Existence and uniqueness for variational data assimilation in continuous time

Jochen Bröcker

doi:10.3934/mcrf.2021050

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2023, Volume 13, Issue 1: 94-115. Doi: 10.3934/mcrf.2021050

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Existence and uniqueness for variational data assimilation in continuous time

Jochen Bröcker

School of Mathematical, Physical, and Computational Sciences, University of Reading, Reading RG6 6AX, United Kingdom

Received: August 2020

Revised: May 2021

Early access: October 2021

Published: March 2023

The author was supported by the UK Engineering and Physical Sciences Research Council under grant agreement EP/L012669/1. Fruitful discussions with Horatio Boedihardjo, Tobias Kuna, and Dan Crisan are gratefully acknowledged

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Abstract

A variant of the optimal control problem is considered which is nonstandard in that the performance index contains "stochastic" integrals, that is, integrals against very irregular functions. The motivation for considering such performance indices comes from dynamical estimation problems where observed time series need to be "fitted" with trajectories of dynamical models. The observations may be contaminated with white noise, which gives rise to the nonstandard performance indices. Problems of this kind appear in engineering, physics, and the geosciences where this is referred to as data assimilation. The fact that typical models in the geosciences do not satisfy linear growth nor monotonicity conditions represents an additional difficulty. Pathwise existence of minimisers is obtained, along with a maximum principle as well as preliminary results in dynamic programming. The results also extend previous work on the maximum aposteriori estimator of trajectories of diffusion processes.

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Mathematics Subject Classification: Primary 49J55, 49K45; Secondary 86A10, 93E99, 60G35.

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