Nonlinear boundary stabilization of parallelly connected Kirchhoff plates
In this very important paper a system of nonlinearly coupled Kirchhoff plates is considered. The dynamic problem is modelled by a coupled system of two nonlinearly coupled Kirchhoff equations: \[ \begin{cases} v_{tt} - \gamma^2 \Delta v_{tt} + \Delta^2 v = k_1 (u - v) \\ u_{tt} - \gamma^2 \Delta u_{tt} + \Delta^2 u = k_2 (v - u) \end{cases} \quad \text{in } Q_\infty = (0, \infty) \times \Omega \] and boundary conditions on \(\Sigma_\infty \equiv \Gamma \times (0, \infty)\) \[ \begin{cases} \Delta v + (1 - \mu) B_1 v = - f_1 \left( {\partial \over \partial \nu} v_t \right) \\ {\partial \over \partial \nu} \Delta v + (1 - \mu) B_2 v - \gamma^2 {\partial \over \partial \nu} v_{tt} - v + {\partial \over \partial \tau} h_1 \left( {\partial \over \partial \tau} (v - u) \right) \\ - \ell_1 (v - u) = g(v_t) - {\partial \over \partial \tau} f_2 \left( {\partial \over \partial \tau} v_t \right) \end{cases} \] \[ \begin{cases} \Delta u + (1 - \mu) B_1 u = - f_1 \left( {\partial \over \partial \nu} u_t \right) \\ {\partial \over \partial \nu} \Delta u + (1 - \mu) B_2 u - \gamma^2 {\partial \over \partial \nu} u_{tt} - u + {\partial \over \partial \tau} h_2 \left( {\partial \over \partial \tau} (u - v) \right) \\ - \ell_2 (u - v) = g (u_t) - {\partial \over \partial \tau} f_2 \left( {\partial \over \partial \tau} u_t \right) \end{cases} . \] The function \(k_i (s)\), \(h_i (s)\), \(\ell_i (s)\) are assumed to be differentiable with the property that for \(s \in R\): \(sk_i (s) \geq 0\), \(sh_i (s) \geq 0\), \(s \ell_i (s) \geq 0\), \(|h_i (s) |\leq C |s |^R\) for some \(K > 0\) and \(|s |> 1\). Main result: It is proved that the boundary dissipation acting via moments and shears applied to the edge of the plate and represented by the nonlinear functions \(f_i\) and \(g\) (the functions \(g\) and \(f_i\) are continuous, monotone, zero at the origin and subject to the following growth conditions: \(m |s |\leq |f_i (s) |\leq M |s |\), for \(|s |> 1\), \(i = 1,2\), \(|g(s) |\leq M |s |^r\) for \(|s |> 1\) for some positive constants \(m,M\) and \(r)\) produces uniform decay of the energy.
scientific article; zbMATH DE number 911966