Quadratic Growth without Quadratic Control Regularization

Definitions and mathematical statement

Let $\Omega\subset{\mathbb{R}}^3$ be a bounded domain. The supplied record concerns distributed optimal control of an $H^1$-critical defocusing wave equation on $\Omega$. Here $H^1(\Omega)$ denotes the Sobolev space of square-integrable functions with square-integrable first weak derivatives. The parameter $\beta_2$ is the coefficient governing a quadratic $L^2$ control penalty in the objective; setting $\beta_2=0$ removes that penalty. The complete state equation, boundary and initial conditions, time interval, admissible controls, and remaining objective terms are not supplied.

For clarity, a quadratic-growth assertion for a reduced objective $J_0$, an admissible set $U_{\mathrm{ad}}$, and a local minimizer $\bar u$ has the general form \[ \exists c>0\ \exists\mathcal N\ni\bar u\quad \forall u\in U_{\mathrm{ad}}\cap\mathcal N, \qquad J_0(u)\ge J_0(\bar u)+c\|u-\bar u\|_X^2. \] Here a reduced objective means the objective evaluated at the state associated with a control. The control norm $\|\cdot\|_X$, the topology defining the neighborhood $\mathcal N$, and even the hypotheses ensuring that this reduced objective is well defined must be specified by the underlying control problem. This display is a definition template, not a reconstruction of the source's particular meaning of strong quadratic growth.

Question supplied by the record. For the stated distributed optimal-control problem, can the strong quadratic-growth property and its resulting stability estimates, described in the supplied assessment as established under positive quadratic control regularization $\beta_2>0$, be extended to $\beta_2=0$? The intended extension must retain the applicable hypotheses of that analysis, with any necessary modifications stated explicitly. Those hypotheses and the precise stability estimates are absent from the supplied record, so a fully quantified formulation of the intended extension remains unresolved.

Short English statement

The question asks whether strong quadratic growth and the associated stability estimates persist for the specified critical defocusing wave control problem after removing the quadratic $L^2$ control penalty. The supplied record labels this case open, but omits the full control problem and the exact growth and stability assertions needed for a precise theorem statement.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004630$, title: Quadratic Growth without Quadratic Control Regularization; exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. The record gives proposed year $2021$ and identifies no individual proposer.
• Supplied source citation: Challenges in Optimization with Complex PDE-Systems (2022), Oberwolfach Reports, pp. 419--506, DOI 10.4171/owr/2021/9. https://doi.org/10.4171/owr/2021/9. The supplied background describes this report as documenting workshop discussions and open problems.
• Supplied literature source, labeled "Oberwolfach report": https://doi.org/10.4171/owr/2021/9.
• Supplied literature source, labeled "Challenges in optimal control": https://ems.press/content/serial-article-files/46884?nt=1.
• Supplied literature assessment, dated exactly 2026-08-22: the cited optimal-control analysis requires a positive quadratic control regularization coefficient, and the case $\beta_2=0$ remains explicitly open. This is supplied dated curation derived from the OWR record, not an independently verified assessment; the record requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report for preservation of hypotheses, notation, and the open target. This is supplied verification metadata, not independent verification in this draft.