Averaging of pulse evolution systems
Для импульсной эволюционной системы dx/dt + Ах = f (ωt, х), t ≠ω-1ti (х), Δx| t =ω-1ti (х) = ω-1Ii (х), где А — секториальный оператор в банаховом пространстве X, обоснован принцип усреднения....
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| Date: | 1992 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | Russian |
| Published: |
Institute of Mathematics, NAS of Ukraine
1992
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7797 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512760212750336 |
|---|---|
| author | Mitropolsky , Yu. A. Rogovchenko , Yu. V. Митропольский, Ю. А. Роговченко, Ю. В. |
| author_facet | Mitropolsky , Yu. A. Rogovchenko , Yu. V. Митропольский, Ю. А. Роговченко, Ю. В. |
| author_sort | Mitropolsky , Yu. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-09-14T12:03:56Z |
| description | Для импульсной эволюционной системы
dx/dt + Ах = f (ωt, х), t ≠ω-1ti (х),
Δx| t =ω-1ti (х) = ω-1Ii (х),
где А — секториальный оператор в банаховом пространстве X, обоснован принцип усреднения. |
| first_indexed | 2026-03-24T03:33:54Z |
| format | Article |
| fulltext |
Yll.K 517.928+ 517.986.7
IO. A. MHTponoJthCKuii, a rrni:1.,
IO. B. Poron'ICIIKO, KaHJ:1. qJH:J.-MaT. nayr, (JIH-T MaTeMaTHIHI AH YKpaHHbl, KRen)
Y cpe]];HeHHe HMIIYJibCHblX 3BOJIIO~HOHHbIX CHCTeM
.Llm1 HMilYJibCHOH 3BOJI!OUHOHHOH CHCTeMbl
dx/dt +Ax= f (rot, x), t =I= ro-\ (x),
'1x I = ro- 11 (x)
l=w-lt;(x) ' '
rite A - CCKTOpH3JlbHblH onepaTOp B 6attaXOBOM npocTpaHCTBe X, OOOCHOBaH npum.1,un ycpe,!l•
HeHHS! .
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dx/dt + Ax= f (rot, x), t =I= ro-\ (x),
llxl 1 =ro- 1/.(x),
t=w- t;(z) ,
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qaeMblX B nocJie11Hee BpeM51 nmp<j:>epeHUH3JlbHb!X ypaBHeHHH C HMnyJibCHblM
B03,/leHCTBHeM (3-51. B H3CTOHmeH pa6oTe npHB0.llHTCH o6oCHOB3 HHe npHHUH
na ycpenHeHH51 H. H. 6oroJ1106oaa llJIH HeJIHHeHHblX 3B0JIIOUH0HHbIX CHCTeM C
6bICTpo OCUHJIJIHPY!Omei\ npasoi\ lJ3CTblO H M3Jlb!M HMnyJibCHb!M B03MymeHHeM Ha
rttnep nosepXH0CTHX. O6ocHOB3HHIO MeT0lla ycpe111-1eHHH llJIH a6cTpaKTHhIX na
pa6oJIHlJ eCKHX ypaBHeHHH 6e3 ytJeTa HMnyJibCHoro B03lleHCTBH51 n0C!3511UeHa
pa6oTa [6].
PacCMOTpHM HMilYJibCHYIO 3B0JIIOUH0HHYIO CHCTeMy (J.13C)
dxldt+Ax=f(wt. x), t=;i=cu- 1t;(x), (I)
/).x lt=w- lt;<x> = w- 1/; (X),
r.zte t > 0, ro > 0, A - ceKTOpHaJihHhlH oneparnp a 6aHaXOBOM npoCTpaHCTBe
X [7], Re o (A)>'\'> 0, a cJie.noaaTeJihHo, .llJIH a:> 0 onpe.neJieHhI 11po6Hh1e
C) IO. A. MHTPOilOJibCKHI'I, IO. B . P0f0BlJEHKO, 1992
76 ISSN 0041-6053. ~11,p. Mar. t11CypH.., 1992, T. 44. M I
nenem1 oneparnpa Aa H npocTpaHcrna Xa=D (Aa) c HOpMoi-'1 Ix \xa=\x \x+
+\Aax/x; OT06pa)KeI-IH5! f:IR+xu-x, 'i·u-xa, (.:V--+IR, iE~, on
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THX, V = {xEXa:lxlxa~P}-
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(]) Ha UHmepaaAe 10, s) 6yce.M nOHUMCl!llb KljC01tHO-Henpepbt6HYFO C pa3pbt6Cl
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011106paJ1CeHue f (wt, x,,, (t)) : [0, ws) __,,.. X 1-1en pepbt6HO, Xw (t) E D (A) OAR npou3-
60Abl-lOZo w E (w 0,+oo) u x,,,(t) yooriAenwopRem npu t=;i=cu-·1t,(x) nepaoMy
yparmeHwo e (I), a npu t = w-·1t;(x) - amopoMy ypaaHeHuto.
