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Знакоопределенность решений и устойчивость линейных дифференциальных уравнений с переменным распределенным запаздыванием.

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???????? ?????. ??????????
2008, ? 8, c. 73?77
http://www.ksu.ru/journals/izv_vuz/
e-mail: izvuz.matem@ksu.ru
??????? ?????????
?.?. ????????, ?.?. ?A????????
??????????????????? ??????? ? ????????????
???????? ???????????????? ????????? ? ??????????
?????????????? ?????????????
?????????. ??????? ??????????? (? ???????? ?????????? ???????? ??????) ??????? ??????????????? ??????? ???? ?????? ?????? ???????? ???????????????? ????????? ? ?????????? ?????????????? ?????????????. ?? ?????? ????? ?????????? ????????? ??????????? ???????? ???????????????? ???????????? ??????????? ?????????.
???????? ?????: ????????? ? ?????????????, ????????????, ???????????????? ????????????,
??????? ????.
???: 517.929
Abstract. We study one class of linear di?erential equations with varying distributed delay. We
obtain an e?ective (in terms of parameters of the initial problem) criterion for the positiveness of
the Cauchy function of this class of equations. On the base of this result we establish e?ective
criteria for the exponential stability of equations under consideration.
Keywords: equations with delay, nonoscillation, exponential stability, Cauchy function.
????? R = (??, +?), R+ = [0, ?), ? = {(t, s) ? R2+ : t s}.
?????????? ?????????????-???????????????? ?????????
t?? (t)
x(s) ds = f (t), t ? R+
x?(t) + ax(t) + b
(1)
t?? (t)?h(t)
(x(?) = 0, ???? ? < 0),
??? a, b ? R, h, ? : R+ ? R+ ? ????????? ???????, ? ??????? f ?????????????? ??????????? ?? ?????? ???????? ??????? ?? R+ .
??????? ????????? (1) ??????? ([1], c. 10) ??????? ????????????? ?????? ?????????
??????????? ?? ?????? ???????? ??????? ???????. ??? ???????? ([1], ?. 84, ??????? 1.1),
????????? (1) ?????????? ????????? ? ??? ??????? ????? ?????????????
t
C(t, s)f (s) ds,
x(t) = C(t, 0)x(0) +
0
??? C : ? ? R ? ??????? ???? ????????? (1).
????????? 26.03.2007
73
74
?.?. ????????, ?.?. ?A????????
???????? ??????? ???? ????????? ???? ??????????? ????????? ? ????? ???????? ????????????: ??? ????????? ???????? ???????????????? ??????? ?? ?????????????, ??????????? ??????? ?? ????????????, ???????? ?????? ?????? ? ?. ?.
?????????? ????????? ???? (1) ??? f (t) = 0 ?? ????????? t s
t?? (t)
x(?) d? = 0, t s
(2)
x?(t) + ax(t) + b
t?? (t)?h(t)
(x(?) = 0, ???? ? < s).
??????? ????? ????????? ????? ???????????? ????? ??????? ???? x(t) = C(t, s)x(s). ??????? ? ????????? (2) ?????? eat x(t) = y(t). ????? y(t) ???????? ???????? ?????????
t?? (t)
ea(t??) y(?) d? = 0, t ? s
(3)
y?(t) + b
t?? (t)?h(t)
(y(?) = 0, ???? ? < s),
??????? ????? ????? ???? ???????? ????? ??????? ???? Y (t, s) ????????? (3) ? ????
y(t) = Y (t, s)y(s). ????????? ????????????? ??????? ??? ????????? (2) ? (3), ????????, ???
?? ??????? ???? ??????? ??????? ???????????? ea(t?s) C(t, s) = Y (t, s). ????????, ???
??????????????? ??????? C ???????????? ??????????????? ??????? Y , ?????????????,
?????? ????????? (2) ????? ??????? ????????? (3).
??? b 0 ??????? ????????? (3) ????????????, ??? ??????? ?????? b > 0 ?????????
??????? ? ???????????????? ??????????? ([2], c. 65). ? ?????????? ???????? ?????????
??????? ??????????????? ??????? ???? ????????? (1).
????? h = supt h(t), ? = supt ? (t).
??????? 1. ??? ??????????????? ??????? ???? ????????? (1) ??????????, ?????
