Exponential Stabilization for Timoshenko Beam with Distributed Delay in the Boundary Control

Abstract

We consider the exponential stabilization for Timoshenko beam with distributed delay in the boundary control. Suppose that the controller outputs are of the form ${\alpha }_{\mathrm{1}}{u}_{\mathrm{1}}(t)+{\beta }_{\mathrm{1}}{u}_{\mathrm{1}}(t-\tau )+{\int }_{-\tau }^{\mathrm{0}}\mathrm{‍}{g}_{\mathrm{1}}(\eta )u_{\mathrm{1}}(t+\eta )d\eta$ and ${\alpha }_{\mathrm{2}}{u}_{\mathrm{2}}(t)+{\beta }_{\mathrm{2}}{u}_{\mathrm{2}}(t-\tau )+{\int }_{-\tau }^{\mathrm{0}}\mathrm{‍}{g}_{\mathrm{2}}(\eta )u_{\mathrm{2}}(t+\eta )d\eta$; where $u_{\mathrm{1}}(t)$ and $u_{\mathrm{2}}(t)$ are the inputs of boundary controllers. In the past, most stabilization results for wave equations and Euler-Bernoulli beam with delay are required ${\alpha }_i>{\beta }_i>\mathrm{0},i=\mathrm{1,2}$. In the present paper, we will give the exponential stabilization about Timoshenko beam with distributed delay and demand to satisfy the lesser conditions for ${\alpha }_i,{\beta }_i,i=\mathrm{1,2}$.