The exact controllability of EulerBernoulli beam systems with small delays in the boundary feedback controls

This work is concerned with the exact controllability of an EulerBernoulli beam system with small delays in the boundary feedback controls w_{tt}(x,t)+w_xxxx (x,t)=0,\quad xƐ(0,1), t 0, w(0,t)=wx(0,t)=0, t≥ 0, w_xx(1,t-Ɛ)=-k_2^2 w_tx(1,t)-c2w_t(1,t-Ɛ) ,Ɛ 0, k^2_2 + k^2_2 ≠ 0, w_xxx(1,t)=k_1^2w_t(1,t-Ɛ)c_1w_tx(1,t-Ɛ), k_i,c_iƐ R,(i=1,2), with boundary conditions w(x,t)=Φ(x,t), w_t(x,t)=Φ(x,t), -Ɛ 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.