  A procedure for the numerical approximation of high-dimensional
Hamilton-Jacobi-Bellman (HJB) equations associated to optimal feedback control
problems for semilinear parabolic equations is proposed. Its main ingredients
are a pseudospectral collocation approximation of the PDE dynamics, and an
iterative method for the nonlinear HJB equation associated to the feedback
synthesis. The latter is known as the Successive Galerkin Approximation. It can
also be interpreted as Newton iteration for the HJB equation. At every step,
the associated linear Generalized HJB equation is approximated via a separable
polynomial approximation ansatz. Stabilizing feedback controls are obtained
from solutions to the HJB equations for systems of dimension up to fourteen.
