Abstract
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.
Highlights
In this paper, we study the exponential stability property of a damped Euler-Bernoulli beams with variable coefficients under a force feedback in rotation and velocity rotation
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 study the exponential stability property of a damped Euler-Bernoulli beams with variable coefficients under a force feedback in rotation and velocity rotation
Summary
We study the exponential stability property of a damped Euler-Bernoulli beams with variable coefficients under a force feedback in rotation and velocity rotation. In (Wang, 2004), a question has been raised and is valid for our system (1)-(4): Due to the nonuniform physical thickness and/or density of the Euler-Bernoulli beam with the variable coefficient damping γ(x) in equation (1), what conditions are needed to put onto the damping term to guarantee exponential stability? In order to study the eigenvalues of systems with variable coefficients, we will used the two steps provided by Birkhoff’s works (Birkhoff, 1908) and Naimark’s works (Naimark, 1967) This approach was used by many authors for study the Euler-Bernoulli beams equations with variable coefficients
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