Existence of a uniform positive lower bound on entry-exit durations for general planar models

Establish, for the more general planar fast-slow systems considered in Theorem \ref{teo:general}, the existence of a positive lower bound \tau_{\min} such that any strategy reaching the target and containing an entry-exit process of duration below \tau_{\min} cannot have total time arbitrarily close to the optimal infimum.

Background

The existence proof for an optimal strategy in the toy model relies on a lemma asserting that near-optimal target-reaching strategies cannot contain entry-exit processes with arbitrarily small positive durations. This lower bound prevents minimizing sequences from collapsing toward the origin through infinitely many entry-exit processes in finite total slow time.

When extending the argument to more general planar fast-slow systems, the authors explain that compactness of the strategy spaces and the remaining existence argument can be established, but the analogue of the positive lower-bound assumption depends on the particular model. Consequently, extending the full optimal-strategy existence theorem requires proving this property separately for each broader class or identifying general conditions that guarantee it.

References

Importantly, note that an additional assumption regarding $\tau_{\min}$ (recall Lemma \ref{lemma:tau_min}) is necessary. Indeed, we are not able to prove such result in this general setting, since it depends on the particular structure of the model under consideration.

— On the optimal control of entry-exit phenomena in planar fast-slow dynamical systems  (2609.25747 - Borsotti et al., 22 Sep 2026) in Section 5, item 5, final paragraph of “Extension to more general planar fast-slow models”