5yneM npennoJiaraTb, tJTO l19C ( I) 51BJI HeTC51 onHoy.napHoi-i, T. e. HH
T e rpaJJbHaH KpHB35! ITJ)OH3B0JlbHOro pemeHHH l19C (I) nepeceKaeTC51 C
Ka n<noi-i H3 nmepnoBepx1-1ocTei-i t=w-1t; (x) He 6oJJee o,nHoro pa3 a (rrpttMe,
pbl on1rny napIIb!X CHCTeM H ycJJOBH51, 06ecnec11rna1omue OllHOY napHOCTb,
CM. B [8, 9]).
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BhlITOJJHeHbl ycJJOBH51
0 < 0 1 ~ inf ti+1 (x) - sup ti (x) ~ 0 2 < + oo;
xEV xEV
4) paBHOMepHO no XEV cymecrny10T npe)].eJlhl
N
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CH B TOM, lJTO aHaJIOf'lil!HhlMH CBOHCTB3Mli 6yn_yT o6Jia}l3Th li OTo6pa)KeHHH
f O li ] O·
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w.eHue 1aoa'tU Kowu z (0) = Xo OAR ycpeoHeliH,Ofi cucmeMbt (2) z = z (t) V I pe-
aeAeHO IAR 11cex t > 0 u pacnoAoJJCe1-to (1 o6Aacmu U BM.ecme c He,wmopou
ceoe ; -nKoecmt-tocmb,o, mo aAR npou3aOAb1-lbtX e > 0 u Q > 0 HatloemcR
maKoe w., > 0, 'lmo npu w > (J)0 u t E (0, QI 6yiJern cnpaaeiJAueo HepaaeHcmeo
I Xw (!) - ? (t) I xa ~ e.
ll O K a 1 a T e Jl h CT B 0. Ey neM HenocpencTBeHHO oueHHBaTb yKJJOHe
HHe peLUeHH5! Xro(l) l13C (l) OT pememrn z(t) ycpe11HeHHOf1 CHCTeMbl (2)
Ha 0Tpe11<e !E [O, Q]' 3aMeHHB CHCTeMbl (I) H (2) HHTerpaJJbHhlMH ypaBHeHH
HMH. no aHaJJOrHH C (8) HaXO)!{}leHHe peweHHH 3a}lalJH KomH X (0) =Xo l!JIH
l13C (I) paBHOCHJlbHO H3XO)l{,lleHHIO pellleHHH HHTerpaJibHOro ypaBHeHHSl
I
(t , 1, <2, <P>) -A, + r -A<t-·t)t < ( o> <2> <Pl)) d + X,o , y , y , ... y = e x0 } e W't, x,,, 't, y , y , ... , y 't
ISSN 0041-6053. Y1'p. MaT. ~ypn., 1992, r. 44, M - l 71
(3 )
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Bb!TIOJIHHJIHCb COOTH011Jemrn
ct) f ( (i)) -1 ( (i )) (I) (2) (p)) Y + ; !/ = Xw(W f ;lj ,y ,y , ... ,y , i = l,p.
a p - '-IHCJIO rHn epnoBepxHocTei'i t = w- 1 t; (x), nona.uaiom,Hx n HI-nep n aJI
(0 , Q).
