On a control problem for the wave equation in R3

Solutions of the wave equation (waves) initiated by infinitely distant sources (controls) are considered, and the L2-completeness of reachable sets consisting of such waves is stidued. This problem is a natural analog of the control problem for a bounded domain where the completeness (local approximate controllability) in subdomains filled with waves generated by boundary controls occurs. It is shown that, in contrast to the latter case, the reachable sets formed by waves incoming from infinity are not complete in filled subdomains and describe the associated defect. Next, extending the class of controls to a set of special polynomials, the completeness is gained. A transform defined by jumps that arise in projecting functions to reachable sets is introduced. Its relevance to the Radon transform is clarified. Bibliography: 7 titles. © Springer Science+Business Media, Inc. 2007.