The well-posedness and stability of a beam equation with conjugate variables assigned at the same boundary point

A Euler-Bernoulli beam equation subject to a special boundary feedback is considered. The well-posedness problem of the system proposed by G. Chen is studied. This problem is in sharp contrast to the general principle in applied mathematics that the conjugate variables cannot be assigned simultaneously at the same boundary point. We use the Riesz basis approach in our investigation. It is shown that the system is well-posed in the usual energy state space and that the state trajectories approach the zero eigenspace of the system as time goes to infinity. The relaxation of the applied mathematics principle gives more freedom in the design of boundary control for suppression of vibrations of flexible structures.