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On a control problem for the wave equation in \(\mathbb R^3\). (Russian, English) Zbl 1162.35013

Zap. Nauchn. Semin. POMI 332, 19-37 (2006); translation in J. Math. Sci., New York 142, No. 6, 2528-2539 (2007).
The authors consider the solutions of the wave equation (waves) initiated by the infinitely far controls and study the \(L_2\)-completeness of the reachable sets consisting of such waves. It is shown that the reachable sets formed by the waves incoming from the 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, they obtain the completeness. A transform defined by jumps appearing in result of projecting functions on the reachable sets is introduced and its relation to the Radon transform is analized.

MSC:

93C20 Control/observation systems governed by partial differential equations
35L05 Wave equation
93B03 Attainable sets, reachability
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References:

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