Geodesic
On a control problem for the wave equation in~$\mathbf R^3$
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 35, Tome 332 (2006), pp. 19-37

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We consider the solutions of the wave equation (waves) initiated by the infinitely far sources (controls) and study the $L_2$-completeness of the reachable sets consisting of such waves. This problem is a natural analog of the control problem for a bounded domain where the completeness (local approximate controllability) in the subdomains filled with waves generated by boundary controls occurs. We show that, in contrast to the latter case, the reachable sets formed by the waves incoming from infinity, aren't complete in the filled subdomains and describe the corresponding defect. Then, extending the class of controls on a set of special polynomials, we gain the completeness. A transform defined by jumps appearing in result of projecting functions on the reachable sets is introduced. Its relation to the Radon transform is clarified. 
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     author = {M. I. Belishev and A. F. Vakulenko},
     title = {On a control problem for the wave equation in~$\mathbf R^3$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {19--37},
     publisher = {mathdoc},
     volume = {332},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_332_a1/}
}
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M. I. Belishev; A. F. Vakulenko. On a control problem for the wave equation in~$\mathbf R^3$. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 35, Tome 332 (2006), pp. 19-37. http://geodesic.mathdoc.fr/item/ZNSL_2006_332_a1/