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Exponential Stability of a Degenerate Euler-Bernoulli Beam with Axial Force and Delayed Boundary Control

This paper establishes the global exponential stability of a degenerate Euler-Bernoulli beam under non-uniform axial force and delayed boundary control by proving the system's well-posedness via the Lumer-Phillips theorem and deriving a precise decay rate using a novel weighted Lyapunov functional.

Original authors: Ben Bakary Junior Siriki, Adama Coulibaly

Published 2026-02-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ben Bakary Junior Siriki, Adama Coulibaly

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a long, flexible diving board (a beam) sticking out over a pool. In the real world, these boards aren't perfect; they might be made of a material that gets weaker or "degenerate" near the base, or they might be under tension from a heavy weight pulling on them (axial force).

Now, imagine we want to stop this board from wobbling forever after someone jumps off. We attach a smart controller at the far end that tries to push the board back to a flat position. But here's the catch: the controller is a bit slow. It takes a tiny amount of time (a "delay") to react.

This paper is a mathematical investigation into whether we can still stop the wobbling (stabilize the beam) even when:

The board is weak or "degenerate" at one end.
There is a heavy weight pulling on it.
The controller is slow to react.

Here is the story of how the authors solved this puzzle, explained simply.

1. The Problem: A Wobbly, Weak, and Slow System

Usually, engineers assume beams are perfect and controllers are instant. But in reality, materials wear out (degeneracy), and computers take time to process data (delay).

The Degeneracy: Think of the beam as a tree branch. Near the trunk (the base), the wood might be softer or hollow. The math gets tricky here because the usual rules of physics break down at that weak spot.
The Axial Force: Imagine someone pulling the beam tight like a guitar string. This changes how it vibrates.
The Delay: The controller sees the beam move, thinks about it, and then pushes back. If the beam moves very fast, the controller might push at the wrong time, making the wobble worse instead of better.

The big question the authors asked: "Can we design a controller that stops the beam from shaking, even if the beam is weak, the force is uneven, and the controller is slow?"

2. The Solution: Building a New "Energy Map"

To answer this, the authors didn't just guess; they built a rigorous mathematical framework.

Step 1: Proving the System Exists (Well-Posedness)
First, they had to prove that the system actually behaves logically. They created a special "mathematical playground" (a weighted space) where the weak parts of the beam are treated with extra care. They showed that if you start with a specific wobble, the system will evolve in a predictable way and won't explode into chaos. It's like proving that a car with a wobbly wheel will still drive down the road without falling apart, provided you drive it correctly.

Step 2: The "Brake" Test (Exponential Stability)
The core of the paper is proving that the energy of the wobble dies out quickly.

The Analogy of the Brake: Imagine the beam has kinetic energy (the wobble). The controller acts as a brake.
The Danger of Delay: If the brake is applied too late (delay), it might push the car forward instead of stopping it.
The Secret Sauce: The authors found a specific rule: The immediate braking force must be stronger than the delayed force.
- Mathematically, they proved that if the "instant" gain ( $\kappa_1$ ) is bigger than the "delayed" gain ( $\kappa_2$ ), the system is safe.
- They built a special "Lyapunov Functional." Think of this as a super-battery meter. A normal meter just shows how much energy is left. This special meter also accounts for the "memory" of the delay and the "weakness" of the beam. They proved that this meter always goes down, and it goes down fast (exponentially).

3. The Results: How Fast Does It Stop?

The authors didn't just say "it stops." They calculated how fast it stops.

Weak vs. Strong Degeneracy: They found that if the beam is "weakly degenerate" (a little soft at the base), it stops vibrating faster than if it is "strongly degenerate" (very soft/hollow at the base). It's like trying to stop a wobbly table with slightly loose legs vs. a table with one leg made of jelly. The jelly leg makes it harder to stabilize.
The Delay Factor: The longer the delay ( $\tau$ $τ$ ), the slower the stopping process.
- If the delay is tiny, the beam stops very quickly.
- If the delay is huge, the beam still stops (it's stable), but it takes a very long time to settle down. It's like trying to steer a ship with a rudder that reacts 10 minutes late; you can eventually get to the right direction, but you'll be zig-zagging for a long time.

4. Why This Matters

This research is a big deal for engineering because:

Aging Infrastructure: Bridges and buildings get old and develop weak spots (degeneracy). This math helps engineers understand how to keep them stable.
Robotics and Nanotech: Tiny robots and nanomaterials often have delays in their sensors and actuators. This paper provides a safety guarantee for controlling them.
Robustness: It proves that even with imperfections (weakness, tension, and lag), we can still design systems that are safe and stable, as long as we follow the rule: React instantly with more force than you react late.

Summary

In simple terms, the authors took a very difficult physics problem—a wobbly, weak beam controlled by a slow computer—and proved that it can be tamed. They showed that as long as the "instant" control is strong enough to overpower the "slow" control, the beam will eventually stop shaking, and they gave a precise formula for how fast that happens. It's a victory for mathematical engineering, ensuring that even imperfect, aging, or delayed systems can be kept safe and steady.

1. Problem Statement

The paper investigates the global exponential stabilization of a degenerate Euler-Bernoulli beam subjected to two specific physical complexities:

Non-uniform Axial Force: The beam is subjected to an axial force $q(x)$ that varies along the length of the beam.
Degenerate Flexural Rigidity: The flexural rigidity function $\sigma(x)$ vanishes at the boundary $x=0$ ( $\sigma(0)=0$ ), making the system degenerate. The authors distinguish between Weak Degeneracy (WD) and Strong Degeneracy (SD) based on the behavior of $\sigma(x)$ near the singularity.
Delayed Boundary Control: The control law applied at the free end ( $x=1$ ) involves a time delay $\tau > 0$ . The boundary condition at $x=1$ combines instantaneous velocity feedback ( $\kappa_1 u_t$ ) and delayed velocity feedback ( $\kappa_2 u_t(t-\tau)$ ).

