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

Output error minimizing back and forth nudging method for initial state recovery