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Identification of the initial function for discretized delay differential equations. (English) Zbl 1072.65109
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.

##### MSC:
 65L09 Numerical solution of inverse problems involving ordinary differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010) 34K29 Inverse problems for functional-differential equations
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