We present a model predictive control (MPC) framework for linear switched evolution equations arising from a parabolic partial differential equation (PDE). First-order optimality conditions for the resulting finite-horizon optimal control problems are derived. The analysis allows for the incorporation of convex control constraints and sparse regularization. Then, to mitigate the computational burden of the MPC procedure, we employ Galerkin reduced-order modeling (ROM) techniques to obtain a low-dimensional surrogate for the state-adjoint systems. We derive recursive a-posteriori estimates for the ROM feedback law and the ROM-MPC closed-loop state and show that the ROM-MPC trajectory evolves within a neighborhood of the true MPC trajectory, whose size can be explicitly computed and is controlled by the quality of the ROM. Such estimates are then used to formulate two ROM-MPC algorithms with closed-loop certification.

翻译：本文针对由抛物型偏微分方程（PDE）导出的线性切换演化方程，提出了一种模型预测控制（MPC）框架。我们推导了由此产生的有限时域最优控制问题的一阶最优性条件。该分析允许纳入凸控制约束和稀疏正则化。随后，为减轻MPC过程的计算负担，我们采用Galerkin降阶建模（ROM）技术，为状态-伴随系统获得一个低维代理模型。我们推导了ROM反馈律以及ROM-MPC闭环状态的递归后验估计，并证明ROM-MPC轨迹在真实MPC轨迹的一个邻域内演化，该邻域的大小可被显式计算，并由ROM的质量所控制。这些估计随后被用于构建两种具有闭环认证的ROM-MPC算法。