  In this paper, we consider simultaneous reconstruction of the diffusion
coefficient and initial state for a one-dimensional heat equation through
boundary control and measurement. The boundary measurement is known to make the
system exactly observable, and both coefficient and initial state are shown to
be identifiable by this measurement. By a Dirichlet series representation for
observation, we can transform the problem into an inverse process of
reconstruction of the spectrum and coefficients for Dirichlet series in terms
of observation. This happens to be the reconstruction of spectral data for an
exponential sequence with measurement error. This enables us to develop an
algorithm based on the matrix pencil method in signal analysis. An error
analysis is made for the proposed method. The numerical simulations are
presented to verify the proposed algorithm.
