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arxiv: 2605.18520 · v1 · pith:BIB2HUHPnew · submitted 2026-05-18 · 🧮 math.OC

Boundary Stabilization for the Rayleigh Beam System under Event-triggered Controls

Siwen Wang , Yi Cheng , Yuhu Wu , Yongxin Wu (UMLP , ENSMM , FEMTO-ST) , Wen Kang (BIT) This is my paper

Pith reviewed 2026-05-20 08:49 UTC · model grok-4.3

classification 🧮 math.OC
keywords Rayleigh beamevent-triggered controlboundary stabilizationexponential stabilityintegral multiplier techniqueenergy perturbation methoddistributed parameter systems
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The pith

Event-triggered boundary controls stabilize the Rayleigh beam system exponentially with a precisely tunable decay rate.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops two event-triggered control laws for boundary stabilization of the Rayleigh beam system. It derives a sufficient condition on the control parameters that ensures exponential stability of the closed-loop system. The analysis applies the integral multiplier technique together with an energy perturbation method, which allows the exponential decay rate to be determined exactly from the parameters. This matters for systems where continuous actuation is costly or impractical, such as vibration suppression in flexible structures.

Core claim

The authors propose two event-triggered control laws incorporating an event-triggering mechanism to tackle the boundary stabilization for the Rayleigh beam system. Under these event-triggered controls, a sufficient condition for parameter determination is constructed to guarantee the exponential stability of the closed-loop system by using the integral multiplier technique and energy perturbation method, wherein the desired exponential decay rate can be precisely determined.

What carries the argument

Event-triggered control laws whose triggering condition, combined with the integral multiplier technique and energy perturbation method, yields an explicit sufficient condition for exponential stability and decay rate.

If this is right

  • The closed-loop Rayleigh beam system is exponentially stable under the proposed event-triggered laws.
  • The exponential decay rate is explicitly determined by the choice of control parameters satisfying the sufficient condition.
  • Actuation or communication occurs only at discrete triggering instants rather than continuously.
  • Numerical examples confirm that the designed controls achieve the predicted stabilization performance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same multiplier-plus-perturbation strategy could be tested on other Euler-Bernoulli or Timoshenko beam models with boundary actuation.
  • Implementation on a physical flexible beam would require verifying that sensor noise does not induce spurious triggers that violate the derived decay bound.
  • The approach suggests a route to event-triggered stabilization for related distributed-parameter systems such as plates or strings.

Load-bearing premise

The event-triggering mechanism is well-defined, excludes Zeno behavior, and preserves the stability estimates obtained from the multiplier and perturbation analysis.

What would settle it

Numerical simulation in which the inter-event times accumulate to a finite limit (Zeno behavior occurs) or in which the beam energy fails to decay at the rate predicted by the sufficient condition.

Figures

Figures reproduced from arXiv: 2605.18520 by ENSMM, FEMTO-ST), Siwen Wang, Wen Kang (BIT), Yi Cheng, Yongxin Wu (UMLP, Yuhu Wu.

Figure 1
Figure 1. Figure 1: Transverse displacements of the closed-loop system [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Time evolution of the energy E(t) of the closed￾loop system under the event-triggered control [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Release instants and release interval by event [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of U1(t) for the continuous-time con￾troller in red dashed line and the event-triggered con￾troller in purple line [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of U2(t) for the continuous-time con￾troller in red dashed line and the event-triggered con￾troller in purple line. conservation laws. Automatica, 70, 275–287. Espitia, N., Karafyllis, I., and Krstic, M. (2021). Event-triggered boundary control of constant-parameter reaction-diffusion pdes: A small-gain approach. Auto￾matica, 128, 109562. Fan, X., Xu, C.Z., Zhou, H.C., et al. (2025). Event￾trigge… view at source ↗
read the original abstract

In this paper, we propose two event-triggered control laws incorporating an eventtriggering mechanism to tackle the boundary stabilization for the Rayleigh beam system. Under this event-triggered controls, a sufficient condition for parameter determination is constructed to guarantee the exponential stability of the closed-loop system by using the integral multiplier technique and energy perturbation method, wherein the desired exponential decay rate can be precisely determined. Numerical examples are presented to demonstrate the efficacy of the event-triggered control methodology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes two event-triggered boundary control laws for the Rayleigh beam system. It constructs a sufficient condition on the control parameters to guarantee exponential stability of the closed-loop system via the integral multiplier technique and energy perturbation method, with the property that the desired exponential decay rate can be precisely determined. Numerical examples are included to illustrate the efficacy of the approach.

