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

Abstract

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.

Keywords:

• beam equation
• stability
• backstepping
• boundary control

Acknowledgements

This work was carried out with the support of the National Natural Science Foundation of China and the National Research Foundation of South Africa.