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Методы решения проблемы устойчивости квазиполиномов и семейств квазиполиномов.

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Y.LI:K 517.962
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BecTHHK CI16ry. Cep. 10, 2006,
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1
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METO,l1;hI PEIIIEHI1.H
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1.
BBe,n;eHHe H
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IIpH npOeKTHpOBaHHH CHCTeM ynpaBJIeHHR
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np06JIeMY YCTOH'-IHBOCTH 3THX CHCTeM. ECJm CHCTeMa ynpaBJIeHHR B npOCTpaHCTBe COCTOR­
HHH MO,n;eJIHpyeTcR JIHHeHHbIM ,n;H<p<pepeHlJ,HaJIbHO-Pa3HOCTHbIM ypaBHeHHeM BH,n;a
±(t) = Aox(t)
+L
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j=1
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r,n;e
= 0,
... , s) -
nOCTORHHble KBa,n;paTHble MaTpHlJ,bI nopR,n;Ka
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h j (j
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nOJIO)KHTeJIbHble BeJIH'-IHHbI, Ha3bIBaeMble 3 ana3,lJ;bIBaHHRMH, TO np06JIeMa YCTOH'-IHBOCTH
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TH'-IeCKOro KBa3HnOJIHHOMa
m
f()..)
= det()..E -
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+L
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j
))
= PO()..) + LPj()..)e-
j=1
B KOTOPOM nOJIHHOMbI
(j
BeJIH'-IHHbI7j
n,
A o, .. . ,As,
onpe,n;eJIRIOTCR 3JIeMeHTaMH MaTpHIJ,
a CTeneHH nOJIHHOMOB
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nOJIHHOMbI YCTOH'-IHBOH CHCTeMbI Ha3bIBaIOTCR YCTOH'-IHBbIMH.
K03<P<PHIJ,HeHTHble KpHTepHH YCTOH'-IHBOCTH KBa3HnOJIHHOMOB, COCTO.HllJ,He H3 KOHe'-IHOrO
'-IHCJIa lIlarOB, K HaCTO.HllJ,eMY BpeMeHH He H3BeCTHbI.
Bce HMeIOIIJ,HeCR OCHOBHble KpHTe­
pHH YCTOH'-IHBOCTH KBa3HnOJIHHOMOB RBJI.HIOTCR pacnpocTpaHeHHeM aHaJIOrH'-IHbIX KpHTe­
pHeB ,n;JI.H nOJIHHOMOB..
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C.
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= Re {e~W.Tm
f ()..)
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= 1m {e~wTm f(iW)}
- npocm'bte, 6e'l1{,ecm­
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51.
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1 pacnpocTpaHReT pe3YJIbTaT TeopeMbI
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HeOTpHIJ,aTeJIbHa.
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COBna,n;aIOT C HyJI.HMH HCXO,ll;HOrO KBa3HnOJIHHOMa
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t-l T06bI npHpa~eHHe aprYMeHTa <PYHKIJ;HH
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0
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f ('x).
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[5].
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CJIY'-Ia5lX, Kor11.a napaMeTpbI CHCTeMbI (K03<pqmIJ;lleHTbI nOJIHUOMOB Po
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9TH OrpaHH'-IeHH5I Onpe11.eJI5IIOTC5I 06JIaCTbIO YCTOM'-IHBOCTH no H3MeH5IeMbIM napaMeTpaM,
KOTOPYIO MO)KHO 3a1l,aTb 6eCKOHe'-IHbIM '-IHCJIOM HepaBeucTB, COrJIaCHO TeopeMe fpOMMepa­
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TeopeMa
2 [3].
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(3)
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po('x), ... ,Pm('x) u3ana3a'bt6a'H.UJl.MUTl, ... ,Tm (co> 0). EC.I1.u1C6a3UnOJl,U'H.OMf(,X) 'H.eUMeem
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~JI,a6'H.'bte MU'HOP'bt 6eC1CO'He"t'HOu MampU'4'bt
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.. J
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npH6JIH)KaIOTC5I K 11.0CTaTO'-IHbIM.
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f (.-\) ,
HanpHMep, 11.JI5I KBa3HnOJIHUOMa
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paccMaTpHBaeTC5I BcnOMOraTeJIbHOe ceMeMCTBO KBa3HnOJIHHOMOB, BKJIIO'-IaIO~ee
B ce651 KBa3HrrOJIHHQM
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149
B Ka'IeCTBe TaKOrO BCrrOMOraTeJlbHOrO CeMeHCTBa MO)KHO HCrrOJlb30BaTb CJle~IQllJ,ee:
(4)
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Po (.~)
+ ... + Pm ().)
(4)
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= ... = Tm = 0).
(npH Tl
B
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T~,
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C POCTOM T~, •. • , T~
•.. ,
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(4)
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m
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+ LPi(iw)e- i9j
j=l
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E
+00) X
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<I> (0 1 ,
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0
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rYPBHII,a YCTOH'IHBOrO rrOJlHHOMa
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Om)
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0) > O.
