Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square
The paper studies the observability problem of the semi-discrete approximation -- in space -- of the wave equation: \[ u_{tt}-\Delta u=0,\quad \text{in }\Omega\times (0,T), \] \[ u(x,0)= u^0,\quad u_t(x,0)= u^1,\quad\text{in }\Omega,\quad u=0,\quad\text{in }\partial\Omega\times (0,T), \] where \(\Omega\) denotes the rectangle: \((0,\pi)\times (0,\pi)\subset \mathbb{R}^2\). Let \(\Gamma_0\) denote a subset of \(\partial\Omega\) constituted by two consecutive sides. The observability of this system means the inequality of the energy: \[ E(0)= {1\over 2} \int_\Omega\{|\nabla u^0|^2+|u^1|^2\} dx\leq C(T) \int^T_0 dt\int_{\Gamma_0} \Biggl|{\partial u\over\partial n}\Biggr|^2 d\sigma,\quad T> 2\sqrt 2\pi. \] Given \(J, K\in\mathbb{N}\), let \(h_1= \pi/(J+ 1)\) and \(h_2= \pi/(K+ 1)\) be the mesh size. The finite-difference semi-discretization of the wave equation with states \(u_{jk}(t)\), \(1\leq j\leq J\), \(1\leq k\leq K\) and the corresponding energy \(E_{h_1h_2}(t)\) are introduced. The observability inequality to this discretization contains a constant \(C_{h_1h_2}(T)\), \(T>0\), which corresponds to the above \(C(T)\). It is shown that \(C_{h_1h_2}(T)\) goes to infinity for a particular choice of \(h_1\) and \(h_2\) with \(h_1,h_2\to\infty\). A class of solutions generated by low frequencies is extracted so that the uniform observability inequality would hold.
scientific article; zbMATH DE number 1344459