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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 1991 Volume 182, Number 3, Pages 307–331 (Mi sm1296)

This article is cited in 37 papers

Boundary control and a matrix inverse problem for the equation $u_{tt}-u_{xx}+V(x)u=0$

S. A. Avdonin, M. I. Belishev, S. A. Ivanov


Abstract: The authors solve the problem of recovering the matrix-valued potential $V(x)$, $x>0$, from the given reaction operator $R\colon u(0,t)\mapsto u_x(0,t)$, $t>0$. They show the connections between this problem and the theory of boundary control, which allows them to obtain analogues of the classical Gel'fand–Levitan–Krein equations. They establish the basis property for a family of vector-valued exponentials; this property is connected with the spectral characteristics of the boundary value problem. They prove the controllability of the corresponding system under a boundary control $u(0,t)=f(t)$.

UDC: 517.9

MSC: Primary 35L20, 35B37; Secondary 35R30, 49N50, 34B24, 93B05

Received: 15.01.1990


 English version:
Mathematics of the USSR-Sbornik, 1992, 72:2, 287–310

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