Exact controllability of damped coupled Euler-Bernoulli and Timoshenko beam model

The zero controllability problem for the system of two coupled hyperbolic equations which governs the vibrations of the coupled Euler-Bernoulli and Timoshenko beam model is studied in the paper. The system is considered on a finite interval with a two-parameter family of physically meaningful boundary conditions containing damping terms. The controls are introduced as separable forcing terms gi (x) f(i) (t), i = 1, 2, on the right-hand sides of both equations. The force profile functions gi (x), i = 1, 2, are assumed to be given. To construct the controls fi (t), i = 1, 2, which bring a given initial state of the system to zero on the specific time interval [0, T], the spectral decomposition method has been applied. The approach, used in the present paper, is based on the results obtained in the recent works by the author and the collaborators. In these works, the detailed asymptotic and spectral analyses of the non-self-adjoint operators generating the dynamics of the coupled beam have been carried out. It has been shown that for each set of the boundary parameters, the aforementioned operator is Riesz spectral, i.e. its generalized eigenvectors form a Riesz basis in the energy space. Explicit asymptotic formulas for the two-branch spectrum have also been derived. Based on these spectral results, the control problem has been reduced to the corresponding moment problem. To solve this moment problem, the asymptotical representation of the spectrum and the Riesz basis property of the generalized eigenvectors have been used. The necessary and/or sufficient conditions for the exact controllability are proven in the paper and the explicit formulas for the control laws are given. The case of the approximate controllability is discussed in the paper as well.