Nonlinear Boundary Stabilization for Timoshenko Beam System

• Published in: arXiv.org 2014-09
• Main Authors: Oliveira, M L, Feitosa, A J R, M Milla Miranda
• Format: Article
• Language: English
• Subjects: Arrays; Boundary conditions; Continuity (mathematics); Decay; Galerkin method; Initial conditions; Timoshenko beams
• ISSN: 2331-8422

This paper is concerned with the existence and decay of solutions of the following Timoshenko system: $$ \left\|\begin{array}{cc} u"-\mu(t)\Delta u+\alpha_1 \displaystyle\sum_{i=1}^{n}\frac{\partial v}{\partial x_{i}}=0,\, \in \Omega\times (0, \infty),\\ v"-\Delta v-\alpha_2 \displaystyle\sum_{i=1}^{n}\frac{\partial u}{\partial x_{i}}=0, \, \in \Omega\times (0, \infty), \end{array} \right. $$ subject to the nonlinear boundary conditions, $$ \left\|\begin{array}{cc} u=v=0 \,\, in \,\Gamma_{0}\times (0, \infty),\\ \frac{\partial u}{\partial \nu} + h_{1}(x,u')=0\, in\,\, \Gamma_{1}\times (0, \infty),\\ \frac{\partial v}{\partial \nu} + h_{2}(x,v')+\sigma (x)u=0 \, in\, \,\Gamma_{1}\times (0, \infty), \end{array} \right. $$ and the respective initial conditions at \(t=0\). Here \(\Omega\) is a bounded open set of \(\mathbb{R}^n\) with boundary \(\Gamma\) constituted by two disjoint parts \(\Gamma_{0}\) and \(\Gamma_{1}\) and \(\nu(x)\) denotes the exterior unit normal vector at \(x\in \Gamma_{1}\). The functions \(h_{i}(x,s),\,\, (i=1,2)\) are continuous and strongly monotone in \(s\in \mathbb{R}\). The existence of solutions of the above problem is obtained by applying the Galerkin method with a special basis, the compactness method and a result of approximation of continuous functions by Lipschitz continuous functions due to Strauss. The exponential decay of energy follows by using appropriate Lyapunov functional and the multiplier method.}