Exponential stability of laminated Timoshenko beams with boundary/internal controls
The authors deal with the initial-boundary value problem describing the behavior of the two-layered cantilever Timoshenko beam configured over the interval \(I=(0,\ell)\), composed of the two identical beams of uniform thickness bonded by an adhesive layer of negligible thickness. After introducing the variable \(\xi=3s-\psi\) denoting the effective rotation angle they obtain the system \begin{align*} &mw_{tt}+G(3s-\xi-w_x)_x=0,\\ &I_m\xi_{tt}-G(3s-\xi-w_x)-D\xi_{xx}=0,\\ &I_m s_{tt}+G(3s-\xi-w_x)+\frac {4\gamma}3s+\frac {4\beta}3s_t=0 \ \text{in}\ (0,\ell) \times (0,\infty) \end{align*} with the dissipative boundary conditions \begin{align*} &w(0,t)=\xi(0,t)=s(0,t)=0,\\ &(3s-\xi-w_x)(\ell,t)=\kappa_1w_t(\ell,t),\ \xi_x(\ell,t)=-\kappa_2\xi_t(0,t),\ s_x(\ell,t)=0,\ t\ge 0 \end{align*} and the initial condition \[ (w,w_t,\xi,\xi_t,s,s_t)=(w_0(x),w_1(x),\xi_0(x),\xi_1(x),s_0(x),s_1(x))\ \text{in}\ (0,\ell). \] Using the semigroup framework the authors prove the existence, uniqueness and continuous dependence of a solution with respect to initial data. The main result is the exponential stability of the solution semigroup.
scientific article; zbMATH DE number 7138374
scientific article