Uniqueness of the optimal strategy in the general target regime

Determine whether the optimal strategy is unique when the target threshold satisfies y^\textnormal{stop} > y_\infty^\textnormal{cycle}, thereby resolving the authors’ conjecture that the optimal strategy is not unique in this regime.

Background

The paper studies minimum-time control of repeated entry-exit processes in a planar fast-slow toy model. An admissible strategy is represented by a sequence of parameters determining the exit point at each entry-exit process, and the objective is to reach a prescribed target \hat{z} in minimum slow time.

The authors analytically establish uniqueness in the regime where y_{\infty,1} \le -\sqrt{u_m/u_M}\,\Pi_{\textnormal{fast}}^{-1}(y^\textnormal{stop}) \le y^\textnormal{stop} \le y_\infty^\textnormal{cycle}. For the complementary regime y^\textnormal{stop} > y_\infty^\textnormal{cycle}, however, they neither prove nor disprove uniqueness and conjecture that multiple optimal strategies exist. A counterexample would resolve the conjecture in the non-uniqueness direction.