  We show that for linear dynamical systems with skew-adjoint generators, the
initial state estimate given by the back and forth nudging method with
colocated feedback, converges to the minimizer of the discrepancy between the
measured and simulated outputs - given that the observer gains are chosen
suitably and the system is exactly observable. If the system's generator A is
essentially skew-adjoint and dissipative (with not too much dissipation), the
colocated feedback has to be corrected by the operator e^{At}e^{A*t} in order
to obtain such convergence. In some special cases, a feasible approximation for
this operator can be found analytically. The case with wave equation with
constant dissipation will be demonstrated.
