Optimal energy decay rate of Rayleigh beam equation with only one boundary control force

We consider a clamped Rayleigh beam equation subject to only one boundary control force. Using an explicit approximation, we first give the asymptotic expansion of eigenvalues and eigenfunctions of the undamped underlying system. We next establish a polynomial energy decay rate for smooth initial data via an observability inequality of the corresponding undamped problem combined with a boundedness property of the transfer function of the associated undamped problem. Finally, by a frequency domain approach, using the real part of the asymptotic expansion of eigenvalues of the infinitesimal generator of the associated semigroup, we prove that the obtained energy decay rate is optimal.