Differentical equations, dynamical systems and optimal control

Asymptotic of regularized solutions of the ill-posed Cauchy problem for a linear nonautonomous system with delay

P. G. Surkov<sup>ab

a</sup> N.N. Krasovskii Institute of Mathematics and Mechanics of UB RAS, 16 S.Kovalevskaya Str., 620990, Ekaterinburg, Russian Federation
^b Yeltsin Ural Federal University, 19 Mira Str., 620002, Ekaterinburg, Russian Federation

Abstract: We consider an ill-posed Cauchy problem for a linear nonautonomous system of differential equations with delay. We find asymptotic formulas describing analytical dependances of regularized solutions on the regularization parameter on a finite interval of the negative half-line. The Tikhonov regularization method is used. We chose the stabilizing functional such that it does not generate a compact set in the space of states. The asymptotic formulas are found for smooth enough initial functions which do not provide backward continuation.

Keywords: differential equations with delay, ill-posed problem, asymptotic methods.

UDC: 517.929

MSC: 34K06,34K25

Received October 19, 2015, published January 24, 2017

DOI: 10.17377/semi.2017.14.006

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