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Dynamic output-based feedback stabilizability for linear parabolic equations with memory

Arbaz Khan, Sumit Mahajan, Sérgio S. Rodrigues

TL;DR

This work develops a dynamic output-based feedback framework for linear parabolic equations with memory, enabling stabilization through explicit projections-based input and output injections realized by finite actuator/sensor arrays. A dynamic Luenberger observer supplies state estimates used in the control law, with stability proven under a set of verifiable assumptions and kernel properties; for exponential kernels, the authors prove exponential stabilizability with rate equal to the kernel rate $\gamma$, and validate the results via thorough numerical experiments using finite elements and IMEX time integration. The study demonstrates that large enough actuator/sensor configurations and gain choices render the coupled system exponentially stable, while memory terms influence the achievable rate and require careful discretization of the memory integral. Notably, a static-output variant is shown to be feasible when actuators also carry sensors, yielding a purely output-driven stabilizing law with explicit gains. The results have practical significance for control of distributed-parameter systems with history effects, and open doors to extensions to nonlinear dynamics and optimized sensor/actuator placements.

Abstract

The stabilizability of a general class of linear parabolic equations with a memory term, is achieve by explicit output feedback. The control input is given as a function of a state-estimate provided by an exponential dynamic Luenberger observer based on the output of sensor measurements. The numbers of actuators and sensors are finite. The feedback input and output injection operators are given explicitly involving appropriate orthogonal projections. For exponential kernels, exponential stabilizability can be achieved with the rate of the exponential kernel. The discretization and simulation of the controlled systems are addressed as well and results of simulations are reported showing the performance of the proposed dynamic output-based control feedback input. We include simulations for both exponential and weakly singular Riesz kernels, showing the success of the strategy in obtaining a stabilizing input.

Dynamic output-based feedback stabilizability for linear parabolic equations with memory

TL;DR

This work develops a dynamic output-based feedback framework for linear parabolic equations with memory, enabling stabilization through explicit projections-based input and output injections realized by finite actuator/sensor arrays. A dynamic Luenberger observer supplies state estimates used in the control law, with stability proven under a set of verifiable assumptions and kernel properties; for exponential kernels, the authors prove exponential stabilizability with rate equal to the kernel rate , and validate the results via thorough numerical experiments using finite elements and IMEX time integration. The study demonstrates that large enough actuator/sensor configurations and gain choices render the coupled system exponentially stable, while memory terms influence the achievable rate and require careful discretization of the memory integral. Notably, a static-output variant is shown to be feasible when actuators also carry sensors, yielding a purely output-driven stabilizing law with explicit gains. The results have practical significance for control of distributed-parameter systems with history effects, and open doors to extensions to nonlinear dynamics and optimized sensor/actuator placements.

Abstract

The stabilizability of a general class of linear parabolic equations with a memory term, is achieve by explicit output feedback. The control input is given as a function of a state-estimate provided by an exponential dynamic Luenberger observer based on the output of sensor measurements. The numbers of actuators and sensors are finite. The feedback input and output injection operators are given explicitly involving appropriate orthogonal projections. For exponential kernels, exponential stabilizability can be achieved with the rate of the exponential kernel. The discretization and simulation of the controlled systems are addressed as well and results of simulations are reported showing the performance of the proposed dynamic output-based control feedback input. We include simulations for both exponential and weakly singular Riesz kernels, showing the success of the strategy in obtaining a stabilizing input.
Paper Structure (34 sections, 4 theorems, 96 equations, 15 figures)

This paper contains 34 sections, 4 theorems, 96 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Dynamic and static output-feedback
  3. Dynamic output-based feedback stabilization
  4. Literature review
  5. Outline of paper and notation
  6. Assumptions
  7. Stability of the closed-loop coupled system
  8. Satisfiability of the assumptions
  9. Satisfiability of Assumptions \ref{['ass:A']} and \ref{['ass:Arc']}
  10. Satisfiability of Assumption \ref{['ass:KMLS']}
  11. Satisfiability of Assumption \ref{['ass:Xi']}
  12. Numerical Results
  13. Parameters
  14. Discretization
  15. Finite element spatial discretization
  16. ...and 19 more sections

Key Result

Theorem 3.1

Let Assumptions ass:A--ass:Xi hold true. Then there exists a pair $(M_*,S_*)\in{\mathbb N}_+\times{\mathbb N}_+$ such that, for all $M\ge M_*$ and $S\ge S_*$ there exist $(\lambda_{1*},\lambda_{2*})=(\lambda_{1*}(M),\lambda_{2*}(S))\in\overline{\mathbb R}_+\times\overline{\mathbb R}_+$ such that, fo In the particular case where the memory term is given by the kernel exk-intro as we have that the

Figures (15)

  • Figure 1: Supports of actuators (slash-$\slash$-lines) and sensors (backslash-$\backslash$-lines).
  • Figure 2: Coarsest spatial triangulations ${\mathfrak T}={\mathfrak T}_0^{\ell\times\ell}$, $\ell\in\{2,4,6\}$.
  • Figure 3: Closed-loop free dynamics \ref{['sys-num-2']}, for several values of $\eta$.
  • Figure 4: Using 2 actuators and 2 sensors as in \ref{['sys-num-2']}, for several choices of $\lambda$.
  • Figure 5: Using 2 actuators and 2 sensors as in \ref{['sys-num-2']}, for several values of $\eta$.
  • ...and 10 more figures

Theorems & Definitions (14)

  • Remark 1.1
  • Remark 2.5
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • Corollary 4.3
  • proof
  • Remark 4.4
  • Remark 4.5
  • ...and 4 more