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

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