????? ? ???????????? M (ah, bh2 , ? /h) ???????????? ???????, ?????????? ????? ah = 0,
bh2 = 0, ? /h = 0 ? ???????????? ????????????, ???????? ?????????? ????????????????
???????????:
ah = ? +
e? (?(? /h
?(1 ? e? )
,
+ 1) ? 1) + 1 ? ?? /h
? 2 e??? /h
,
bh = ?
e (?(? /h + 1) ? 1) + 1 ? ?? /h
2
(4)
? ? R.
???????, ??? ??? ??????????????? ??????? ???? ??????????? ????????? (1) ???????
??????? 1 ???????? ???????????? ? ????????????.
? ?????????? ???????????? ah, bh2 , ? /h ??????? ??????????????? ??????? ???? ?????????? ?? ???????.
????????? 1. ????? ah = 0. ??????? ???? ????????? (1) ????????????, ???? ?????
? ???????????? M (bh2 , ? /h) ??????????? ???????, ???????????? ???? bh2 = 0 ? ??????,
???????? ?????????? ???????????????? ???????????:
?2+
?
1?e??
?2e
,
bh2 =
e? ? 1
1
2
, ? ? R+ .
? /h = ?
? 1 ? e??
?????????? ?? ?????? ?? ???????? (5) ? = 0, ????????
(5)
??????????????????? ??????? ? ????????????
75
????????? 2 ([3]). ????? a = 0 ? ? = 0. ??????? ???? ????????? (1) ????????????,
???? ??????????? ??????????? bh2 �(2 ? �), ??? �? ????????????? ?????? ?????????
e?� = 1 ? �/2.
???????? ?????????? ?????????? ? ???????????? ???????????? ?????? ?????? ????????? ? ?????????????? ?????????? ?????????????. ??????? ? ????????? (1) ? (t) = 0 ?
?????????? ?????????
t
x(s) ds = f (t), t ? R+ .
(6)
x?(t) + ax(t) + b
t?h(t)
????? ? ????????? (6) a = ?, b = 1, h(t) ? ?. ?????? ?? ??????? 1 ??????? ??????????????? ??????? ???? ????????? (6). ?? ??????? (4) ???????, ??? ??? ??????? ????????
?????? ? ??????????????? ?????
?(1 ? e? )
,
1 + e? (1 ? ?)
?2
.
?2 =
?
1 + e (1 ? ?)
?? = ? +
(7)
????????? (7) ??????? ? > ?2 ?????? ? ???????????? ???????????? ? > 0, ?. ?. ? ? ?
??????? ?????????????? ????????????, ??????? ?????????
? = ?(?). ???????? ?????????,
?
??? ??????? ? = ?(?) ????????? ???????, ? ??? ? > ? 2 ?? ?????? ??????? ???????????
??????? ???????????? ???????????????? ?????????.
??????? 2. ????? a, b ? R, h : R+ ? R+ ? ????????? ??????? ? ????????? ???? ??
???? ?? ????????? ???? ???????:
?) b > 0, ? ab < lim h(t) lim h(t) < ab + ?2b ? ?ab ,
t??
t??
?) b = 0, a > 0,
?) b < 0, lim h(t) < ? ab .
t??
????? ????????? (6) ??????????????? ?????????.
76
?.?. ????????, ?.?. ?A????????
??? ?????????????? ??????? 2 ???????????? ?????, ???????????? ?.?. ????????? ? ?????? [4]. ?? ??????????? ? ????????? ??????? ???????????? ????????? ? ???????? ??????????? ????????? (6) (???? a < 0, ?? ? ???????? ????????????? ?????????????????
?????????), ??? ???????? ????????? ??????? ??????? 1.
???????????????? ?????? ??????? 2 ?? ????????? ???????.
??????. ????? a = 1,2, b = 0,25. ??? ???
?a
b
= 2,4, ??, ???????? ??????? 2 ? ????? ?),
????? ?2,4 < lim h(t) lim h(t) < 2,4 + 4?(2,4). ??? ??? h(t) 0, ?? ?????? ?????
t??
t??
?????????????, ?????????????, ????????? (6) ??????????????? ?????????, ???? h(t) 0 ?
lim h(t) < 6,91. ?????? h(t) = 0 ?? ???????????, ??? ??? a > 0.
t??
????????? 3 ([3]).
????? ? ?
