The Well-Posedness and Regularity of the Euler-Bernoulli Equation with Variable Coefficients

The open loop system of an Euler-Bernoulli plate with variable coefficients and partial boundary Neumann control and collocated observation is considered. The geometric multiplier method on Riemannian manifolds is adopted in investigation. It is shown that the system is well-posed in the sense of D. Salamon and regular in the sense of G. Weiss and the feedthrough operator is computed to be zero. The result implies particularly that the exact controllability of the open-loop system is equivalent to the exponential stability of the closed-loop system under output proportional feedbacks.