Trajectory existence from the unresolved subset of S_2 for k_0>4

Determine whether trajectories of the differential inclusion F exist from points in S_2\setminus\overline{S_+\cup S_-} when the first order at which the vector fields can produce different Lie derivatives satisfies k_0>4.

Background

Under assumptions (A1) and (A2), the paper defines S_2 as the subset of the singular set where the first k_0-1 common Lie derivatives vanish and the k_0-th Lie derivatives satisfy L_f{k_0}h<0<L_g{k_0}h. The main theorem gives necessary and sufficient existence criteria for k_0=2 and k_0=4.

For k_0>4, the theorem proves existence for starting points outside S_2\setminus\overline{S_+\cup S_-}, but does not settle points in that remaining subset. The unresolved issue concerns whether trajectories can exist from precisely those singular points not lying in the closure of the switching regions S_+ and S_-.

References

In this case, it remains an open problem whether there exist trajectories of $F$ starting from points in $S_2 \setminus \overline{S_+ \cup S_-}$.

— Differential inclusions and the chattering phenomenon  (2610.03561 - Ivanov, 2 Oct 2026) in Section 1, subsection “Main result and steps of the proof”