Optimal uniform observation time for mixed finite element waves
Fix a bounded measurable potential $a : [0,1] \to [0,\infty)$, with its pointwise values specified. For $N \ge 1$ put $h = 1/(N+1)$ and let $$M_h = \frac h4 \operatorname{tridiag}(1,2,1), \qquad K_h = \frac1h \operatorname{tridiag}(-1,2,-1), \qquad L_h = h \operatorname{diag}\bigl(a(h), \dots, a(Nh)\bigr),$$ all of size $N \times N$: the mixed finite element semi-discretisation of Castro and Micu of the wave equation $u'' - u_{xx} + a u = 0$ on $(0,1)$ with homogeneous Dirichlet conditions. For solutions $U = (u_1, \dots, u_N)^T$ of $M_h \ddot U + (K_h + L_h) U = 0$ with initial data $(U^0, U^1)$ let $$\mathcal{E}_h(U^0, U^1) = (U^0)^* (K_h + L_h) U^0 + (U^1)^* M_h U^1, \qquad \mathcal{O}_{h,T}(U) = \int_0^T \Bigl( \Bigl|\frac{u_1(t)}{h}\Bigr|^2 + \Bigl|\frac{\dot u_1(t)}{2}\Bigr|^2 \Bigr) dt,$$ and $$T_*(a) = \inf\bigl\{ T > 0 : \exists\, c(a,T) > 0 \ \forall N \ge 1 \ \forall (U^0, U^1),\ \mathcal{O}_{h,T}(U) \ge c(a,T)\, \mathcal{E}_h(U^0, U^1) \bigr\},$$ the constant being independent of the mesh. For the wave equation itself, $\int_0^T |u_x(t,0)|^2\, dt$ is bounded below by a constant times the energy of the initial data for every $T > 2$. Castro and Micu proved that $T_*(a)$ is finite: there are $T_0, \kappa_0 > 0$ independent of $h$ such that the inequality holds with $c = \kappa_0$ for every $T > T_0$.
Problem. Determine $T_*(a)$. In particular, is $T_*(a) = 2$ for every such potential?
Source: Castro and Micu (2025), after Theorem 1, who call the optimal value of the uniform observation time an open problem. Stated by AIM — Open Applied Problems as the question whether it equals 2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- AP, OC
- Keywords
- AIM Open Applied Problems, boundary observability, mixed finite elements, wave equation
References
@article{castro2025,
author = {Castro, Carlos and Micu, Sorin},
title = {A mixed finite elements approximation of inverse source problems for the wave equation with variable coefficients using observability},
journal = {Numer. Math.},
volume = {157},
number = {5},
year = {2025},
pages = {1847--1895},
doi = {10.1007/s00211-025-01489-0},
eprint = {2501.11352},
archivePrefix = {arXiv},
}
@misc{aim-mixed-wave-optimal-observation-time,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Optimal uniform observation time for mixed finite element waves}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/605-mixed-wave-optimal-observation-time.md},
note = {Entry 605 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
T_*(a) = 2 for every bounded measurable a >= 0: uniform observability holds for every T > 2 and fails for every T < 2..
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, 606-mixed-wave-observation/PROOF.md. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00696, "Optimal uniform observation time for mixed finite element waves", clopen.solutions.