Iterative algorithms for data assimilation problems (Q696195)

The following constrained minimum problem is considered: minimize the functional \(J(\varphi)\) with respect to the variable \(u\) \[ J(\varphi)={1\over 2}\int_0^T(C(\varphi-\widehat\varphi), \varphi-\widehat\varphi)_{X} dt+{\alpha\over 2}(\varphi|_{t=0}-\widehat\varphi^0, \varphi|_{t=0}-\widehat\varphi^0), \] subject to the following condition \[ {d\over dt}\varphi +A(t)\varphi=f, \quad t\in(0,T),\tag{1} \] \[ \varphi(0)=u, \] where the differential equation is, in general, a partial one, and \(C\) is a linear operator in some functional Hilbert space \(X\). A necessary optimality condition [see \textit{J. L. Lions}, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. (Études mathématiques) (1968; Zbl 0179.41801)] reduces this problem to the solution of a certain boundary value problem with respect to the variable \(t\in [0,T]\) for a system of 2 non-homogeneous differential equations with differential operator of the equation (1), and with the operator formally adjoint to it. The authors discuss the properties of obtained boundary value problem, propose certain iterative process to solve it (numerically) and state a theorem on convergence of the proposed procedure.