Well-posedness and regularity of Euler-Bernoulli equation with variable coefficient and Dirichlet boundary control and collocated observation

Two types of open‐loop systems of an Euler–Bernoulli equation with variable coefficient and Dirichlet boundary control and collocated observation are considered. The uncontrolled boundary is either hinged or clamped. It is shown, with the help of multiplier method on Riemannian manifold, that in both cases, systems are well‐posed in the sense of D. Salamon and regular in the sense of G. Weiss. In addition, the feedthrough operators are found to be zero. The result implies that the exact controllability of open‐loop is equivalent to the exponential stability of closed‐loop under a proportional output feedback for these systems. Copyright © 2013 John Wiley & Sons, Ltd.