Relation Between the Work Functional and the Observability Gramian in Earlier Iterative Analysis

Clarify the relation between the power-type quantity identified in the convergence analysis of the iterative viscosity-identification scheme and the observability Gramian.

Background

The paper introduces the work functional W and its time-averaged version P as central quantities for analyzing viscosity sensitivity in the nudging-based data-assimilation framework. The authors note that a related power-type quantity had appeared in earlier convergence analysis of an iterative viscosity-identification method.

Although both quantities play an identifiability-related role, the paper states that the relationship between the earlier power-type quantity and the observability Gramian has not been established. Determining this relationship could connect the earlier iterative analysis with the Gramian-based interpretation developed in the present work.

References

We point out that a power-type quantity was also identified in the convergence analysis of the iterative scheme def:CHL that was carried out in through the derivative of an energy-type quantity, which was crucial to obtaining the identifiability criterion there, though its relation to the observability Gramian is not clear; in kind, the quantities $W(\hat{\nu};\tau)$ and $P(\hat{\nu};\tau)$ will hold an analogously distinguished place in the present analysis.

— Excitation and Identifiability in the 2D Navier-Stokes equations  (2609.31230 - Martinez et al., 25 Sep 2026) in Remark \ref{rmk:work:power}, Section 3, “Model Error Analysis”