Geodesic
Approximate solutions to Dirichlet control problems for the wave equation in Sobolev classes and dual observation problems
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 12, pp. 2191-2208

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Dual control and observation problems for the wave equation with variable coefficients subject to Dirichlet boundary conditions are solved by a variational method. This method was earlier proposed by the author for an approximate analysis of linear equations with nonuniform perturbations of the operator. Explicit bounds on the constant that are required to implement the method are obtained using the correct solvability property of the dual observation problem. Finite-dimensional approximations of the control and observation problems are obtained by the difference method preserving the duality relation. The convergence of approximate solutions is established in the norms of the corresponding dual spaces. 
@article{ZVMMF_2006_46_12_a7,
     author = {M. M. Potapov},
     title = {Approximate solutions to {Dirichlet} control problems for the wave equation in {Sobolev} classes and dual observation problems},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {2191--2208},
     publisher = {mathdoc},
     volume = {46},
     number = {12},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_12_a7/}
}
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M. M. Potapov. Approximate solutions to Dirichlet control problems for the wave equation in Sobolev classes and dual observation problems. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 12, pp. 2191-2208. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_12_a7/