WIAS Preprint No. 442, (1998)

Reconstruction of source terms in evolution equations by exact controllability



Authors

  • Yamamoto, Masahiro

2010 Mathematics Subject Classification

  • 35L55 35R30 93B05

Keywords

  • Source term, reconstruction, operator equation of second kind, Fredholm equation of second kind, Hilbert Uniqueness method, exact controllability

Abstract

For fixed $rho = rho(x,t)$, we consider the solution $u(f)$ to $$ u''(x,t) + Au(x,t) = f(x)rho(x,t), quad x in Omega, thinspace t > 0 $$ $$ u(x,0) = u'(x,0) = 0, qquad x in Omega, qquad B_ju(x,t) = 0, quad x in partialOmega, thinspace t > 0, thinspace 1 le j le m, $$ where $u'= fracpartial upartial t$, $u'' = fracpartial^2 u partial t^2$, $Omega subset R^r$, $r ge 1$ is a bounded domain with smooth boundary, $A$ is a uniformly symmetric elliptic differential operator of order $2m$ with $t$-independent smooth coefficients, $B_j$, $1 le j le m$, are $t$-independent boundary differential operators such that the system $ A, B_j _1le j le m$ is well-posed. Let $ C_j _1 le j le m$ be complementary boundary differential operators of $ B_j _1 le j le m$. We consider a multidimensional linear inverse problem : for given $Gamma subset partialOmega$, $T > 0$ and $n in 1, ..., m$, determine $f(x)$, $x in Omega$ from $C_ju(f)(x,t)$, $x in Gamma$, $0

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