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

TL;DR

This paper demonstrates that for certain linear systems, the back and forth nudging method with colocated feedback converges to the initial state estimate that minimizes output discrepancy, with specific corrections needed for dissipative systems.

Contribution

It introduces conditions under which the back and forth nudging method converges for skew-adjoint systems and proposes corrections for dissipative cases, including wave equations.

Findings

Convergence to the output discrepancy minimizer under specific conditions.

Necessary correction operator for dissipative systems.

Application to wave equations with constant dissipation.

Abstract

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.