Geodesic
Asymptotic of regularized solutions of the ill-posed Cauchy problem for a linear nonautonomous system with delay
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 41-58.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{SEMR_2017_14_a83,
     author = {P. G. Surkov},
     title = {Asymptotic of regularized solutions of the ill-posed {Cauchy} problem for a linear nonautonomous system with delay},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {41--58},
     publisher = {mathdoc},
     volume = {14},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2017_14_a83/}
}
TY  - JOUR
AU  - P. G. Surkov
TI  - Asymptotic of regularized solutions of the ill-posed Cauchy problem for a linear nonautonomous system with delay
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2017
SP  - 41
EP  - 58
VL  - 14
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2017_14_a83/
LA  - ru
ID  - SEMR_2017_14_a83
ER  - 
%0 Journal Article
%A P. G. Surkov
%T Asymptotic of regularized solutions of the ill-posed Cauchy problem for a linear nonautonomous system with delay
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2017
%P 41-58
%V 14
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2017_14_a83/
%G ru
%F SEMR_2017_14_a83
P. G. Surkov. Asymptotic of regularized solutions of the ill-posed Cauchy problem for a linear nonautonomous system with delay. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 41-58. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a83/

[1] N. N. Krasovskii, Certain problems in the theory of stability of motion, Gosudarstv. Izdat. Fiz.-Mat. Lit., M., 1959 | MR

[2] J. K. Hale, S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993 | DOI | MR | Zbl

[3] A. D. Myshkis, Linear differential equations with retarded argument, Nauka, M., 1972 | MR | Zbl

[4] Yu. F. Dolgii, E. N. Putilova, “Backward of Solutions of a Linear Differential Equation with Delay as an Ill-Posed Problem”, Differential Equations, 29:8 (1994), 1141–1146 | MR

[5] Yu. F. Dolgii, P. G. Surkov, “Asymptotic behavior of regularized solutions of a linear nonautonomous system of differential equations with anticipation”, Differential Equations, 46:4 (2010), 470–488 | DOI | MR | Zbl

[6] A. N. Tikhonov, V. Ya. Arsenin, Solutions of ill-posed problems, Scripta Series in Mathematics, John Wiley Sons, New York–Toronto, Ont.–London, 1977 | MR | Zbl

[7] P. G. Surkov, “Regularization of ill-posed Cauchy problem for an autonomous system with delay using one class of stabilizing functionals”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 20, no. 3, 2014, 234–245

[8] V. K. Ivanov, V. V. Vasin, V. P. Tanana, Theory of linear ill-posed problems and its applications, Nauka, M., 1978 | MR | Zbl

[9] I. M. Rapoport, On Some Asymptotic Methods in the Theory of Differential Equations, Izdat. Akad. Nauk Ukrain. SSR, Kiev, 1954 | MR

[10] M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations, Nauka, M., 1983 | MR | Zbl

[11] F. R. Gantmacher, The Theory of Matrices, Chelsea Publishing Company, New York, 1959 | MR

[12] L. S. Pontryagin, Ordinary differential equations, Nauka, M., 1982 | MR | Zbl

[13] K. Yosida, Functional Analysis, Grundlehren der Mathematischen Wissenschaften, 123, Springer-Verlag, Berlin–New York, 1980 | MR | Zbl