Maximum principle for optimal control of SPDEs with locally monotone coefficients

The aim of this paper is to derive a maximum principle for a control problem governed by a stochastic partial differential equation (SPDE) with locally monotone coefficients. To reach our goal we adapt the method which uses the relation between backward stochastic partial differential equation (BSPDE) and the maximum principle. In particular, necessary conditions for optimality for this stochastic optimal control problem are obtained. In spite of the fact that the method used here was used by several authors before, our adaptation is not immediate. It applies a trick which is used to get estimates for solutions of SPDE with Locally Monotone Coefficients as in the proof of the Lemmas 5.1 and 5.3. This adaptation permits us to apply our results to get a maximum principle for the optimal control to the cases when the system is governed by the 2D stochastic Navier-Stokes equation and by a stochastic Burgers' equation.