  In this paper we study by probabilistic techniques the convergence of the
value function for a two-scale, infinite-dimensional, stochastic controlled
system as the ratio between the two evolution speeds diverges. The value
function is represented as the solution of a \textit{backward stochastic
differential equation} (BSDE) that it is shown to converge towards a
\textit{reduced} BSDE. The noise is assumed to be additive both in the slow and
the fast equations for the state. Some non degeneracy condition on the slow
equation is required. The limit BSDE involves the solution of an
\textit{ergodic} BSDE and is itself interpreted as the value function of an
auxiliary stochastic control problem on a reduced state space.

Optimization and Control