Stabilization of Parabolic Time-Varying PDEs using Certified Reduced-Order Receding Horizon Control

TL;DR

This paper develops a method for stabilizing linear, time-varying parabolic PDEs using reduced-order models within a receding horizon control framework, ensuring stability and providing performance guarantees.

Contribution

It introduces a ROM-based RHC algorithm with rigorous error analysis, guaranteeing stability and suboptimality for controlling PDEs.

Findings

Proves exponential stability of the control scheme.

Provides a posteriori error bounds for reduced-order models.

Demonstrates effectiveness on unstable systems with numerical experiments.

Abstract

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.