Multidimensional Null Controllability for Bilinear Evolution Equations

Definitions and mathematical statement

The supplied record asks for controllability results for the formal bilinear evolution equation \[ \frac{{\,\mathrm{d}} u}{{\,\mathrm{d}} t}+Au+p(t)Bu=0. \] Here $u$ denotes the evolving state, $p$ the time-dependent control, and $A,B$ the operators appearing in the supplied equation. Their state spaces, domains, regularity assumptions and admissible control classes are not specified in the record.

The requested research targets are:

1. Establish the property called $j$-null controllability in spatial dimensions $d\in\{2,3\}$, using strategies other than the moment method. The original excerpt motivates this request by saying that use of the moment method restricts the cited results to one-dimensional or radial settings. It refers specifically to results relying on an unavailable condition labelled $(3)$.
2. Establish controllability for a Fokker--Planck equation subject to perfectly reflecting boundary conditions.

Unresolved formulation. The record does not define $j$, $j$-null controllability, the spatial domain, initial and terminal states, the control action, or the time and neighborhood quantifiers. In particular, it does not determine whether the desired property is local or global, or whether the target is zero, an eigenstate trajectory, or another specified state. The Fokker--Planck equation, its coefficients, its control mechanism, the precise reflecting boundary operator and its desired controllability property are also absent. Thus the two requests above are research targets, not complete quantified mathematical assertions. No particular realization of these missing data is imposed here.

Qualifications from the original excerpt. The phrase "such boundary conditions" has no explicit antecedent in the supplied original excerpt; the Fokker--Planck and perfectly reflecting boundary formulation occurs in the supplied clean statement. The unavailable condition $(3)$ is said to include a nonvanishing condition, abbreviated NVC. The surrounding discussion also mentions asymptotic lower estimates, abbreviated AB, for an appropriate unspecified $q$, spaces $X$, and an algorithm producing potential functions $\mu$. None of these hypotheses or constructions is supplied sufficiently to impose it on the requested multidimensional result.

The excerpt then presents a neighboring one-dimensional Schrödinger controllability result labelled Theorem 4 and attributed to "A.-B., Urbani 2019--2020." Its displayed reference trajectory and terminal neighborhood are \[ \psi_1(t,x)=e^{-i\lambda_1t}\varphi_1(x) \quad(t\geq0,\ x\in[0,1]), \qquad R_T=\left\{u_f\in S\cap D(A): \|u_f-\psi_1(T)\|_{H^2}<\delta\right\}. \] The excerpt reports that, for a given $T>0$ and an admissible $\mu$, there exist $\delta>0$ and a map $\Gamma:R_T\to L^2(0,T)$ with $\Gamma(\psi_1(T))=0$, such that every $u_f\in R_T$ is reached at time $T$ from $u_0=\varphi_1=\psi_1(0)$ with control $p=\Gamma(u_f)$. However, the admissibility notation for $\mu$ and the differentiability designation for $\Gamma$ are corrupted, and $S,A,\lambda_1,\varphi_1$ and the Schrödinger equation's boundary conditions are undefined here. This is supplied contextual material, not a complete theorem adopted as a hypothesis or a definition of the requested $j$-null controllability.

Short English statement

The record requests methods beyond the moment method for $j$-null controllability in two and three spatial dimensions, together with controllability for a Fokker--Planck equation with perfectly reflecting boundaries. Essential definitions, hypotheses and quantifiers are missing, so a precise theorem cannot yet be extracted. The supplied open-status label and verification claims are recorded as source metadata; this statement-only draft requires human mathematical review.

Sources

• Local UnsolvedMath registry record, permanent ID $30005730$, source problem code catalogue entry, title "Multidimensional Null Controllability for Bilinear Evolution Equations," in the supplied Oberwolfach Reports Open Problems collection. Project registry references: data/unsolvedmath/problems.json and catalog/problems.json. This draft uses the supplied complete record; it does not claim an independent inspection of those files.
• Supplied primary report citation: Control Methods in Hyperbolic PDEs (2024), Oberwolfach Reports, pp. 2941--3011, DOI https://doi.org/10.4171/owr/2023/52. The record gives proposed year 2023 and also labels this reference "Oberwolfach Report 52/2023." These supplied year designations are retained without independently resolving their relationship. No individual proposer is identified.
• Supplied literature source: Boundary controllability for degenerate or singular hyperbolic equations, 2025, https://doi.org/10.1007/s00245-025-10236-8.
• Supplied literature source: Oberwolfach Report 52/2023, https://doi.org/10.4171/owr/2023/52.
• Supplied literature assessment dated exactly 2026-08-22: the record labels the problem open, says the report leaves $j$-null controllability under other two- and three-dimensional boundary strategies open, and says later work establishes special one-dimensional degenerate cases rather than the extracted general question. This is supplied dated curation, not an independently verified assessment of the literature or current problem status.
• Supplied statement-verification metadata labels the statement corrected_verified and reports comparison with the official Oberwolfach report. That verification claim is part of the supplied record and was not independently checked here. The neighboring Theorem 4 is attributed only to "A.-B., Urbani 2019--2020"; no separate bibliographic reference or expansion of the abbreviated name is supplied.