Normality of infinite chattering extremals

Determine whether Case C, in which the boundary-contact set consists of infinitely many isolated contacts accumulating at a limiting contact, necessarily implies an abnormal extremal with multiplier \(\lambda=0\), and whether Case C can occur only for a measure-zero set of boundary values.

Background

The paper studies time-optimal chain-of-integrators, interpreted as elevator-control systems, subject to higher-order state constraints. In Case C, the contact set with the state-constraint boundary contains infinitely many isolated contact points converging to a limiting point; this behavior is identified as chattering.

The authors establish that Case C requires the degree condition $k-s>1$, but they do not prove whether genuine infinite chattering is compatible with normal extremals. They explicitly raise the unresolved issue of whether infinite chattering is instead a singular or abnormal phenomenon and conjecture that it can occur only when $\lambda=0$, for a specific measure-zero set of boundary values.