Significance. If the sufficient condition rigorously secures both exponential stability and a Zeno-free event-triggering mechanism, the work would contribute to event-triggered control for infinite-dimensional systems by adapting standard multiplier and perturbation techniques to handle sampling errors while allowing prescription of the decay rate. The numerical examples provide useful validation of practical performance.

major comments (1)
  1. [Proof of the sufficient condition for exponential stability and event-triggering] The central claim constructs a sufficient condition asserted to guarantee both exponential stability (via integral multiplier and energy perturbation) and a well-defined event-triggering mechanism without Zeno behavior. In boundary-controlled PDEs the perturbation analysis bounds the effect of the sampling error only when inter-event times are bounded below by a positive constant; that lower bound is typically obtained by using the exponential decay rate that the condition is supposed to deliver. If the paper does not first fix a positive dwell-time estimate independently of the decay (or choose the triggering threshold so that the two estimates close simultaneously without circular invocation), the sufficient condition does not rigorously secure the closed-loop result. Please clarify the logical order in the proof of the main theorem.
minor comments (1)
  1. [Abstract] The abstract refers to 'two event-triggered control laws' without indicating what distinguishes them; a short clarifying sentence would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comment on the logical structure of the proof. We address the major comment below.

read point-by-point responses

  1. Referee: [Proof of the sufficient condition for exponential stability and event-triggering] The central claim constructs a sufficient condition asserted to guarantee both exponential stability (via integral multiplier and energy perturbation) and a well-defined event-triggering mechanism without Zeno behavior. In boundary-controlled PDEs the perturbation analysis bounds the effect of the sampling error only when inter-event times are bounded below by a positive constant; that lower bound is typically obtained by using the exponential decay rate that the condition is supposed to deliver. If the paper does not first fix a positive dwell-time estimate independently of the decay (or choose the triggering threshold so that the two estimates close simultaneously without circular invocation), the sufficient condition does not rigorously secure the closed-loop result. Please clarify the logical order in

    Authors: We appreciate the referee's identification of this important point concerning the order of estimates. In the proof of the main theorem, the sufficient condition on the control parameters and triggering threshold is constructed so that a positive lower bound on the inter-event times can be obtained first from the event-triggering rule and the a-priori energy estimates (which rely on the multiplier technique but do not yet invoke the final decay rate). With this dwell-time in hand, the energy perturbation analysis is then applied to establish the exponential decay with the prescribed rate. The parameters are chosen to make these two estimates compatible. We will revise the manuscript to make this logical sequence explicit by adding a dedicated remark immediately before the perturbation step, thereby removing any possible ambiguity about circular reasoning. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard multiplier and perturbation techniques

full rationale

The paper derives a sufficient condition on parameters for exponential stability of the closed-loop Rayleigh beam system via the integral multiplier technique and energy perturbation method, with the desired decay rate stated as determinable from that condition. The event-triggering mechanism is posited to be well-defined and Zeno-free while preserving the stability estimates. No quoted step reduces the central stability claim to a fitted input, self-citation chain, or definitional tautology; the analysis remains self-contained relative to the PDE dynamics and does not exhibit the load-bearing interdependence that would force the result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard assumptions of the Rayleigh beam PDE and well-posedness of the boundary control problem; no new free parameters or invented entities are introduced beyond the control gains and triggering threshold.

axioms (1)
  • domain assumption The Rayleigh beam satisfies the standard Euler-Bernoulli-type PDE with appropriate boundary conditions at the controlled end.
    Invoked implicitly when applying the integral multiplier technique to the closed-loop system.

pith-pipeline@v0.9.0 · 5609 in / 1125 out tokens · 41624 ms · 2026-05-20T08:49:28.211252+00:00 · methodology

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Reference graph

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