The exact controllability of Euler-Bernoulli beam systems with small delays in the boundary feedback controls

Volume 10, Issue 5, pp 2778--2787 http://dx.doi.org/10.22436/jnsa.010.05.44

Publication Date: May 25, 2017 Submission Date: February 15, 2017

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Authors

Zhang Zhuo - Basic Course Department, Business College of Shanxi University, Taiyuan 030031, Shanxi, P. R. China.

Abstract

This work is concerned with the exact controllability of an Euler-Bernoulli beam system with small delays in the boundary feedback controls \[w_{tt}(x,t)+w_{xxxx}(x,t)=0,\quad x\in (0,1)\quad t>0, \] \[w(0,t)=w_x(0,t)=0,\quad t\geq 0,\] \[w_{xx}(1,1-\varepsilon)=-k_2^2 w_{tx}(1,t)-c_2w_t(1,t-\varepsilon),\quad \varepsilon>0,\quad k_1^2+k_2^2\neq 0,\] \[w_{xxx}(1,t)=k_1^2w_t(1,t-\varepsilon)-c_1w_{tx}(1,t-\varepsilon),\quad k_i,c_i\in R,\quad (i=1,2),\] with boundary conditions \[w(x,t)=\varphi(x,t), \quad w_t(x,t)=\psi(x,t), \quad -\varepsilon\leq t\leq 0.\] Our analysis relies on the exact controllability on Hilbert space M and state space H. Our results based on formulating the original system as a state linear system. We formulate the system as the state feedback control systems \(\Sigma(A, B,C)\), and we get the generalized eigenvectors of the operator A. Then we prove that they can form a Riesz basis for the state space H. In the end, the system is proved to be exactly controllable on H.

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Keywords

• Euler-Bernoulli beam
• delay
• boundary feedback control
• exact controllability.

MSC

•  35J30
•  35J35

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