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Finite-Dimensional RHC Control of Linear Time-Varying Parabolic PDEs: Stability Analysis and Model-Order Reduction

Behzad Azmi, Jan Rohleff, Stefan Volkwein

Abstract

This chapter deals with the stabilization of a class of linear time-varying parabolic partial differential equations employing receding horizon control (RHC). Here, RHC is finite-dimensional, i.e., it enters as a time-depending linear combination of finitely many indicator functions whose total supports cover only a small part of the spatial domain. Further, we consider the squared l1-norm as the control cost. This leads to a nonsmooth infinite-horizon problem which allows a stabilizing optimal control with a low number of active actuators over time. First, the stabilizability of RHC is investigated. Then, to speed-up numerical computation, the data-driven model-order reduction (MOR) approaches are adequately incorporated within the RHC framework. Numerical experiments are also reported which illustrate the advantages of our MOR approaches.

Finite-Dimensional RHC Control of Linear Time-Varying Parabolic PDEs: Stability Analysis and Model-Order Reduction

Abstract

This chapter deals with the stabilization of a class of linear time-varying parabolic partial differential equations employing receding horizon control (RHC). Here, RHC is finite-dimensional, i.e., it enters as a time-depending linear combination of finitely many indicator functions whose total supports cover only a small part of the spatial domain. Further, we consider the squared l1-norm as the control cost. This leads to a nonsmooth infinite-horizon problem which allows a stabilizing optimal control with a low number of active actuators over time. First, the stabilizability of RHC is investigated. Then, to speed-up numerical computation, the data-driven model-order reduction (MOR) approaches are adequately incorporated within the RHC framework. Numerical experiments are also reported which illustrate the advantages of our MOR approaches.
Paper Structure (8 sections, 4 theorems, 62 equations, 6 figures, 2 tables, 3 algorithms)

This paper contains 8 sections, 4 theorems, 62 equations, 6 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Well-posedness
  3. Stability of RHC
  4. Model-Order Reduction (MOR)
  5. The POD method
  6. POD-based receding horizon algorithm
  7. Numerical experiments
  8. Discussion

Key Result

Proposition 2.1

For every $(\bar{t}_0,T)\in\mathbb R^2_+$ and $(\bar{y}_0, {\bm u})\in H \times {\mathscr U}_T(\bar{t}_0)$, equation e17 admits a unique weak solution $y \in W(\bar{t}_0,\bar{t}_0+T)$ satisfying with $c_1$ depending on $(T,\nu,a,b,\Omega)$.

Figures (6)

  • Figure 1: Spatial domain $\Omega$ and sub-rectangles $R_1,\ldots,R_{13}$ for the control actuators.
  • Figure 2: Evolution of $\log(|\bm y_\mathsf{rh}(t)|_M)$ (solid line) along with the corresponding reduced-order model $\log(|{\bm y_\mathsf{rh}^\ell}(t)|_M)$ (dashed line) for various choices of $T$. For the reduced models, $T^\mathsf{train}$ is equal to $T$ in the first five cases. The finite element model for $T=4$ is numerically extremely expensive, and for this reason, it is not computed in this article.
  • Figure 3: Evolution of the absolute $L^2$-state error $|(\bm y_\mathsf{rh})_i(t)-({\bm y_\mathsf{rh}^\ell})_i(t)|_M$ for various choices of $T$.
  • Figure 4: Evolution of the absolute FOM-based control error $t\mapsto|({\bm u_\mathsf{rh}})_i(t)|$ for $i=1,...,13$.
  • Figure 5: Evolution of the absolute control error $t\mapsto|({\bm u_\mathsf{rh}})_i(t)-({\bm u_\mathsf{rh}^\ell})_i(t)|$ for $i=1,...,13$. It turns out that the sparsity of the FOM-based RHC control ${\bm u_\mathsf{rh}}$ is very well aproximated by the POD-based RHC control ${\bm u_\mathsf{rh}^\ell}$.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Remark 2.1
  • Definition 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Remark 2.2
  • Definition 3.1
  • Theorem 3.1: Suboptimality and exponential decay
  • ...and 2 more