scholarly journals Riesz basis property and exponential stability for one-dimensional thermoelastic system with variable coefficients

2021 ◽  
Author(s):  
Bao-Zhu Guo ◽  
Han-Jing Ren
Keyword(s):  
Exponential Stability ◽  
Riesz Basis ◽  
Basis Property ◽  
Coupled System ◽  
Variable Coefficients ◽  
Operator Pencil ◽  
Riesz Basis Property ◽  
Asymptotic Expressions ◽  
The Matrix ◽  
Constant Coefficients

In this paper, we study Riesz basis property and stability for a nonuniform thermoelastic system with Dirichlet-Dirichlet boundary condition, where the  heat subsystem is considered as a control to the whole coupled system. By means of the matrix operator pencil method, we obtain the asymptotic expressions of the eigenpairs, which are exactly coincident to the constant coefficients case} We then show that there exists a sequence of generalized eigenfunctions of the system,  which forms a Riesz basis for the state space and the spectrum determined growth condition is therefore proved. As a result, the exponential stability of the system is concluded.

Stability and regularity transmission for coupled beam and wave equations through boundary weak connections

2020 ◽  
Vol 26 ◽  
pp. 73
Author(s):  
Bao-Zhu Guo ◽  
Han-Jing Ren
Keyword(s):  
Wave Equation ◽  
Real Axis ◽  
Wave Equations ◽  
Basis Property ◽  
Coupled System ◽  
Finite Point ◽  
The Real ◽  
Riesz Basis Property ◽  
Asymptotic Expressions ◽  
The Stability

In this paper, we consider stability for a hyperbolic-hyperbolic coupled system consisting of Euler-Bernoulli beam and wave equations, where the structural damping of the wave equation is taken into account. The coupling is actuated through boundary weak connection in the sense that after differentiation of the total energy for coupled system, only the term of the wave equation appears explicitly. We first show that the spectrum of the closed-loop system consists of three branches: one branch is basically along the real axis and accumulates to a finite point; the second branch is also along the real line; and the third branch distributes along two parabola likewise symmetric with the real axis. The asymptotic expressions of both eigenvalues and eigenfunctions are obtained by means of asymptotic analysis. With an estimation of the resolvent operator, the completeness of the root subspace is proved. The Riesz basis property and exponential stability of the system are then concluded. Finally, we show that the associated C0-semigroup is of Gevrey class, which shows that not only the stability but also regularity have been transmitted from regular wave subsystem to the whole system through this boundary connections.

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.

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