Existence and Uniqueness for Four-Dimensional Variational Data Assimilation in Discrete Time

Variational techniques for data assimilation, i.e., estimating orbits of dynamical models from observations, are revisited. It is shown that under mild hypotheses a solution to this variational problem exists. Using ideas from optimal control theory, the problem of uniqueness is investigated and a number of results (well known from optimal control) are established in the present context. The value function is introduced as the minimal cost over all feasible trajectories starting from a given initial condition. By combining the necessary conditions with an envelope theorem, it is shown that the solution is unique if and only if the value function has a derivative at the given initial condition. Further, the value function is Lipschitz and hence has a derivative for almost all (with respect to the Lebesgue measure) initial conditions. Several examples are studied which demonstrate that points of nondifferentiability of the value function (and hence nonuniqueness of solutions) are nevertheless to be expected in practice.