Complementarity Conditions in Nonlinear PDE Control

Definitions and mathematical statement

Let $\Omega$ denote the spatial domain and let $I=(0,T)$, where $T>0$. Write $z$ for the state, $g$ for the control, and $\dot z$ for the time derivative of $z$. The supplied control problem has state inclusion \[ 0\in-\Delta z-g-\sigma\Delta\dot z+\partial|\dot z| \quad\text{in }H^{-1}(\Omega), \qquad z(0)=0, \] and objective to be minimized \[ J(z,g)=j_1(z)+j_2(z(T))+ \frac12\|g\|_{H^1(I;L^2(\Omega))}^{2}. \] Here $H^1(I;L^2(\Omega))$ is the space of square-integrable $L^2(\Omega)$-valued functions with square-integrable weak time derivative. The notation $H^{-1}(\Omega)$ denotes a negative Sobolev space; its precise convention and the realization of $\Delta$ must be specified together with the spatial boundary conditions.

For a locally optimal pair $(\bar z,\bar g)$, put $\bar y=\dot{\bar z}$. The source introduces a subgradient $\bar q\in\partial|\bar y|$ and an adjoint variable $p$. For the scalar absolute value, the convex subdifferential is \[ \partial|r|= \begin{cases} \{1\},&r>0,\\ [-1,1],&r=0,\\ \{-1\},&r<0. \end{cases} \] This describes the scalar notation; the source does not specify whether its state inclusion uses the pointwise graph or a subdifferential of an integral functional on a particular function space.

Question as stated in the source. Is \[ \langle p,1-|\bar q|\rangle=0 \] a necessary optimality condition at a locally optimal pair for the displayed control problem?

Unresolved qualifications. The record does not specify the dimension or regularity of $\Omega$, spatial boundary conditions, the value or allowed range of $\sigma$, the state space, admissible controls beyond the displayed norm, hypotheses on $j_1,j_2$, or the topology of local optimality. It also omits the adjoint equation, adjoint boundary and terminal conditions, the spaces and integration region defining $\langle\cdot,\cdot\rangle$, and whether the claimed condition concerns existence of suitable multipliers or every admissible choice. The relation of $\bar q$ to the state inclusion must also be specified. Consequently, the displayed question is a statement-only reconstruction, not a fully specified quantified assertion.

The supplied dated assessment says that this complementarity condition is already recorded as necessary and that sufficiency or a full characterization remains unresolved. This differs from the explicit question about necessity. Both formulations are retained here without selecting a replacement target. The source calls the system rate-independent, but does not explain the role of the term $-\sigma\Delta\dot z$ or whether a limiting problem is intended.

Short English statement

The source asks whether a locally optimal state and control must satisfy the displayed complementarity relation involving an adjoint and a subgradient of the rate. Essential analytic hypotheses and multiplier quantifiers are missing. The supplied literature assessment instead describes necessity as recorded and sufficiency or a full characterization as unresolved; the intended target therefore requires review.

Sources

• Local registry: data/unsolvedmath/problems.json, with catalog/problems.json as the supplementary registry; supplied permanent UnsolvedMath ID $30003760$, exact source problem code catalogue entry, title "Complementarity Conditions in Nonlinear PDE Control." These identify the supplied record; no independent registry inspection is claimed.
• Supplied source citation: Challenges in Optimal Control of Nonlinear PDE-Systems (2019), Oberwolfach Reports, pp. 941--1020, DOI 10.4171/owr/2018/16. Supplied source and background link: https://doi.org/10.4171/owr/2018/16. Attribution is to the report's workshop open-problem discussion; no individual proposer is supplied. The record separately gives proposed year 2018.
• Supplied literature source: "Oberwolfach Report 16," https://doi.org/10.4171/OWR/2018/16. Literature checked at: 2026-08-21. This is supplied dated curation, not independently verified: it labels the problem open, records the complementarity condition as necessary, and describes sufficiency or full characterization as unresolved. The background explicitly requests independent verification of time-sensitive status and cited claims.
• Supplied verification metadata labels the statement corrected_verified and reports that the official report was checked to restore the state inclusion, initial condition, objective, subgradient, and adjoint variables. This is a supplied verification claim, not an independent verification by this draft.