Uniform stabilization and simultaneous exact boundary controllability for a pair of hyperbolic systems
The following mixed problem in the cylinder \(\Omega\times (0, T)\) is considered: \[ {\partial^2 u\over \partial t^2}- {\partial\over \partial x_i} \Biggl(A_{ij} {\partial u\over \partial x_j}\Biggr)= 0,\quad {\partial^2 v\over \partial t^2}- {\partial\over \partial x_i} \Biggl(A_{ij} {\partial v\over \partial x_j}\Biggr)= 0, \] \[ u|_{t= 0}= \varphi(x),\;v|_{t= 0}= \psi(x),\;{\partial u\over \partial t}\Biggl|_{t= 0}= \varphi_1(x),\;{\partial v\over \partial t}\Biggr|_{t= 0}= \psi_1(x), \] \[ A_{ij} {\partial u\over \partial x_j} \nu_i+ au+ bu_t+ \xi v_t|_{S_0}= 0,\quad u|_{S_1}= 0, \] \[ A_{ij} {\partial v\over \partial x_j} \nu_i+ cv+ dv_t+ \xi u_t|_{S_0}= 0,\quad v|_{S_1}= 0, \] where the smooth boundary \(S\) of \(\Omega\) consists of two disjoint closed surfaces \(S_0\) and \(S_1\), \(A_{ij}= A^*_{ji}\) are square matrices of order \(m\), \(\nu= (\nu_1, \nu_2,\dots, \nu_n)\) is the unit outward normal to the boundary \(S\) and \(a\), \(b\), \(c\), \(d\), \(\xi\) are constants. Using a basic energy identity, the author constructs some stabilizing operators for the above pair of second-order hyperbolic systems; namely, for \(b> 0\), \(d> 0\) the stabilizing boundary operator represents \(2n\) scalar conditions and for \(b> 0\), \(d= 0\), the stabilizing operator is determined by only \(n\) scalar conditions. In this way, he proves the uniform simultaneous stabilization of solutions to the corresponding mixed problems. According to the uniform stabilization with a stabilizing operator having \(n\) scalar conditions, one obtains the exact controllability of the corresponding evolution system with control in only \(n\) boundary conditions on \(S_0\). Finally, the author considers the case of a pair of systems from elasticity theory.
scientific article; zbMATH DE number 917092
scientific article