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Event-triggered boundary control of the linearized FitzHugh-Nagumo equation

Víctor Hernández-Santamaría, Subrata Majumdar, Luz de Teresa

Abstract

In this paper, we address the exponential stabilization of the linearized FitzHugh-Nagumo system using an event-triggered boundary control strategy. Employing the backstepping method, we derive a feedback control law that updates based on specific triggering rules while ensuring the exponential stability of the closed-loop system. We establish the well-posedness of the system and analyze its input-to-state stability in relation to the deviations introduced by the event-triggered control. Numerical simulations demonstrate the effectiveness of this approach, showing that it stabilizes the system with fewer control updates compared to continuous feedback strategies while maintaining similar stabilization performance.

Event-triggered boundary control of the linearized FitzHugh-Nagumo equation

Abstract

In this paper, we address the exponential stabilization of the linearized FitzHugh-Nagumo system using an event-triggered boundary control strategy. Employing the backstepping method, we derive a feedback control law that updates based on specific triggering rules while ensuring the exponential stability of the closed-loop system. We establish the well-posedness of the system and analyze its input-to-state stability in relation to the deviations introduced by the event-triggered control. Numerical simulations demonstrate the effectiveness of this approach, showing that it stabilizes the system with fewer control updates compared to continuous feedback strategies while maintaining similar stabilization performance.

Paper Structure

This paper contains 17 sections, 7 theorems, 105 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Setting of the problem
  4. Outline of the paper
  5. Brief description on the structure of Backstepping under the Event-triggered strategy
  6. Well-posedness result
  7. Exponential stabilizability
  8. Event-triggered rule
  9. Avoidance of Zeno behavior
  10. Stability analysis
  11. Estimate of $u_n(t)$
  12. Estimate of $z_n(t)$
  13. Numerical experiments
  14. Some considerations on the system parameters
  15. Discretization of the coupled model and numerical implementation of the backstepping control
  16. ...and 2 more sections

Key Result

Theorem 3.1

For every $\left(v(t_j), w(t_j)\right)\in L^2(0,1)\times L^2(0,1)$, system FHNlin ET possess a unique solution $(v,w)\in \mathcal{C}^0([t_j, t_{j+1}]; L^2(0,1)\times L^2(0,1))\cap L^2(t_j, t_{j+1}; H^1(0,1)\times L^2(0,1)).$

Figures (5)

  • Figure 1: Evolution in time of the uncontrolled system \ref{['eq:heat_memory_simple']}.
  • Figure 2: Evolution in time of the controlled system \ref{['eq:heat_memory_simple']} with backstepping control \ref{['control_num']}.
  • Figure 3: Norm of the closed-loop dynamics $t\mapsto \|v(t)\|_{L^2}+\|w(t)\|_{L^2}$ of \ref{['eq:heat_memory_simple']} with different design parameters $\lambda$. For comparison, we have added the exponential function $Ce^{-t}$ for some $C>0$, which is the best theoretical decay rate for system \ref{['eq:heat_memory_simple']} with coupling parameters \ref{['coupling']}.
  • Figure 4: Evolution in time of the controlled system \ref{['eq:heat_memory_simple']} with event-triggered control with design parameter $\beta=0.001$.
  • Figure 5: Event-triggered control with design parameter $\beta=0.05$.

Theorems & Definitions (17)

  • Definition 1.1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • ...and 7 more