On the stability of a laminated Timoshenko problem with interfacial slip in the whole space under frictional dampings or infinite memories

Abstract

Abstract The author of the present paper considered in [16] a model describing a vibrating strucure of an interfacial slip and consists of three coupled hyperbolic equations in one-dimensional bounded interval, where the dissipation is generated by either a frictional damping or an infinite memory, and it is acting only on one component. Some strong, polynomial, exponential and non exponential stability results were proved in [16] depending on the values of the parameters and the regularity of the initial data. The objective of the present paper is to compelete the study of [16] by considering this model in the whole line ℝ and under only one control given by a frictional damping or an infinite memory. When the system is controled via its second or third component (rotation angle displacement or dynamic of the slip), we show that this control alone is sufficient to stabilize our system and get different polynomial stability estimates in the L 2-norm of the solution and its higher order derivatives with respect to the space variable. The decay rate depends on the regularity of the initial data, the nature of the control and the parameters in the system. However, when the system is controled via its first component (transversal displacement), we found a new stability condition depending on the parameters in the system. This condition defines a limit between the stability and instability of the system in the sense that, when this condition is staisfied, the system is polynomially stable. Otherwise, when this condition is not satisfied, we prove that the solution does not converge to zero at all. The proofs are based on the energy method and Fourier analysis combined with judicious choices of weight functions.