# Fundamental Control Functions and Error Analysis

• V. Shutyaev
Conference paper
Part of the NATO Science Series book series (NAIV, volume 26)

## Abstract

Consider mathematical model of a physical process that is described by the evolution problem
$$\left\{ {\begin{array}{*{20}c} {\frac{{\partial \phi }} {{\partial t}} = F(\phi ) + f,t \in (0,T)} \\ {\phi |t = 0 = u,} \\ \end{array} } \right.$$
(1.1)
where ϕ = ϕ(t) is the unknown function belonging for any (t) to the Hilbert space X, uX, f is a nonlinear operator mapping X into X. Let Y = L 2(0, T, X), ‖ · ‖Y = (·,·) 1 2/Y , fY. Let us introduce the functional
$$S(u) = \frac{\alpha } {2}||u - u_0 ||_X^2 + \frac{1} {2}||C\phi - \phi _{obs} ||_{Y_{obs} }^2 ,$$
(1.2)
where α = const ≥ 0, u 0X, Y obs are prescribed functions (observational data), Y obs is a Hilbert space (observational space), C : YY obs a linear continuous operator.

## Keywords

Data Assimilation Singular Vector Adjoint Equation Linear Continuous Operator Observational Space
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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