We are interested in the control problem of a partially observable diffusion process, which is initialized at a fixed point of ⁿ, and we want to characterize the associated value function. To resort to the theory of viscosity solutions depends on the possibility to translate such a problem on Hilbert spaces like L²(ⁿ), and so it can not be used here. Nevertheless, a result of N. Bouleau and F. Hirsch allows us to introduce a broadened problem which fulfills the condition. The fact remains to link these two control problems.