Optimal vibration quenching for an Euler-Bernoulli beam (Q1818992)

The authors study the problem of the optimal vibration quenching for Euler-Bernoulli beam under tension with general linear homogeneous boundary condition. The problem can be written in the form \(m(x)W_{tt}+ L[W]= u(x,t)\), \(0<x<l\), \(0<t<T\), where \(L[W]= [EI(x)W_{xx}]_{xx}- [P(x)W_x]_x\), \(W(x,t)\) -- the displacement, \(m\) -- the unit length mass, \(P\) -- the axial tension force, \(E\) -- the Young's modulus, \(I\) -- the inertia moment, and \(T\) -- the terminal time. The initial and boundary conditions are \(W(x,0)= f(x)\), \(W_t(x,0)= g(x)\), \(f\in H^1(0,l)\), \(g\in L^2(0,l)\); \(B_j[W(x,t,u)]_{x= 0}= 0\), \(j= 1,2\); \(B_j[W(x,t,u)]_{x=l} =0\), \(j= 3,4\), \(B_j\) are some functions. Using the distributed control method, the authors construct the optimal quencher and the quenching modes in the form \(u(x,t)= \sum^\infty_{n=1} U_n(t)\varphi_n(x)\), where \(\varphi_n(x)\in L^2[0,l]\) are complete set of orthogonal eigenfunctions, \(U_n(t)= \sigma_n\cos\lambda_n t+\mu_n\sin \lambda_n t\), \(\sigma_n,\mu_n\in \mathbb{R}\). The authors show that, under some conditions, there exists \(u(x,t)\) that quenches the solution \(W(x,t)\) for all modes, and that this solution is unique. Moreover, it is proved that there exists a number \(N\) and a control \(u_N(x,t)\) which quench the first \(N-1\) modes of \(W\) at \(t=T\), and that the proposed method can be extended to determine a control to quench all modes of the vibration. These theoretical results are applied to some particular cases, which leads to numerical and graphical results.