Stabilization of Parabolic Time-Varying PDEs using Certified Reduced-Order Receding Horizon Control
Provides a theoretically grounded, computationally efficient method for stabilizing infinite-dimensional systems, relevant to control engineers working with PDEs.
The paper proves exponential stability and suboptimality for a reduced-order model-based receding horizon control scheme applied to linear time-varying parabolic PDEs, with rigorous a posteriori error bounds. Numerical tests show effectiveness for unstable systems.
We address the stabilization of linear, time-varying parabolic PDEs using finite-dimensional receding horizon controls (RHCs) derived from reduced-order models (ROMs). We first prove exponential stability and suboptimality of the continuous-time full-order model (FOM) RHC scheme in Hilbert spaces. A Galerkin model reduction is then introduced, along with a rigorous a posteriori error analysis for the associated finite-horizon optimal control problems. This results in a ROM-based RHC algorithm that adaptively constructs reduced-order controls, ensuring exponential stability of the FOM closed-loop state and providing computable performance bounds with respect to the infinite-horizon FOM control problem. Numerical experiments with a non-smooth cost functional involving the squared l1-norm confirm the methods effectiveness, even for exponentially unstable systems.