Nonlinear Boundary Stabilization for Timoshenko Beam System

TL;DR

This paper investigates the existence and exponential decay of solutions for a nonlinear boundary-controlled Timoshenko beam system using Galerkin, compactness, and Lyapunov methods.

Contribution

It introduces a novel approach combining Galerkin, compactness, and Strauss's approximation to prove solution existence and decay in a nonlinear boundary setting.

Findings

Proved existence of solutions using Galerkin and compactness methods.

Established exponential energy decay via Lyapunov functional.

Handled nonlinear boundary conditions with strong monotonicity assumptions.

Abstract

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.}