Geodesic
A more accurate threshold in bilateral control and observation problems for the wave equation
Numerical methods and programming, Tome 8 (2007) no. 2, pp. 147-153

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Bilateral Dirichlet and Neumann boundary control problems with strong generalized solutions are considered for the wave equation with variable coefficients. The corresponding dual observation problems are formulated in dual classes of weak generalized solutions. The main results are obtained in the form of estimates with explicit constants and are ready to be used for finding stable approximate solutions. The value of the most important parameter called the controllability-observability threshold is brought into proximity with the known optimal level and coincides with it in the case of constant coefficients.
Keywords: wave equation, controllability, observability, duality, finite-dimensional approximation. 
@article{VMP_2007_8_2_a0,
     author = {M. M. Potapov},
     title = {A more accurate threshold in bilateral control and observation problems for the wave equation},
     journal = {Numerical methods and programming},
     pages = {147--153},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2007_8_2_a0/}
}
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M. M. Potapov. A more accurate threshold in bilateral control and observation problems for the wave equation. Numerical methods and programming, Tome 8 (2007) no. 2, pp. 147-153. http://geodesic.mathdoc.fr/item/VMP_2007_8_2_a0/