Exact controllability of damped Timoshenko beam

We study the zero controllability problem for the system of two coupled hyperbolic equations which govern the vibrations of the Timoshenko beam with spatially nonhomogeneous coefficients: mass density, flexural rigidity and shear stiffness. The system is considered on a finite interval with a two-parameter family of boundary conditions containing damping terms. The controls are introduced as separable forcing terms gi(x)fi(t), i = 1,2, in the right-hand sides of both equations. We apply the spectral decomposition method 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 force profile functions, gi(x), i = 1,2, are assumed to be given. Our approach is based on the results obtained in the recent work by the author. In this work, a detailed asymptotic and spectral analysis of the nonselfadjoint operator, which generates the dynamics of the Timoshenko beam with the dissipative boundary conditions, was given. It was shown that the aforementioned operator was Riesz spectral, i.e. its generalized eigenvectors formed a Riesz basis in the energy space. Explicit asymptotic formulas for two branches of the Timoshenko beam spectrum were also presented. Based on these spectral results, we reduce the control problem to the corresponding moment problem. We solve this moment problem using the asymptotics of the spectrum and give explicit formulas for the controls.