BBe.11.eM o6o3 Ha '-lemrn
<I J) to t ( c2J o ( <Pl) to ti (y = 1, 2 y ) = t 2, . . . , tp Y = o,
IlpH 3TOM
0 = w- 1tg < w- 1t~ < ffi- 1t~ < ... < w- 1t~ < w- 11~+1 = Q
H Tor.ua ypanHeHHe (3) nepe1rnmeTCH B mine
I
Xw(t) =e-A 1x 0 + \ e-A(t-,Jf(ffiT,Xw(T))dT +
0
+ w-1
-A(t-c,,-lt~)
e l;(Xro(ffi- 1t~)), t E [0,Q]. (4)
B TO :>Ke npeMH pewe1rne 33Jl34H Koum z (0) = x0 JlJIH ycpem-1eHHoi'i CHCTeMhl
(2) YJlOBJieTBop HeT HHTerpaJibHOMY ypaBHeHH!O [7]
I
z (t) = e-Arx0 + S e-Au-, i [/0 (z (i:)) + /0 (z (i:))J di:,
()
t E [O, QJ.
(5)
H eTpyn1-10 3aMeTHTh , LJTO npu npmrnThIX npennono:>Kemrnx ypamremrn (4) H
(5) 6yJlyT HMeTb eJUil-lCTBeHHOe peme1rne.
B Jl3JlbHeflmeM H3M !10H3Jl0051TC51 cneny1011.u1e BCI10MOraTeJibl-Ible y-rnep>K
)leHml.
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> y > 0. Toeaa cyu1ecnw y ,om mo•mbie aepxr1ue epaHu
su p \ e-A<t-,J I a = F,
t-,;;;,'11>0 X
l,'tE[O,!l]
sup 1e- 11u- ,> Ix= G,
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l ,T E [0,Q7
sup I e-A<t-"t> - EI ·a = H.
t- ,;;;,'11>0 X
t , ,E[O ,ffl
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I -At I --- C -vt I Aa -At I --- C 1- a - vt
e x"""' e , e x"""' a e ,
I - At E l C a I Aa (e - ) Xix~ a I- at Xix,
rne C, Ca, CI-a - HeK0 Topb1e noJIO:>KHTeJibHbre nocToHHHbre , npH4eM Ca or
pa1rn<1eHa no Cl Ha l<alli).1.OM KOMilaKTHOM HHTepBaJJe, JJe}I<am,eM B (0, + 00 ).
Jl e MM a 2. Ilycmb Xw (f)- pewer1ue sa8a'iu l(owu x (0) = x 0 8Afl 113C
( l ), z (t) - peweHue sa8a•w l(owu z (0) = xll 8 //fl ycpeiJ1-1eHHOU cucmeMbt (2) .
Toe8a (/JyHKU,Ufl Z (t) 6y8em A0KG/1.bH0 eeAb8ep08Q U3 (0, Q) 8 X a_ a r/Jy HK
l {Ufl Xw(f ) -AOKQJlbHO eCAb8ep06Q U3 (ffi - 1t? , (l)-lf~+1 )8 X a 8Afl acex i =0 ,p,
m. e. cyw,ecmey,om maKue no110:J1Cume11bHbte nocmoRHHbte Q u R, 'imo
I z (t + h) - z (t) lxa ~ Qh6', 0 < <'>1 < 1 - a.,
I x6) (t + h) - xw (t) lxa ~ Rh6' , O < 62 ~ 1.
78 ISSN 0041 -6053. Yx:p. MOT. ,ttypH., 1992, T. 44, N! 1
TTepBaH qacrh yrBep,Kii_emrn JieMMhl 2 06oc1-10Bmia B xone noKa3areJihCTB3'
JieMMbl 3.3.2 [71, a BTOpaH qacTb ycraHaBJIHBaeTCH c nOMOI..UblO reopeMhl 3.5. 2"
H3 [7] aHaJiornqHo noKa3aTeJibCTBY JJeMMbl 3. 1 H3 pa60Tbl [8 ].
