Sharp equality of the auxiliary-function variational bound on minimal periods

Determine whether, for an autonomous ordinary differential equation ẋ = f(x) on a domain Ω ⊂ ℝ^n and for all periodic solutions x(t) with period T, the inequality max over periodic orbits of (2π/T)^2 ≤ inf over all C > 0 and functions φ,V ∈ C^1(Ω,ℝ) satisfying C|φ(x)|^2 − |ℒ_f φ(x)|^2 + ℒ_f V(x) ≥ 0 for all x ∈ Ω can be written as an equality, i.e., whether max_{periodic orbits}(2π/T)^2 = inf_{C>0, φ,V ∈ C^1(Ω)}{C : C|φ(x)|^2 − |ℒ_f φ(x)|^2 + ℒ_f V(x) ≥ 0 ∀ x ∈ Ω}.

Background

The paper develops a Lyapunov-like framework for lower-bounding periods of periodic orbits in autonomous ODEs via auxiliary functions φ and V that satisfy a global pointwise inequality C|φ|^2 − |ℒ_f φ|^2 + ℒ_f V ≥ 0. This yields T ≥ 2π/√C for any periodic orbit on which φ averages to zero, and can be optimized using sum-of-squares and semidefinite programming to produce rigorous bounds.

The authors raise the question of sharpness: whether the best possible constant C obtainable by optimizing over φ and V (subject to the inequality) exactly matches the supremum of (2π/T)^2 over all periodic orbits, turning their inequality into an equality. Establishing this would imply that increasing polynomial degrees in the SOS approach could yield arbitrarily sharp bounds.