Boundary controllability of coupled degenerate Euler-Bernoulli beam equations

Abstract

The aim of this paper is to study the null controllability property for two coupled degenerate beam equations where the control function acts only on one of the two equations. The beam equations are coupled via velocity damping terms, where the velocity of the second equation appears in the first equation and vice versa. As a first step, we analyze the well-posedness of the associated homogeneous adjoint problem in suitable functional spaces using semigroup theory. Then, we prove the validity of an indirect observability estimate for this kind of problem in a sufficiently large time, provided that the coupling coefficient is assumed to be positive and small. By means of this estimate and applying the classical Hilbert uniqueness method, we deduce that the original system is null controllable. We also give the explicit controllability time which depends both on the degree of the degeneration and the coupling term.