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Initial function estimation for scalar neutral delay differential equations. (English) Zbl 1172.34052
The paper is concerned with the linear delay differential equation
$y'(t)=ay(t)+by(t-\tau)+cy'(t-\tau)+f(t), \;t\in[0,T]$
subject to the condition $$y(t)=\varphi(t)$$ for $$t\in[-\tau,0]$$. The coefficients $$a,b,c\in R$$ and $$y(t)=y(\varphi,t)$$ denotes a solution. The main problem considered here is the identification of an initial function $$\varphi_*$$ such that the solution $$y(\varphi_*,t)$$ approximates the given function $$\widehat{y}$$. The authors formulate the identification problem in terms of minimization problem of a certain quadratic cost function defined on the specified linear space $$\mathcal{F}$$, and next they derive a set of equations defining the minimizer. Finally, an iteration procedure is described which is proved to be convergent to $$\varphi_*$$. The results generalize those presented in the authors’ earlier paper [J. Comput. Appl. Math. 181, No. 2, 420–441 (2005; Zbl 1072.65109)] for the simpler case $$c=0$$.

##### MSC:
 34K29 Inverse problems for functional-differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010) 34K40 Neutral functional-differential equations 34K06 Linear functional-differential equations
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