BepHeMCH r enepb K .l(OKa3aTeJJbCTBY reopeMbl 1. CorJiaCI-10 npHHHTbIM
npennoJJO)KeHHHM orpe30K [0, Q] H3MeHeHHH t pa3farnaeTCH TO'-!KaMH (J)-lt~ Ha
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npeBbIWaer ro- 10 2. TTocKOJibKY BropaH qacrb yrBep)K.11.emrn JJeMMhI 2 Bepi-i a
TO.'IbKO Ha 111-!TepBaJiaX HenpepbIBI-IOCTH pewe1-111H Xw (t) l13C (1 ), 6y .n.eM Y llH-
,h ( -ltO -ltO ) . Q-ThIBaTh 3TOT 't-'aKT, Bb!IlOJIHHH OIJ,eHKl1 I-Ia 11HTepBaJiax ,ro -i, (I} i+ I , £= ,p.
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0
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0 O<w-ltO < t
'
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O< w-ltO<t xa
'
I
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' t 5
+ I f e-A<t--r> II o (z (ro-lt~)) - I o (z (-r)) j d-r lxa = L Si
o /=I
H OIJ,e Hl1M Ka)K)I.Qe 113 CJiaraeMhlX B OT)l.eJihHOCTl1.
llm1 nepBOfO CJiarae~10ro 11MeeM
"'-110
µ k+I
+ L I f e-Act--r> [f (ro't', Xro (ro-1ti)) - fo (xro (ro-1ti))] d-u +
k= O "'-It¾
w-110
p k+I
+ L / S e-A<t-,;) lfo (xw (ro-1ti)) - fo (Xro ('t))l d't \xa· \7)
k=O ,,)-ltZ
TTepBoe 11 rperbe CJiaraeMble 113 (7) C llOMOlll,blO JieMMbl 2 oue1mBaIOTCl'I CJie,uyIO
W.HM o6pasoM (p0 = p + 1):
"'-11¾+1
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k=O ,,,-11
/!
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k=O
ISSN 0041-6053. :/1.p. Ata1. ;Jt(;ypH. •• 1992, 'l'. 44, ~ J 79·
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0 xa O<t,- <t xa
no:=noMy BTopoe cJiaraeMoe B (7) on eHHBaeTCH TaK:
~ [ ti I O t% O ] . ~ ' F _+_ g (tk+ 1) + - g( t k) ~ 2fl2p 0w- 1g (Q) .
..:..i (t) (I)
k = O
T aKHM o6pa30M, ,n:JIH rrepsoro c.rraraeMor o S1 HMeeM o u e1-1Ky
S 1 ~ 2Fp0 [N R (w- 10 2)1+0 • + Qw- 1g (Q)].
O u e1rna ,D,JIH S2 oqem1,n:Ha:
I
s2 ~ FN s I Xw (.) - Z (.) lxad • .
0
O u eH HM Terrepb T])eTbe CJiaraeMoe:
p
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(8)
(9)
(10)
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80 ISSN 0041-6053. YKp. AtaT. rHCljpH ., 1992, T. 44, M 1
nepBoe CJiaraeMoe B npaBOH GaCTH ( 11) oueHHBaeTCH TaK:
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k=O k+I O<lo <,o X
i k+ I
+ t~ I t~ ~ I; (xw {ffi- 1t~)) - / 0 (x"' (ffi- 1t~)) I J ~ ro- 1Gp0 [ti+1g (ti+1) +
O<t~<ti X
+ t%g (ti)] ~ 2(J)- 1Gp 0Qg(Q).
BTopoe cJiaraeMoe B ( 11 ) oueHJrnaeTCH otieBHJHlhIM o6pa3oM:
I '
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0 0
TaKHM o6pa30M, .UJIH S4 HMeeM cJie,[(y1ow.y10 0ueHKy:
I
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0
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06ne.UHHHH oueHKH (8) - ( 11 ), (13), nOJiytiaeM
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t
+ 2p0(J)- 1 (F + G) Qg (Q) + (F N + GK) I I x6) (T) - z (T) lxad• + GH Lp/32(1)-2.
0
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0
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ISSN 0041-6053. Y,w Mar. J1Cyp1t. , 1992, r. 44, .N!! I 6-1-812 81
ou.eHKY npH t E [O, QI:
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0
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0
=8,
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2. \lumoono11bCKUU IO. A. M eTOJl ycpeirneHHH B HeJJntteiittoH MexattHKt,.- KneB : Hayx,
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C HMTTYJJbCHblM B03JleiicTBHeM // YKp. MaT. 1KypH.- 1985.- 37, N2 1.- C. 56-64.
