Exponential and polynomial stabilization of the Kirchhoff string by nonlinear boundary control

Yuhu Wu,Jianmin Wang

doi:10.1186/1687-2770-2013-215

Yuhu Wu, Jianmin Wang

Open Access

https://doi.org/10.1186/1687-2770-2013-215

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Abstract

This paper addresses the stabilization problem of the nonlinear Kirchhoff string using nonlinear boundary control. Nonlinear boundary control is the negative feedback of the transverse velocity of the string at one end, which satisfies a polynomial-type constraint. Employing the multiplier method, we establish explicit exponential and polynomial stability for the Kirchhoff string. The theoretical results are assured by numerical results of the asymptotic behavior for the system.

Highlights

1 Introduction Stabilization and vibration controllability of string or beam systems arising from different engineering backgrounds has attracted attention of many researchers [ – ]
Boundary feedback stabilization of string and beam systems has become an important research area [ – ]. This is because, in a practice system, vibration is more controlled through a boundary point than using point sensors or actuators away from the boundaries [, ]
There are several nonlinear mathematical models that describe the transversal vibration of stretched strings

Summary

Introduction

Stabilization and vibration controllability of string or beam systems arising from different engineering backgrounds has attracted attention of many researchers [ – ]. Shahruz and Krishna [ ] investigated the stabilization of Kirchhoff string ( ) with a linear negative velocity control, which means the boundary control u has a linear negative velocity feedback form u(t) := u yt( , t) = –Lyt( , t) for all t ≥ , where L is a positive constant. In the literature mentioned above, such as [ , ] and [ ], the exponential stabilization result for various string systems by linear boundary control mainly relies on the Lyapunov direct method. For any constant q ≥ , the energy function E along the solution of system ( a)-( d) satisfies the following estimation, for all S > T ≥ ,. According to hypothesis (H), we know that s ≤ – u(s)s, L u (s) ≤ –L u(s)s for all s ∈ R

Recalling q

Applying the with α