Governing Equations:
The system is modeled by the partial differential equation:
$u_{tt} + (\sigma(x)u_{xx})_{xx} - (q(x)u_x)_x = 0$
subject to clamped or hinged conditions at $x=0$ (depending on WD/SD) and mixed boundary conditions at $x=1$ involving the delay.

The Core Challenge:
Time delays are known to induce instabilities in control systems. The authors address the critical question: Can a degenerate beam with a non-uniform axial force be stabilized exponentially despite the presence of a time delay in the boundary control?

2. Methodology

The authors employ a rigorous mathematical framework combining semigroup theory and Lyapunov stability analysis.

A. Functional Setting and Well-Posedness

Weighted Sobolev Spaces: To handle the degeneracy at $x=0$ , the authors construct appropriate Hilbert spaces ( $H^2_{\sigma,0}$ and $Q_{\sigma,0}$ ) equipped with weighted inner products involving $\sigma(x)$ and $q(x)$ .
Augmented System: To manage the time delay $\tau$ , they introduce an auxiliary variable $w(s,t) = u_t(1, t-s\tau)$ to transform the delay differential equation into a hyperbolic transport equation. This converts the original problem into an augmented system.
Semigroup Theory: The augmented system is formulated as an abstract Cauchy problem $\frac{dY}{dt} = AY$ in a Hilbert state space. Using the Lümer-Phillips theorem, they prove that the linear operator $A$ is $m$ -dissipative and generates a $C_0$ -semigroup of contractions, thereby establishing the well-posedness (existence and uniqueness) of the solution.

B. Stability Analysis (Lyapunov Method)

To prove exponential stability, the authors construct a novel Lyapunov functional $L(t)$ :
$L(t) = E(t) + \epsilon G(t)$

$E(t)$ : The natural energy of the system, which includes kinetic energy, potential energy from bending and axial force, and a term accounting for the delay energy.
$G(t)$ : An auxiliary functional constructed using weighted integral terms (involving $x u_x$ and $u$ ) and exponential weights for the delay term. This term is crucial for handling the degeneracy and the delay simultaneously.
Key Estimates:
1. They prove the equivalence between $L(t)$ and $E(t)$ .
2. They derive refined estimates for the time derivative $\frac{d}{dt}G(t)$ , utilizing integration by parts and the specific boundary conditions.
3. They utilize a variational approach (Lax-Milgram theorem) to solve an associated degenerate elliptic problem, allowing them to bound the boundary terms $u(1,t)$ and $u_x(1,t)$ by the energy $E(t)$ .

3. Key Contributions

Unified Framework for Degeneracy and Delay: This is one of the first works to simultaneously address flexural rigidity degeneracy, non-uniform axial forces, and time-delayed boundary control in a single Euler-Bernoulli model.
Novel Lyapunov Functional: The construction of the functional $G(t)$ with specific weights ( $\iota_{\sigma,q}$ ) tailored to the degeneracy index $\iota_\sigma$ and the axial force gradient allows for the derivation of sharp decay estimates.
Stability Condition: The authors establish that exponential stability is achievable if the direct damping gain dominates the delayed gain:
$\kappa_1 > |\kappa_2|$
This condition ensures that the destabilizing effect of the delay is overcome by the instantaneous feedback.
Quantitative Decay Rate: Unlike qualitative stability proofs, this paper provides an explicit estimate for the decay rate constant $M$ , showing its dependence on system parameters ( $\sigma, q, \tau, \kappa_1, \kappa_2$ ).

4. Main Results

Well-Posedness: The system admits a unique solution for any initial data in the appropriate energy space.
Exponential Stability: Under the condition $\kappa_1 > |\kappa_2|$ , the energy of the system decays exponentially to zero:
$E(t) \leq E(0) e^{1 - t/M}, \quad \forall t \geq M$
where $M$ is a positive constant independent of the initial data.
Impact of Degeneracy Type:
- Weak Degeneracy (WD): The energy dissipates faster.
- Strong Degeneracy (SD): The energy dissipates slower. The authors show that stronger degeneracy impedes the transmission of energy to the dissipative boundary, resulting in a larger decay constant $M$ .
Impact of Time Delay:
- The system remains stable for any $\tau > 0$ provided $\kappa_1 > |\kappa_2|$ .
- However, as $\tau \to \infty$ , the decay rate deteriorates significantly (the constant $M \to \infty$ ), effectively losing the practical exponential decay property.
- As $\tau \to 0$ , the system becomes highly efficient with a rapid decay rate.

5. Significance and Future Perspectives

Engineering Relevance: The results are critical for the control of aging infrastructure (bridges, nanotubes, aerospace structures) where material degradation leads to degeneracy and where sensor/actuator delays are unavoidable.
Theoretical Advancement: The work extends stability theory to complex distributed parameter systems where standard non-degenerate models fail.
Future Work: The authors identify the case where $\kappa_1 \leq |\kappa_2|$ (delay gain exceeds direct damping) as an open problem requiring frequency domain analysis. They also propose numerical studies to quantify sensitivity and extensions to models with distributed singular damping.

In summary, this paper provides a robust analytical proof that a degenerate beam with axial forces can be exponentially stabilized via delayed boundary control, provided the instantaneous feedback is sufficiently strong, and quantifies exactly how degeneracy and delay magnitude affect the speed of stabilization.