Asymptotic approximation of sensitivities in finite dimensional continuous data assimilation with application to parameter estimation

We develop a rigorous justification for a parameter estimation algorithm which couples continuous data assimilation to generic optimization. For finite dimensional systems, we provide a rigorous justification of an asymptotic approximation of the sensitivity for the underlying modeled dynamical system, prove that the $L^2$ loss function satisfies an approximate Polyak-Lojasiewicz inequality, and use that result to justify convergence of gradient descent for the proposed algorithm. Numerical examples are provided that demonstrate the precision of the rigorous results.