# A Hybrid Finite-Dimensional RHC for Stabilization of Time-Varying Parabolic Equations

## TL;DR

This paper develops a finite-dimensional receding horizon control method for stabilizing time-varying parabolic equations, emphasizing the use of an $ ext{l}_1$-norm control cost to achieve sparse actuator activation and stability.

## Contribution

It introduces a novel RHC framework using $ ext{l}_1$-norm control costs for stabilization of time-varying parabolic PDEs, highlighting stability and suboptimality analysis.

## Key findings

- Stable control with few active actuators demonstrated.
- Numerical results confirm theoretical stability and control sparsity.
- Comparison shows qualitative differences between $ ext{l}_1$ and $ ext{l}_2$ control costs.

## Abstract

The present work is concerned with the stabilization of a general class of time-varying linear parabolic equations by means of a finite-dimensional receding horizon control (RHC). The stability and suboptimality of the unconstrained receding horizon framework is studied. The analysis allows the choice of the squared $\ell_1$-norm as control cost. This leads to a nonsmooth infinite-horizon problem which provides stabilizing optimal controls with a low number of active actuators over time. Numerical experiments are given which validate the theoretical results and illustrate the qualitative differences between the $\ell_1$- and $\ell_2$-control costs.