  In this paper we propose the design of an iterative observer using space as a
time-like variable and prove its convergence. The iterative observer algorithm
solves boundary estimation problem for a steady-state elliptic equation system
namely Cauchy problem for Laplace equation. The Laplace equation is formulated
as a first order state space-like system in one of the space variables and an
iterative observer is developed that sweeps over the whole domain to recover
the unknown data on the boundary. State operator matrix is proved to generate
strongly continuous semigroup under certain conditions and the system is shown
to be observable. Convergence results of proposed algorithm are established
using semigroup theory and concepts of observability for distributed parameter
systems. The algorithm is implemented using finite difference discretization
schemes and numerical implementation is detailed. Further, the simulation
results are presented towards the end to show efficiency of the algorithm.
