Existence of Additional Pontryagin Extremals for the Reinhardt Optimal Control Problem

Determine whether any Pontryagin extremals of the Reinhardt optimal control problem exist beyond the explicitly constructed family (the smoothed 6k+2-gons and the circle). Prove that no other extremals exist or classify all remaining possibilities; establishing nonexistence would complete a proof of the Reinhardt conjecture.

Background

The book constructs and analyzes a large explicit family of Pontryagin extremals for the Reinhardt optimal control problem: smoothed 6k+2‑gons (including the smoothed octagon) and the circle. It notes that these cover all currently found extremals.

However, the authors explicitly state that they do not have a proof that no other extremals exist, and they connect this gap to the overall resolution of the Reinhardt conjecture: proving nonexistence of any other extremals would complete the conjecture.