Riesz basis generation and boundary stabilization of two strings connected by a point mass with variable coefficients

2019 ◽  
Vol 43 (5) ◽  
pp. 2322-2336
Author(s):  
Jamel Ben Amara ◽  
Walid Boughamda
Keyword(s):  
Riesz Basis ◽  
Variable Coefficients ◽  
Point Mass ◽  
Boundary Stabilization

scholarly journals Stabilization of Variable Coefficients Euler-Bernoulli Beam with Viscous Damping under a Force Control in Rotation and Velocity Rotation

2017 ◽  
Vol 9 (6) ◽  
pp. 1
Author(s):  
Bomisso G. Jean Marc ◽  
Tour\'{e} K. Augustin ◽  
Yoro Gozo
Keyword(s):  
Exponential Stability ◽  
Force Control ◽  
Riesz Basis ◽  
Closed Loop ◽  
Variable Coefficients ◽  
Viscous Damping ◽  
Closed Loop System ◽  
Bernoulli Beam ◽  
Spectral System ◽  
Euler Bernoulli Beam

This paper investigates the problem of exponential stability for a damped Euler-Bernoulli beam with variable coefficients clamped at one end and subjected to a force control in rotation and velocity rotation. We adopt the Riesz basis approach for show that the closed-loop system is a Riesz spectral system. Therefore, the exponential stability and the spectrum-determined growth condition are obtained.

scholarly journals Boundary Stabilization of a Semilinear Wave Equation with Variable Coefficients under the Time-Varying and Nonlinear Feedback

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Bei Gong ◽  
Xiaopeng Zhao
Keyword(s):  
Wave Equation ◽  
Riemannian Geometry ◽  
Nonlinear Term ◽  
Variable Coefficients ◽  
Time Varying ◽  
Nonlinear Feedback ◽  
Boundary Stabilization ◽  
Semilinear Wave Equation ◽  
The Stability ◽  
Stability Results

We study the boundary stabilization of a semilinear wave equation with variable coefficients under the time-varying and nonlinear feedback. By the Riemannian geometry methods, we obtain the stability results of the system under suitable assumptions of the bound of the time-varying term and the nonlinearity of the nonlinear term.

scholarly journals EXPONENTIAL STABILISATION OF A TREE-SHAPED NETWORK OF STRINGS WITH VARIABLE COEFFICIENTS

2011 ◽  
Vol 53 (3) ◽  
pp. 481-499 ◽  
Author(s):  
YAN NI GUO ◽  
GEN QI XU
Keyword(s):  
Riesz Basis ◽  
Imaginary Axis ◽  
Variable Coefficients ◽  
Finite Multiplicity ◽  
Feedback Controllers ◽  
Velocity Feedback ◽  
Root Node ◽  
System Operator ◽  
Exponentially Stable ◽  
Isolated Eigenvalues

AbstractIn this paper, we deal with a tree-shaped network of strings with a fixed root node. By imposing velocity feedback controllers on all vertices except the root node, we show that the spectrum of the system operator consists of all isolated eigenvalues of finite multiplicity and is distributed in a strip parallel to the imaginary axis under certain conditions. Moreover, we prove that there exists a sequence of eigenvectors and generalised eigenvectors that forms a Riesz basis with parentheses, and that the imaginary axis is not an asymptote of the spectrum. Thereby, we deduce that the system is exponentially stable.

scholarly journals Boundary Stabilization of the Wave Equation with Time-Varying and Nonlinear Feedback

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Jian-Sheng Tian ◽  
Wei Wang ◽  
Fei Xue ◽  
Pei-Yong Cong
Keyword(s):  
Wave Equation ◽  
Bounded Domain ◽  
Riemannian Geometry ◽  
Nonlinear Term ◽  
Variable Coefficients ◽  
Time Varying ◽  
Nonlinear Feedback ◽  
Boundary Stabilization ◽  
Suitable Assumption ◽  
Uniform Decay

We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying and nonlinear term. By the Riemannian geometry methods and a suitable assumption of nonlinearity and the time-varying term, we obtain the uniform decay of the energy of the system.

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