Hamilton-Jacobi-Bellman equation and Viscosity solutions for an optimal control problem for stochastic convective Brinkman-Forchheimer equations
Sagar Gautam, Manil T. Mohan
TL;DR
The work advances stochastic control of damped Navier–Stokes–type flows by embedding SCBF dynamics on the torus into an infinite‑dimensional HJB framework. Through a carefully designed viscosity solution approach and a robust comparison principle, it proves the existence and global uniqueness of the viscosity solution to the HJB equation for supercritical nonlinearities (2D: $r>3$, 3D: $r\in(3,5)$, with $2\beta\mu\ge1$ at $r=3$). The methodology hinges on a damping/absorption mechanism that dominates the convective term, enabling uniform energy estimates and a viable infinite‑dimensional DP principle. This yields a rigorous link between the value function and the HJB solution, with potential implications for large‑scale stochastic fluid control and further extensions to infinite horizons and Lévy noise. Overall, the paper provides a rigorous, globally unique viscosity‑solution theory for HJB equations associated with stochastic CBF controls in both two and three dimensions.
Abstract
In this work, we consider the following two- and three-dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations in torus $\mathbb{T}^d,\ d\in\{2,3\}$: \begin{align*} \mathrm{d}\boldsymbol{u}+\left[-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p\right]\mathrm{d}t=\mathrm{d}\mathrm{W}, \ \nabla\cdot\boldsymbol{u}=0, \end{align*} where $μ,α,β>0$, $r\in[1,\infty)$ and $\mathrm{W}$ is a Hilbert space valued $\mathrm{Q}-$Wiener process. The above system can be considered as damped stochastic Navier-Stokes equations. Using the dynamic programming approach, we study the infinite-dimensional second-order Hamilton-Jacobi equation associated with an optimal control problem for SCBF equations. For the supercritical case, that is, $r\in(3,\infty)$ for $d=2$ and $r\in(3,5)$ for $d=3$ ($2βμ\geq 1$ for $r=3$ in $d\in\{2,3\}$), we first prove the existence of a viscosity solution for the infinite-dimensional HJB equation, which we identify with the value function of the associated control problem. By establishing a comparison principle for $r\in(3,\infty)$ and $r=3$ with $2βμ\geq1$ in $d\in\{2,3\}$, we prove that the value function is the unique viscosity solution and hence we resolve the global unique solvability of the HJB equation in both two and three dimensions.