Exact controllability and boundary stabilization of flexural vibrations of an internally damped flexible space structure.
The authors consider damping of vibrations of a flexible rectangular plate, rigidly connected to a hub on one end of its length with the other end free. The usual fourth-order partial differential equation models the dynamics of this plate, but a fifth-order term is present. It represents the Voigt-type internal damping of the plate's motion. The damping constant is assumed to be ``small''. At the hub end \((x=0)\) a control force \(Q(t)\) is opposed by elastic and damping terms represented by first, second, and third-order partial derivatives. The visco-elastic term is assumed to be ``small'' compared to the elastic term. This is the background of this study. The theorem on exact controllability of this system is proved by using the Hilbert uniqueness method (HUM). This technique, first proposed by J. L. Lions and later popularized and improved by Lagnese, is explained in detail, saving some readers the trip to the library. Following this type of approach by application of the multipliers technique, and by pages of clever manipulations, they prove exact controllability of their system. They also show that the system's energy decays exponentially with time. Obviously, if the Voigt term is absent, the results still apply, thus their results routinely are true for the Euler-Bernoulli beam.
scientific article; zbMATH DE number 1861026
scientific article