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Stabilization of nonautonomous linear parabolic equations with inputs subject to time-delay

Karl Kunisch, Sérgio S. Rodrigues

TL;DR

This work addresses stabilization of nonautonomous linear parabolic equations under time-delayed feedback. It develops a predictor-based strategy that uses a finite number of actuators and a Luenberger observer to counteract delays, ensuring stability even when the state is not directly measured. The authors establish theoretical results for asymptotic and exponential stabilization under a structured set of assumptions and demonstrate exponential convergence of the observer; they also provide explicit constructions for input/observer gains and validate the approach via numerical simulations on polygonal domains with sensor and actuator arrays. The findings highlight the practical viability of delay-compensated feedback in high-dimensional, time-varying PDE settings and discuss robustness to measurement and discretization errors.

Abstract

The stabilization of nonautonomous parabolic equations is achieved by feedback inputs tuning a finite number of actuators, where it is assumed that the input is subject to a time delay. To overcome destabilizing effects of the time delay, the input is based on a prediction of the state at a future time. This prediction is computed depending on a state-estimate at the current time, which in turn is provided by a Luenberger observer. The observer is designed using the output of measurements performed by a finite number of sensors. The asymptotic behavior of the resulting coupled system is investigated. Numerical simulations are presented validating the theoretical findings, including tests showing the response against sensor measurement errors.

Stabilization of nonautonomous linear parabolic equations with inputs subject to time-delay

TL;DR

This work addresses stabilization of nonautonomous linear parabolic equations under time-delayed feedback. It develops a predictor-based strategy that uses a finite number of actuators and a Luenberger observer to counteract delays, ensuring stability even when the state is not directly measured. The authors establish theoretical results for asymptotic and exponential stabilization under a structured set of assumptions and demonstrate exponential convergence of the observer; they also provide explicit constructions for input/observer gains and validate the approach via numerical simulations on polygonal domains with sensor and actuator arrays. The findings highlight the practical viability of delay-compensated feedback in high-dimensional, time-varying PDE settings and discuss robustness to measurement and discretization errors.

Abstract

The stabilization of nonautonomous parabolic equations is achieved by feedback inputs tuning a finite number of actuators, where it is assumed that the input is subject to a time delay. To overcome destabilizing effects of the time delay, the input is based on a prediction of the state at a future time. This prediction is computed depending on a state-estimate at the current time, which in turn is provided by a Luenberger observer. The observer is designed using the output of measurements performed by a finite number of sensors. The asymptotic behavior of the resulting coupled system is investigated. Numerical simulations are presented validating the theoretical findings, including tests showing the response against sensor measurement errors.

Paper Structure

This paper contains 23 sections, 9 theorems, 97 equations, 8 figures.

Table of Contents

  1. Introduction
  2. On the novelties
  3. Destabilizing effect of delays
  4. Prediction and time irreversibility
  5. Stabilizability and nonautonomous dynamics
  6. Detectability
  7. Contents
  8. Contents
  9. Assumptions and auxiliary results
  10. Assumptions
  11. Auxiliary results
  12. Destabilizing effect of delayed feedback inputs
  13. Stabilization
  14. Asymptotic null controllability
  15. A predictor-based stabilizing feedback input
  16. ...and 8 more sections

Key Result

Lemma 2.7

If Assumptions A:HV-A:Arc hold true and $f\in L^2_{\rm loc}({\mathbb R}_+;V')$, then the solution of sys-yf satisfies, for any given $s\ge0$ and $T>0$, with constants $C_{0,T}=\overline C_{\left[C_{\rm rc}\right]}\ge0$ and $C_T=\overline C_{\left[C_{\rm rc}\right]}\ge1$ independent of $(s,y_0)$.

Figures (8)

  • Figure 1: Actuators ("/"-pattern) and sensors ("\\"-pattern)
  • Figure 2: Spatial triangulation ${\mathfrak T}$.
  • Figure 3: Instability of the free dynamics and free observer.
  • Figure 4: Evolution for feedback inputs delayed by time-$\tau$.
  • Figure 5: Time-zoom of the norms evolution in Fig \ref{['fig:feed-delay']}.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Definition 2.1
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • proof
  • Remark 3.1
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • ...and 9 more