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

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, x is an element of (0, 1), t > 0, w(0, t) = w(x)(0, t) = 0, t >= 0, w(xx)(1, t - epsilon)= -k(2)(2)wtx(1, t) - c(2)wt(1, t - epsilon), epsilon > 0, k(1)(2) + k(2)(2) not equal 0, w(xxx)(1, t)=k(1)(2)wt(1, t epsilon) c(1)wtx(1, t epsilon), k(i), c(i) subset of R, (i = 1, 2), with boundary conditions w(x, t) - phi(x, t), wt(x, t) - psi(x, t), -epsilon <= t <= 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. (C) 2017 All rights reserved.