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We propose an approach for the synthesis of robust and optimal feedbackcontrollers for nonlinear PDEs. Our approach considers the approximation ofinfinite-dimensional control systems by a pseudospectral collocation method,leading to high-dimensional nonlinear dynamics. For the reduced-order model, weconstruct a robust feedback control based on the $\cH_{\infty}$ control method,which requires the solution of an associated high-dimensionalHamilton-Jacobi-Isaacs nonlinear PDE. The dimensionality of the Isaacs PDE istackled by means of a separable representation of the control system, and apolynomial approximation ansatz for the corresponding value function. Ourmethod proves to be effective for the robust stabilization of nonlineardynamics up to dimension $d\approx 12$. We assess the robustness and optimalityfeatures of our design over a class of nonlinear parabolic PDEs, includingnonlinear advection and reaction terms. The proposed design yields a feedbackcontroller achieving optimal stabilization and disturbance rejectionproperties, along with providing a modelling framework for the robust controlof PDEs under parametric uncertainties.

Robust feedback control of nonlinear PDEs by numerical approximation of high-dimensional Hamilton-Jacobi-Isaacs equations