Well-posedness and regularity for an Euler–Bernoulli plate with variable coefficients and boundary control and observation

The open loop system of an Euler–Bernoulli plate with variable coefficients and partial boundary Neumann control and collocated observation is considered. Using the geometric multiplier method on Riemannian manifolds, we show that the system is well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. Moreover, we determine that the feedthrough operator of this system is zero. The result implies in particular that the exact controllability of the open-loop system is equivalent to the exponential stability of the closed-loop system under proportional output feedback.