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

2025/08/22 by Azmi, Behzad, Kartmann, Michael, Volkwein, Stefan · 1 citation
#35Q93 #49M20 #49M25 #65M15 #93A15 #93C20 #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2508.16801

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.

Citations

• Certified Model Predictive Control for Switched Evolution Equations using Model Order Reduction
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• A simplified Newton method to generate snapshots for POD models of semilinear optimal control problems
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Cited by

• Optimality-Based Control Space Reduction for Infinite-Dimensional Control Spaces

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• A Hybrid Finite-Dimensional RHC for Stabilization of Time-Varying\n Parabolic Equations
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