  We propose an approach for the synthesis of robust and optimal feedback
controllers for nonlinear PDEs. Our approach considers the approximation of
infinite-dimensional control systems by a pseudospectral collocation method,
leading to high-dimensional nonlinear dynamics. For the reduced-order model, we
construct a robust feedback control based on the $\cH_{\infty}$ control method,
which requires the solution of an associated high-dimensional
Hamilton-Jacobi-Isaacs nonlinear PDE. The dimensionality of the Isaacs PDE is
tackled by means of a separable representation of the control system, and a
polynomial approximation ansatz for the corresponding value function. Our
method proves to be effective for the robust stabilization of nonlinear
dynamics up to dimension $d\approx 12$. We assess the robustness and optimality
features of our design over a class of nonlinear parabolic PDEs, including
nonlinear advection and reaction terms. The proposed design yields a feedback
controller achieving optimal stabilization and disturbance rejection
properties, along with providing a modelling framework for the robust control
of PDEs under parametric uncertainties.
