Optimal constant in the quartic functional inequality

Determine the optimal constant in the inequality $\lambda\int_0^1(\varphi')^4\,dt\leq\int_0^1(\varphi\varphi'')^2\,dt$ for all functions $\varphi$ in the specified space $W^{3,1}_0(0,1)$.

Background

The paper defines a threshold constant λ∗>0\lambda^*>0 as the supremum of the parameters for which the inequality holds on W03,1(0,1)W^{3,1}_0(0,1). This constant determines a controllability transition for a scalar-input system involving a competition between two quartic bad brackets: the system is not W−1,∞W^{-1,\infty}-STLC for λ≤λ∗\lambda\leq\lambda^* and is smoothly-STLC for λ>λ∗\lambda>\lambda^*.

The authors establish only the bounds 1/9≤λ∗≤22/91/9\leq\lambda^*\leq22/9 and state that the optimal value is not known. Determining it would sharpen the exact threshold separating the two controllability regimes.

References

Both the lower and upper bounds $1/9 \leq \lambda* \leq 22/9$ are probably not sharp. Nevertheless, to the authors' knowledge, the optimal constant such that ineq:main holds is unknown and does not easily follow from known optimal Sobolev embeddings constants.

— Some quartic control results for scalar-input systems  (2608.28582 - Beauchard et al., 28 Aug 2026) in Remark following Lemma in Section “A functional inequality” (Section “Bad-bad competition”)