Hamilton-Jacobi-Bellman equation and Viscosity solutions for an optimal control problem for stochastic convective Brinkman-Forchheimer equations

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 $\mu ,\alpha ,\beta >0$, $r\in [1,\infty )$ 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 $d=3$ ($2\beta \mu \ge 1$ 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 $2\beta \mu \ge 1$ 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.