Analysis of four-dimensional variational data assimilation problems in low regularity spaces

¹ MODEMAT Research Center for Mathematical Modeling and Optimization, Quito, Ecuador
² Departamento de Matemática, Escuela Politécnica Nacional, Quito, Ecuador
³ Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 29 May 2025
Accepted: 29 January 2026

Abstract

We carry out a rigorous analysis of four-dimensional variational data assimilation (4DVAR) problems for linear and semilinear parabolic partial differential equations. Continuity of the state with respect to the spatial variable is required since pointwise observations of the state variable appear in the cost functional. Using maximal parabolic regularity tools, we prove this regularity for initial conditions with Lβ-regularity guaranteed by control constraints, rather than Sobolev regularity of the controls ensured by artificial cost terms. We obtain existence of optimal controls and first-order necessary optimality conditions for both the convex and nonconvex problem with spatial dimension d = 2, 3, as well as second-order sufficient optimality conditions for the nonconvex problem for d = 2.