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Identification of the Diffusion Coefficient in Linear Evolution Equations in Hilbert Spaces

Volume 2, Number 1 (2011), 14 - 28

Identification of the Diffusion Coefficient in Linear Evolution Equations in Hilbert Spaces

Communicated By: 
Irena Lasiecka
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Abstract

Let H be a real separable Hilbert space and A : D(A)!H be a positive and self-adjoint (unbounded) operator. We consider the identification problem consisting in searching for a function u : [0,T]!H and a positive constant n that fulfill the initial-value problem u0+nAu = 0, t 2 (0,T), u(0) = u0, and the additional condition ku(T)k2 = r, where u0 2 H and r > 0 are given. If the initial datum u0 fulfills the restriction ku0k2 > r, by means of a finite-dimensional approximation scheme, we construct a unique solution (u,n) of suitable regularity on the whole interval [0,T], and exhibit an explicit continuous dependence estimate of Lipschitz-type with respect to the data u0 and r. Also, we provide specific applications to second and fourth-order parabolic initial-boundary value problems.