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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 \(c=0\).

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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