Abstract
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.
Highlights
We consider a clamped Rayleigh beam equation subject to only one boundary control force
Rao [23] studied the stabilization of Rayleigh beam equation subject
Using a methodology introduced in [1], we 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
Summary
We define the linear operators A ∈ L(W, W ), B ∈ L(V, V ), C ∈ L(V, V ), by e23 (2.3). Assume that Ay ∈ V , we can formulate the variational equation (2.2) into the following form e26 (2.6). We introduce the linear unbounded operator A0 by e28 (2.7). We give the following characterization of the linear bounded operator Bβ. Let y0(x) = γ−1/2 cosh−1(γ−1/2) sinh(γ−1/2x) and define the linear bounded operator B by eb (2.13). (3) For all y ∈ V , C−1By = BB∗y. CBB∗y = Cy(1)B1 = y(1)Cy0 = y(1)δ1 = By. we will formulate problem (2.6) into the following closed loop system (2.16). Assume that β > 0 and let γ0 be the solution of the equation (2.17).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
