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

Serge Nicaise,Ali Wehbe,Mohamad Sammoury,Denis Mercier

doi:10.5269/bspm.v35i3.29266

Abstract

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 $\frac{1}{\sqrt{t}}$.

Highlights

In [21], Wehbe considered a Rayleigh beam clamped at one end and subjected to two dynamical boundary controls at the other end, namely ytt − γyxxtt + yxxxx = 0, y(0, t) = yx(0, t) = 0, yxx(1, t) + aη(t) = 0, yxxx(1, t) − γyxtt(1, t) − bξ(t) = 0,0 < x < 1, t > 0, t > 0, t > 0, t > 0, where γ > 0 is the coefficient of moment of inertia, a > 0 and b > 0 are positive constants, η and ξ denote respectively the dynamical boundary control moment and force
We show that the system of eigenvectors of the damped problem form a Riesz basis
The lack of uniform stability was proved by a compact perturbation argument of Gibson and a polynomial energy decay rate of type 1 t is obtained by a multiplier method usually used for nonlinear problems

Summary

Introduction

In [12], Lagnese studied the stabilization of system (1.1)-(1.4) with two static boundary controls (the case a > 0, b > 0, η(t) = yxt(1, t) and ξ(t) = yt(1, t)) He proved that the energy decays exponentially to zero for all initial data. Using an explicit approximation, they gave the asymptotic expansion of eigenvalues and eigenfunctions of the undamped system corresponding to (1.1)-(1.4), they established the optimal polynomial energy decay rate via an observability inequality of solution of the undamped system and the boundedness of the transfer function associated with the undamped problem. In subsection 3.2, we show that the system of eigenvectors of the damped problem forms a Riesz basis and we establish the optimal polynomial energy decay rate of type √1

Rayleigh beam equation with only one dynamical control moment

W and the energy space

Polynomial Stability for smooth initial data

Optimal polynomial decay rate

Rayleigh beam equation with only one dynamical boundary control force

Riesz basis and polynomial stability with optimal decay rate