SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1496-1524, January 2020.
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 $\mathcal{H}_{\infty}$ 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 modeling framework for the robust control of PDEs under parametric uncertainties.

中文翻译：

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高维Hamilton-Jacobi-Isaacs方程数值逼近对非线性PDE的鲁棒反馈控制

SIAM应用动力系统杂志，第19卷，第2期，第1496-1524页，2020年1月。
我们提出了一种用于非线性PDE的鲁棒和最优反馈控制器的综合方法。我们的方法考虑通过伪谱搭配方法逼近无限维控制系统，从而导致高维非线性动力学。对于降阶模型，我们基于$ \ mathcal {H} _ {\ infty} $控制方法构造一个鲁棒的反馈控制，该方法需要求解相关的高维Hamilton-Jacobi-Isaacs非线性PDE 。Isaacs PDE的维数通过控制系统的可分离表示形式以及对应值函数的多项式近似ansatz来解决。我们的方法被证明对于非线性动力学的稳定稳定有效，直到尺寸$ d \大约12 $。我们评估了一类非线性抛物线偏微分方程的设计的鲁棒性和最优性，包括非线性对流和反应项。提出的设计产生了一种反馈控制器，该控制器实现了最佳的稳定性和干扰抑制性能，并提供了在参数不确定性下鲁棒控制PDE的建模框架。