5. CaAIOU/leHKO A. M. MeTOll ycpe)lHeHHH B CHCTeMax c T0Jl'IKaMH // MaT. cjlH3HKa.- 1971.
Bbm. 9.- C. 101-117.
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11e1rnii // MaT. c6.- 1970.- 81, N2 1.- C. 53-61.
7. Xe1-tpu II. r eoMeTpH'IecKaH TeopHH no.~ y.mHeiitthIX napa6oJJH'IecKHX ypaBHeHHH.- M.:
Mttp, 1985.- 376 C.
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TOH'IHBOCTb. - Knen, 1986.- 44 c.- (npenpttm / AH YCCP. 11.tt-T MaTeMaTHKH; 86.65),
9. Ca,1wii11u1KO A. M., flepecm,oK H. A., Tpocpwtit!fK C.H. npo6JieMa <6tteHHii» B HM TTYJibCHhlJII
cttcTeMax.- Knen, 1990.- 46 c.- (TipenpnHT / AH YCCP. I1H-T MaTeMaTHKH ; 90.11).
10 . l(paCHoCellbCKUU M. A. OnepaTop CllBHra no TpaeKT0pHHM 11Hcjl¢epeHUHaJJbHbIX ypaBHe
HHii .- M. : HayKa, 1966.- 332 c.
11 . !11-tmezpallbHbl.e onepaTOpb1 B npocTpattcrnax cyMMttpyeMhlX cjlyHKUHH / M. A. KpaCHoceJib
CKHii, n. n. 3a6peiiKO, E. I1. TiycTb!JlbHHK, n. E. Co6oJieBCKHli.- M. : HayKa, 1966.-
500 c.
OoJiy'leHo 14.06.91
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| id | umjimathkievua-article-7797 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:54Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/90/31a0fa21841b858bf9c871073b81c790.pdf |
| spelling | umjimathkievua-article-77972023-09-14T12:03:56Z Averaging of pulse evolution systems Усреднение импульсных эволюционных систем Mitropolsky , Yu. A. Rogovchenko , Yu. V. Митропольский, Ю. А. Роговченко, Ю. В. Для импульсной эволюционной системы dx/dt + Ах = f (ωt, х), t ≠ω-1ti (х), Δx| t =ω-1ti (х) = ω-1Ii (х), где А — секториальный оператор в банаховом пространстве X, обоснован принцип усреднения. Для імпульсної еволюційної системи dx/dt + Ах = f (ωt, х), t ≠ω-1ti (х), Δx| t =ω-1ti (х) = ω-1Ii (х), де A — векторіальний оператор у банаховому просторі X, обгрунтовано принцип усереднення. Institute of Mathematics, NAS of Ukraine 1992-02-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7797 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 1 (1992); 76-83 Український математичний журнал; Том 44 № 1 (1992); 76-83 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7797/9465 Copyright (c) 1992 Yu. A. Mitropolsky, Yu. V. Rogovchenko |
| spellingShingle | Mitropolsky , Yu. A. Rogovchenko , Yu. V. Митропольский, Ю. А. Роговченко, Ю. В. Averaging of pulse evolution systems |
| title | Averaging of pulse evolution systems |
| title_alt | Усреднение импульсных эволюционных систем |
| title_full | Averaging of pulse evolution systems |
| title_fullStr | Averaging of pulse evolution systems |
| title_full_unstemmed | Averaging of pulse evolution systems |
| title_short | Averaging of pulse evolution systems |
| title_sort | averaging of pulse evolution systems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7797 |
| work_keys_str_mv | AT mitropolskyyua averagingofpulseevolutionsystems AT rogovchenkoyuv averagingofpulseevolutionsystems AT mitropolʹskijûa averagingofpulseevolutionsystems AT rogovčenkoûv averagingofpulseevolutionsystems AT mitropolskyyua usrednenieimpulʹsnyhévolûcionnyhsistem AT rogovchenkoyuv usrednenieimpulʹsnyhévolûcionnyhsistem AT mitropolʹskijûa usrednenieimpulʹsnyhévolûcionnyhsistem AT rogovčenkoûv usrednenieimpulʹsnyhévolûcionnyhsistem |