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Steklov Institute of Mathematics at St.Petersburg

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PREPRINT 19/2005

M.I. BELISHEV AND A.F. VAKULENKO##
ON A CONTROL PROBLEM FOR THE WAVE EQUATION
IN ${\bold R}^3$

This preprint was accepted November 8, 2005

ABSTRACT:
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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