  In this paper we study the limit of the value function for a two-scale,
infinite-dimensional, stochastic controlled system with cylindrical noise and
possibly degenerate diffusion. The limit is represented as the value function
of a new reduced control problem (on a reduced state space). The presence of a
cylindrical noise prevents representation of the limit by viscosity solutions
of HJB equations, while degeneracy of diffusion coefficients prevents
representation as a classical BSDE. We use a vanishing noise regularization
technique.
