A Verification Theorem for an Optimal Control Problem Governed by the Convective Brinkman--Forchheimer Equations

Abstract

This article establishes a verification theorem for an optimal control problem governed by the two- and three-dimensional convective Brinkman--Forchheimer equations on the $d$-dimensional torus, $d\in \{2,3\}$: $$\frac{\partial \mathfrak{u}}{\partial t}-\hat{I}¼\hat{I}”\mathfrak{u}+(\mathfrak{u}\cdot \nabla )\mathfrak{u}+\hat{I}\pm \mathfrak{u}+{\hat{I}}^2|\mathfrak{u}{|}^{r-1}\mathfrak{u}+\nabla \mathfrak{p}=\mathbf{f},\nabla \cdot \mathfrak{u}=0,$$ where $\hat{I}¼,\hat{I}\pm ,{\hat{I}}^2>0$ and $r\in [1,\infty )$. We derive the Pontryagin maximum principle and develop a verification framework for the associated control problem, a topic that has received comparatively little attention for fluid models of Navier--Stokes type. A major challenge in establishing the verification theorem and the corresponding feedback characterization for the CBF system is that the analysis requires a substantially different regularity framework from that used for the two-dimensional Navier--Stokes equations. In particular, the present approach relies on strong solution theory, a delicate treatment of the nonlinear absorption term, novel estimates in negative-order Sobolev spaces, and continuous dependence estimates in stronger topologies, especially in the three-dimensional setting. A distinctive feature of the present work is that the verification framework is developed not only in two dimensions, but also in the three-dimensional supercritical regime, corresponding to $r\in (3,5]$, and in the critical case $r=3$ under the condition $2{\hat{I}}^2\hat{I}¼\ge 1$. Consequently, the feedback characterization and verification arguments can be rigorously justified in both two and three dimensions.