A class of semilinear stochastic partial differential equations and their controls: Existence results

This paper concerns a class of similinear stochastic partial differential equations, of which the drift term is a second-order differential operator plus a nonlinearity, and the diffusion term is a first-order differential operator. When the nonlinearity is only continuous in the state, it is shown that there exist solutions of the equation provided that the Wiener process involved is one-dimensional. The existence of optimal relaxed controls for this class of equations is also proved. Our method is based on a group analysis of the first-order differential operator and a time change technique.