Geodesic
Singular numbers of the monodromy operator and sufficient conditions of the asymptotic stability of periodic system of differential equations with fixed delay
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 13 (2007) no. 2, pp. 66-79 Cet article a éte moissonné depuis la source Math-Net.Ru

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To obtain sufficient conditions for the asymptotic stability of linear periodic systems with fixed delay commensurable with the period of coefficients, singular numbers of the monodromy operator are used. To find these numbers, a self-adjoint boundary value problem for ordinary differential equations is applied. We study the motion of eigenvalues of this boundary value problem under a variation of a parameter. Obtaining sufficient conditions for the asymptotic stability is reduced to finding the bifurcation value of the parameter for the boundary value problem. 
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Yu. F. Dolgii; E. V. Ul'yanov. Singular numbers of the monodromy operator and sufficient conditions of the asymptotic stability of periodic system of differential equations with fixed delay. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 13 (2007) no. 2, pp. 66-79. http://geodesic.mathdoc.fr/item/TIMM_2007_13_2_a5/

[1] Krasovskii N. N., Nekotorye zadachi teorii ustoichivosti dvizheniya, Fizmatgiz, M., 1959 | MR

[2] Kheil Dzh., Teoriya funktsionalno-differentsialnykh uravnenii, Mir, M., 1984 | MR

[3] Shimanov S. N., “K teorii lineinykh differentsialnykh uravnenii s periodicheskimi koeffitsientami i s zapazdyvaniem vremeni”, Prikl. matematika i mekhanika, 27:3 (1963), 450–458 | MR | Zbl

[4] Lilo J. C., “First order periodic differential difference equations”, J. Math. Anal. Appl., 70:2 (1979), 389–398 | DOI | MR

[5] Kulenovic M. R. S., Ladas G., Meimaridou A., “Stability of solutions of linear delay differential equations”, Proc. Amer. Math. Soc., 100:3 (1984), 433–441 | DOI | MR

[6] Bashkirov A. I., “Priznak eksponentsialnoi ustoichivosti uravneniya s posledeistviem i s periodicheskimi parametrami”, Differents. uravneniya, 22:11 (1986), 1994–1997 | MR | Zbl

[7] Malygina V. V., “Nekotorye priznaki ustoichivosti uravnenii s zapazdyvayuschim argumentom”, Differents. uravneniya, 28:10 (1992), 1716–1723 | MR | Zbl

[8] Azbelev N. V., Simonov P. M., Ustoichivost reshenii uravnenii s obyknovennymi proizvodnymi, Permskii un-t, Perm, 2001

[9] Dolgii Yu. F., Ustoichivost periodicheskikh differentsialnykh uravnenii s otklonyayuschimsya argumentom, Avtoref. dis. $\dots$ d-ra fiz.-mat. nauk, Ekaterinburg, 1994, 32 pp.

[10] Dolgii Yu. F., Shimanov S. N., “Suschestvovanie zony ustoichivosti dlya odnogo uravneniya s zapazdyvaniem”, Ustoichivost i nelineinye kolebaniya, UrGU, Sverdlovsk, 1988, 11–18

[11] Gasilov G. L., “O kharakteristicheskom uravnenii sistemy lineinykh differentsialnykh uravnenii s periodicheskimi koeffitsientami i zapazdyvaniem”, Izv. vuzov. Matematika, 1972, no. 4, 60–66 | MR | Zbl

[12] Zverkin A. M., “K teorii differentsialno-raznostnykh uravnenii s zapazdyvaniyami, soizmerimymi s periodom koeffitsientov”, Differents. uravneniya, 24:9 (1988), 1481–1492 | MR | Zbl

[13] Dolgii Yu. F., “Ob ustoichivosti odnoi periodicheskoi sistemy s zapazdyvaniem”, Kraevye zadachi, Permskii politekh. in-t, Perm, 1989, 16–21 | MR

[14] Dolgii Yu. F., Nikolaev S. G., “Ob ustoichivosti periodicheskoi sistemy differentsialnykh uravnenii s zapazdyvaniem”, Differents. uravneniya, 35:10 (1999), 1330–1336 | MR | Zbl

[15] Dolgii Yu. F., “Ispolzovanie samosopryazhennykh kraevykh zadach pri issledovanii ustoichivosti periodicheskikh sistem s zapazdyvaniem”, Tr. In-ta matematiki i mekhaniki UrO RAN, 12, no. 2, Ekaterinburg, 2006, 78–87 | MR

[16] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 | MR

[17] Weyl H., “Inequalities between the two kinds of eigenvalues of a linear transformation”, Proc. Acad. Sci. USU, 35 (1949), 408–411 | DOI | MR | Zbl

[18] Elsgolts L. E., Norkin S. B., Vvedenie v teoriyu differentsialnykh uravnenii s otklonyayuschimsya argumentom, Nauka, M., 1971 | MR

[19] Akhiezer N. I., Glazman I. M., Teoriya lineinykh operatorov v gilbertovom prostranstve, Nauka, M., 1966 | MR | Zbl

[20] Dolgii Yu. F., Putilova E. N., “Prodolzhenie nazad reshenii lineinogo differentsialnogo uravneniya s zapazdyvaniem kak nekorrektnaya zadacha”, Differents. uravneniya, 29:8 (1993), 1317–1323 | MR

[21] Yakubovich V. A., Starzhinskii V. M., Lineinye differentsialnye uravneniya s periodicheskimi koeffitsientami i ikh prilozheniya, Nauka, M., 1972 | MR