighborhood of a singular trajectory, the optimality (strong minimality) of small parts of one of the \(n-1\) possible types of singular curves was proved. In the present paper, we prove that a sufficiently small part of \(any\) first-order singular trajectory yields a weak minimum. Moreover, an \(arbitrary\) singular curve is a singular trajectory, and any (not necessarily small) part of this trajectory yields a weak minimum of the problem. Note, that if a singular control is a boundary control, then the minimum is simultaneously strong.