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Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square. (English) Zbl 0939.93016
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.
Reviewer: T.Nambu (Kobe)

##### MSC:
 93C20 Control/observation systems governed by partial differential equations 93B07 Observability 35L20 Initial-boundary value problems for second-order hyperbolic equations
##### Keywords:
discretized wave equation; observability
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