Riesz basis and exponential stability for Euler-bernoulli beams with variable coefficients and indefinite damping under a force control in position and velocity

Abstract

This article concerns the Riesz basis property and the stability of a damped Euler-Bernoulli beam with nonuniform thickness or density, that is clamped at one end and is free at the other. To stabilize the system, we apply a linear boundary control force in position and velocity at the free end of the beam. We first put some basic properties for the closed-loop system and then analyze the spectrum of the system. Using the modern spectral analysis approach for two-points parameterized ordinary differential operators, we obtain the Riesz basis property. The spectrum-determined growth condition and the exponential stability are also concluded.

Keywords

• Beam equation
• asymptotic analysis
• Riesz basis
• exponential stability