Stabilization for the Kirchhoff type equation from an axially moving heterogeneous string modeling with boundary feedback control

The first objective of this paper is to make the mathematical model for vibration suppression of an axially moving heterogeneous string. In order to describe the geometrical nonlinearity due to finite transverse deformation, the exact expression of the strain is used. The mathematical modeling is derived first by using Hamilton’s principle and variational lemma and the derived nonlinear PDE system is the Kirchhoff type equation with boundary feedback control. Next, we show the existence and uniqueness of strong solutions of the PDE system via techniques of functional analysis, mainly a theorem of compactness for the analysis of the approximation of the Faedo–Galerkin method and estimate a decay rate for the energy. The theoretical results are assured by numerical results of the solution’s shape and asymptotic behavior for the system.