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

Resumo

In \cite{WehbeRayleigh:06}, Wehbe considered the Rayleigh beam equation with two dynamical boundary controls and established the optimal polynomial energy decay rate of type $\dfrac{1}{t}$. The proof exploits in an explicit way the presence of two boundary controls, hence the case of the Rayleigh beam damped by only one dynamical boundary control remained open. In this paper, we fill this gap by considering a clamped Rayleigh beam equation subject to only on dynamical boundary feedback. First, we consider the Rayleigh beam equation subject to only one dynamical boundary control moment. We give the asymptotic expansion of eigenvalues and eigenfunctions of the undamped underlying system and we establish a polynomial energy decay rate of type $\frac{1}{t}$ for smooth initial data via an observability inequality of the corresponding undamped problem combined with the boundedness property of the transfer function of the associated undamped problem. Moreover, using the real part of the asymptotic expansion of eigenvalues of the damped system, we prove that the obtained energy decay rate is optimal. Next, we consider the Rayleigh beam equation subject to only one dynamical boundary control force. We start by giving the asymptotic expansion of the eigenvalues and the eigenfunctions of the damped and undamped systems using an explicit approximation of the characteristic equation determining these eigenvalues. We next show that the system of eigenvectors of the damped problem form a Riesz basis. Finally, we establish the optimal energy decay rate of polynomial type $\frac{1}{\sqrt{t}}$.