????????? (6) a = 0, b > 0, h : R+ ? R+ ? ?????????
?
??????? ? 0 < b lim h(t) b lim h(t) < 2 �(2 ? �), ??? �? ????????????? ??????
t??
t??
????????? e?� = 1 ? �/2. ????? ????????? (6) ??????????????? ?????????.
? ?????????? ???????? ??????? ????????? ? ???????????? ?????? ?????. ? ?????? [5]
??????? ??????????? ??????? ????????????? ??????????????? (?????????????????) ??????? ??? ????????? ???? (1) ??? a = 0 ? ? = 0, ??????? ? ????? ???????????? ????? ???
bh2 1/e ? 0,37. ????????? 2 ???? ?????? ?????? bh2 �(2 ? �) ? 0,65. ? ??????? [6],
[7] ????????? ???????? ???????????? ??? ????????? ???? (6) ??? h(t) ? h, ?? ?????????
??? ????????? (1) ??????? ??????????? ? ??????????? ???????? ???????????? [8]?[10], ??
????????? ??????????? ???? ????? ? ???????? 2 ???????? ?? ????????????.
??????????
[1] ??????? ?.?., ???????? ?.?., ???????????? ?.?. ???????? ? ?????? ????????????????????????????? ?????????. ? ?.: ?????, 1991. ? 280 c.
[2] ??????? ?.?., ??????? ?.?. ???????????? ??????? ????????? ? ????????????? ????????????. ?
?????: ???-?? ??????. ??-??, 2001. ? 230 ?.
[3] Malygina V.V. Positiveness of the Cauchy function and stability of a linear di?erential equation with
distributed delay // Memoirs on Di?. Equat. Math. Phys. ? 2007. ? V. 41. ? P. 87?96.
[4] ????????? ?.?., ??????????? ?.?. ?? ??????????????? ? ?????????????? ????????? ????????
????????? ?????????????-???????????????? ????????? ??????? ??????? // ?????????. ?????????. ?
1989. ? ?. 25. ? ? 12. ? ?. 2090?2103.
[5] Sugie J. Oscillation solutions of scalar delay-di?erential equations with state dependence // Appl. Anal. ?
1988. ? V. 27. ? P. 217?227.
[6] Burton T.A., Hatvani L. On nonuniform asymptotic stability for nonautonomous functional di?erential
equations // Di?. Integral Equat. ? 1990. ? V. 3. ? ? 2. ? P. 285?293.
[7] ?????? ?.?. ???????????? ????????? ????????????????? ????????? ? ?????????????? ?????????????
// ?????????. ?????????. ? 1996. ? ?. 32. ? ? 12. ? C. 1665?1669.
[8] ?????? ?.?. ????????????? ?????? ? ????????????? ??????????? // ?????????? ? ????????????. ?
2003. ? ? 4. ? ?. 167?173.
[9] Funacubo M., Hara T., Sakata S. On the uniform asymptotic stability for a linear integro-di?erential equation
of Volterra type // J. Math. Anal. Appl. ? 2006. ? V. 324. ? P. 1036?1049.
[10] ?????????? ?.?., ???????? ?.?. ????????? ???????? ???????????? ???????? ?????????? ???????????????? ????????? ? ?????????????? ????????????? // ???. ?????. ??????????. ? 2007. ? ? 6. ?
?. 55?63.
?.?. ????????
??????, ??????? ?????????????? ?????????? ? ????????,
???????? ??????????????? ??????????? ???????????,
614000, ?. ?????, ????????????? ????????, ?. 29?,
??????????????????? ??????? ? ????????????
e-mail: mavera@list.ru
?.?. ?a????????
????????, ??????? ?????????????? ?????????? ? ????????,
???????? ??????????????? ??????????? ???????????,
614000, ?. ?????, ????????????? ????????, ?. 29?,
e-mail: sabatulina@do.pstu.ru
V.V. Malygina
Associate Professor, Chair of Computational Mathematics and Mechanics,
Perm State Technical University,
29a Komsomol?skii Ave., Perm, 614000 Russia,
e-mail: mavera@list.ru
T.L. Sabatulina
postgraduate, Chair of Computational Mathematics and Mechanics,
Perm State Technical University,
29a Komsomol?skii Ave., Perm, 614000 Russia,
e-mail: sabatulina@do.pstu.ru
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