Singular Limit of Two-Scale Stochastic Optimal Control Problems in Infinite Dimensions by Vanishing Noise Regularization

In this paper we study the limit of the value function for a two-scale, infinitedimensional, 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 Hamilton-Jacobi-Bellman equations as in [Swiech, ESAIM Control Optim. Calc. Var., to appear] while degeneracy of diffusion coefficients prevents representation as a classical backward stochastic differential equation as in [Guatteri and Tessitore, Appl. Math. Optim., 83 (2021), pp. 1025-1051]. We use a vanishing noise""regularization technique.