Hamilton–Jacobi–Bellman equation and viscosity solutions for the optimal control problem of stochastic convective Brinkman–Forchheimer equations

Source abstract

This work is devoted to the infinite-dimensional second-order Hamilton–Jacobi–Bellman equation associated, via the dynamic programming approach, with an optimal control problem for the two- and three-dimensional stochastic convective Brinkman–Forchheimer equations on the torus, driven by an additive Hilbert space valued [Formula: see text]-Wiener process. In the two-dimensional setting, many existing results (see [F. Gozzi et al., Commun. Pure Appl. Math. 58(5) (2005) 671–700]) rely essentially on the identity [Formula: see text], which is fundamental in closing the estimates. This cancellation fails in three dimensions, where the real analytical challenge arises. Our contribution is precisely a framework that does not rely on it. The damping term [Formula: see text] together with the dissipative term [Formula: see text] allows us to control [Formula: see text] effectively, so that our analysis extends naturally to three dimensions, which cannot be handled for the Navier–Stokes equations. In contrast, for the classical Navier–Stokes equations, the lack of suitable Sobolev embeddings prevents control of the convective term in three dimensions, thereby obstructing the closure of estimates. For the supercritical case, that is, [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] ([Formula: see text] for [Formula: see text] in [Formula: see text]) we first prove the existence of a viscosity solution of the Hamilton–Jacobi–Bellman equation, identified with the value function of the control problem. By establishing a comparison principle for [Formula: see text] and [Formula: see text] with [Formula: see text] in [Formula: see text], we then prove uniqueness.