<p>We consider a hybrid system consisting of a nonhomogeneous Euler–Bernoulli beam under axial force, with an inertial end mass, and subject to a single boundary control moment applied at the end opposite to the mass. The system is governed by the equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\rho (x)y_{tt}(x, t) + (\sigma (x)y_{xx}(x, t))_{xx} - (q(x)y_x(x, t))_x = 0, ~ x \in (0, \ell ), \, t > 0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>y</mi> <mrow> <mi mathvariant="italic">tt</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>y</mi> <mrow> <mi mathvariant="italic">xx</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="italic">xx</mi> </mrow> </msub> <mo>-</mo> <msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>y</mi> <mi>x</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mi>x</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.166667em" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\rho >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sigma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$q \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. By employing a novel method, we prove that the spectrum of the system operator does not intersect the imaginary axis and that all the eigenvalues of the underlying system are geometrically simple. Leveraging these results and the Riesz basis approach, we establish the exponential decay of the system’s energy. This study is motivated by the earlier work of Rao (SIAM J. Control Optim. 1995), where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\rho = \sigma \equiv 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>=</mo> <mi>σ</mi> <mo>≡</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\ell = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$q \equiv 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≡</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, by applying multiple usual dampings.</p>

Exponential stability of a nonhomogeneous Euler–Bernoulli Beam with axial force and a concentrated inertial mass

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发表日期：

2025年11月26日

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Jamel Ben Amara,
Samih Ghnimi,
Samir Toumi

摘要

We consider a hybrid system consisting of a nonhomogeneous Euler–Bernoulli beam under axial force, with an inertial end mass, and subject to a single boundary control moment applied at the end opposite to the mass. The system is governed by the equation $\rho (x)y_{tt}(x, t) + (\sigma (x)y_{xx}(x, t))_{xx} - (q(x)y_x(x, t))_x = 0, ~ x \in (0, \ell ), \, t > 0,$ $ρ (x) y_{tt} (x, t) + {(σ (x) y_{xx} (x, t))}_{xx} - {(q (x) y_{x} (x, t))}_{x} = 0, x \in (0, ℓ), t > 0,$ where $\rho >0$ $ρ > 0$ , $\sigma >0$ $σ > 0$ and $q \ge 0$ $q \geq 0$ . By employing a novel method, we prove that the spectrum of the system operator does not intersect the imaginary axis and that all the eigenvalues of the underlying system are geometrically simple. Leveraging these results and the Riesz basis approach, we establish the exponential decay of the system’s energy. This study is motivated by the earlier work of Rao (SIAM J. Control Optim. 1995), where $\rho = \sigma \equiv 1$ $ρ = σ \equiv 1$ , $\ell = 1$ $ℓ = 1$ , and $q \equiv 0$ $q \equiv 0$ , by applying multiple usual dampings.