Backstepping approach to the arbitrary decay rate for Euler-Bernoulli beam under boundary feedback

In this article, we are concerned with the boundary stabilisation of the Euler-Bernoulli beam equation for which all eigenvalues of the (control) free system are located on the imaginary axis of the complex plane. The fourth-order system in spacial variable is transformed into a coupled heat-like system. This enables us to make a natural backstepping transformation in vector form to transform the system into a target system which has arbitrary decay rate. The state feedback is thus designed. It is shown that the original closed-loop system is exponentially stable with the given arbitrary decay rate.