A Stochastic Maximum Principle for Control Problems Constrained by the Stochastic Navier–Stokes Equations

Abstract

We analyze the control problem of the stochastic Navier–Stokes equations in multi-dimensional domains considered in Benner and Trautwein (Math Nachr 292(7):1444–1461, 2019) restricted to noise terms defined by a Q-Wiener process. The cost functional related to this control problem is nonconvex. Using a stochastic maximum principle, we derive a necessary optimality condition to obtain explicit formulas the optimal controls have to satisfy. Moreover, we show that the optimal controls satisfy a sufficient optimality condition. As a consequence, we are able to solve uniquely control problems constrained by the stochastic Navier–Stokes equations especially for two-dimensional as well as for three-dimensional domains.

Optimal control of two dimensional third grade fluids

“…Then, they proved the existence and uniqueness of optimal control, by analyzing some cost functional depending on stopping time. Recently, they studied in [5], the optimality system and established the necessary and sufficient optimality conditions with a multiplicative noise driven by a Q-Wiener process. It is worth mentioning that the approach in [4,5] was based on the theory of semigroups.…”

“…Recently, they studied in [5], the optimality system and established the necessary and sufficient optimality conditions with a multiplicative noise driven by a Q-Wiener process. It is worth mentioning that the approach in [4,5] was based on the theory of semigroups.…”