  We consider a multidimentional Brownian control problem (BCP) with model
uncertainty that formally emerges from a multiclass M/M/1 queueing control
problem under heavy-traffic with model uncertainty. The BCP is formulated as a
multidimensional stochastic differential game with two players: a minimizer
that has an equivalent role to the decision maker in the queueing control
problem and a maximizer whose role is to set up the uncertainty of the model.
The dynamics are driven by a Brownian motion. We show that a state-space
collapse propery holds. That is, the multidimensional BCP can be reduced to a
one-dimensional BCP with model uncertainty that also takes the form of a
two-player stochastic differential game. Then, the value function of both of
the games is characterized as the unique solution to a free-boundary problem
from which we extract equilibria for both games. Finally, we analyze the
dependence of the value function and the equilibria on the ambiguity
parameters.
