Identification of the initial function for discretized delay differential equations (Q557705)

The paper is concerned with a system of linear delay differential equations \[ {\dot y}(t)-A(t)y(t)-B(t)y(t-\tau)=f(t), \] \(t \in [0,T]\) subject to \(y(t)=\varphi(t)\), \(t \in [-\tau,0]\). The functions \(f(t), \varphi(t) \in \mathbb R^{n \times 1}\) and \(A(t), B(t) \in \mathbb R^{n \times n}\) are supposed to be continuous. Let \(y(t)=y(\varphi;t)\) be a solution to the equation and \({\hat \varphi}(t)\), \({\hat y}(t)\) be two given functions. The authors analyze the data assimilation problem of finding a minimizer to the discrepancy functional \[ \int_{-\tau}^0 \| \varphi(t)- {\hat \varphi}(t)\|^2 dt + \| \varphi(0)- {\hat \varphi}(0) \|^2+ \| y(\varphi;0)-{\hat y}(0)\|^2+ \int_0^T \| y(\varphi;t)-{\hat y}(t) \|^2 dt. \] A discrete version of this problem is derived. The main result is a convergence theorem for an iterative process of solving the suggested finite-dimensional variational problem.