Data Assimilation as a Problem in Optimal Tracking: Application of Pontryagin's Minimum Principle to Atmospheric Science

A data assimilation strategy based on feedback control has been developed for the geophysical sciences - a strategy that uses model output to control the behavior of the dynamical system. Control enters through addition of a forcing term to the dynamical law. The method has similarity to nudging, but the strategy optimally determines the correction term rather than specifying it empirically. The foundation for the method rests firmly on Pontryagin's minimum principle where a least squares fit of idealized path to dynamical law follows from Hamiltonian mechanics. The early history of this method found application in control of vehicle trajectories in space science, but it has now been adapted to geophysical science by forcing the model's trajectory to track observations in accord with their accuracy. Following development of the theory for both linear and nonlinear systems, we apply this principal to Burgers' linear dynamical model that describes the transfer of energy from large- to small-scales of motion as an advective wave steepens. We further use the optimal result to reconstruct the transition matrix associated with the dynamical law as an alternate means for investigation of model error.