Stability analysis of an Euler-Bernoulli beam with joint controls at an arbitrary internal point

This paper considers pointwise stabilization of an Euler-Bernoulli beam equation where two kinds of feedback are jointed with each other for the arbitrary internal point in the boundary conditions. The asymptotic expressions of the eigenvalues are obtained for all sufficient large positive integers. And then the asymptotical stability and exponential stability have been analyzed by the characteristic equation. It is shown that the pointwise control system is asymptotically stable for all of the jointed internal points and not exponentially stable for any joint internal point.