Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

Optimality conditions are provided for a class of control problems driven by a Wiener process, which amounts to a stochastic maximum principle in differential form. The control is considered to act on the drift and the volatility, both of which may be unbounded operators, which allows us to consider SPDEs with control and/or noise on the boundary. By the factorization method, a regularizing property is established for the state equation which is then employed to prove, by duality, a similar result for the backward time costate equation. The costate equation is understood in the sense of transposition. Finally, the cost is shown to be Gateaux differentiable and its derivative is represented in terms of the costate, the optimality condition is deduced using the results of set-valued analysis.