Stability of an interconnected system of euler−bernoulli beam and heat equation with boundary coupling^∗

¹ School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P.R. China.
jmwang@bit.edu.cn
² Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 92093-0411, USA.
krstic@ucsd.edu

Received: 23 January 2013
Revised: 4 January 2014

Abstract

We study the stability of an interconnected system of Euler−Bernoulli beam and heat equation with boundary coupling, where the boundary temperature of the heat equation is fed as the boundary moment of the Euler−Bernoulli beam and, in turn, the boundary angular velocity of the Euler−Bernoulli beam is fed into the boundary heat flux of the heat equation. We show that the spectrum of the closed-loop system consists only of two branches: one along the real axis and the another along two parabolas symmetric to the real axis and open to the imaginary axis. The asymptotic expressions of both eigenvalues and eigenfunctions are obtained. With a careful estimate for the resolvent operator, the completeness of the root subspaces of the system is verified. The Riesz basis property and exponential stability of the system are then proved. Finally we show that the semigroup, generated by the system operator, is of Gevrey class δ> 2.