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1, ... , m,
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=
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Wo > 0 TaKOe,
'ITO
Po ('tWO)
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3=1
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j
= 1, ... ,
m,
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OJ
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8m )
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Om) > 0, mo
{]Io(,\,) +
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150
t,
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=
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1, ... , m, e'btnOJl,'He'HO 'Hepaee'Hcmeo
KO.MnJl,eKC'H'bt.MU K03<fi<fiU'qUe'Hma.MU
27r,
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m}
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..., ()m
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m
Po(iwo)
+ LPj(iwo)e-i(}j
= 0,
j=1
= 0, qTO HeB03MO)KHO B 06JIaCTll {(()I, •.• , ()m)l()j < ()J, j = 1, ... , m}.
TO <1>(()1, •.• , ()m)
2
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na3,lJ,hIBaHllM KBa3llnOJIllHOMa
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'm
(iw )
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Ipo
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I= E
I,
j=1
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Ul, ... , Um -
m
n (U] + 1)
j=1
,
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m
m
j=1
j=1
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k:f.j
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n (U] + 1).
II (W, Ul, ... , Um)
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m
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Ul, ... , Um II npe,lJ,CTaBJI5IeTC5I B Bll,lJ,e <J?(Ul, ... , um)
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3
a M e q
nOJIllHOMOM
a H II e.
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6
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= 1, ... ,m} -
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2
3
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(5)
+ Vj}, (5)
=
1, ... , m, ce­
B MHO)KeCTBO
(6)
151
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HI1~eHI1I1
<1>(UI, "', Um)
06fl3aTeJIbHO 6Y,l1,eT
=
0,
<1>(UI,
"', Um)
Tj
ceMeHCTBO
(4)
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<
> 0,
-
> 0,
3
CJIe,l1,OBaTeJIbHO, IIO TeopeMe
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2
j = -1, "', m,
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-arcctg
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j
J,
w*
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+ Vj >
J-tj
p06acTHo yCTOH~I1BO,
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it
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ECJII1 X* -
MaKCI1MaJIbHbIH
KopeHb cpe,l1,11 BCex KopHeH aJII'e6pal1qeCKI1X ypaBHeHI1H
il
TO
./
M j::::::
X·-Vj
I-'j
IlOJIaI'M B
JIee· IIpOCTyIO,
qepe3
,
,J =
(6)
1
,"',
= 0, "', 2n; , , , ;
im
= 0, "', 2n,
(7)
m,
J-tm
J-tl
=
XOTfl 11 60JIee I'py6yIO
1,
VI
= ", =
Ym
=
0,
MO)KHO
o~eHKy BeJIl1ql1H M I , "', Mm,
A 2 (mn-k) a6COJIIOTHOe 3Ha~eHl1e CyMMbI
= 0, "', mn - 1, a qepe3 A - MaKCI1MYM
X 2k , k
1I0JIy~I1Tb 60­
ECJII1 0603HaqHTb
OTpl1~aTeJIbHblX K03<p<pI1~eHTOB <l>OPMbI
cpe,l1,11 3TI1X BeJIl1ql1H, TO, COI'JIaCHO MeTO,r:t,y
KOIIII1 O~eHKI1 MaKCI1MaJIbHOI'O KOpHfl 1I0JIHHOMa O,l1,Hoii lIepeMeHHOH, HeTpy,l1,HO 1I0JIyqHTb
o~eHKy
M Jl + ':0'
j :::;
P€3YJIbTaTbI TeopeMbI
j
= 1, ,.. ,m,
3 paClIpOCTpaHflIOTCfl H
Ha 60JIee CJIO)KHble ceMeiicTBa, B TOM qHCJIe
C HelIapaMeTpHqeCKOH HeOlIpe,l1,eJIeHHOCTbIO,
TeopeMa 4.
IIycm'b Be.n.U"tU'H'bt
M I , .. "
2
TO
Mm onpeae.n.e'H'bt U B'btnO.ll.'He'H'bt YC.II.OBUJI,
+ Tj < -arcctgMj ,
Tozaa Ce.Mei1.cmBo 1CBaaUnO.ll.U'HOMOB
w*
j
= 1, .. "
m,
6 1COmOpOM 91 (U), ... , 9m (U) - npOU360.a'b'H'bte ifjy'H1C'IJ,UU pacnpeae.tte'HUJ'I., a.aJ£ 1COmOp'btX cy­
~ecm6yrom coom6emcm6yro~ue U'Hmezpa.a'bt JIe6eza-Cmu.ambeca, po6acm'Ho ycmOU"tU60.
,n; 0 K a 3 aTe JI beT B 0 CJIe,n;yeT H3 TeopeMbI
6.5 [7,
534]
c.
H TeopeMbI
3.
TeopeMa
4,
Bo-nepBbIX, n03BOJISIeT nOJIy~HTb ,n;OCTaTO~Hble yCJIOBHSI yCTOR~HBOCTH KBa3HnOJIHHOMa BH,ll;a
(2),
~HCJIO ypaBHeHHR, aHaJIOrH~HbIX cOBoKynHocTH
BO-BTOPbIX, COKpaTHTb
+ 1)4,
(2n
T.
e.
He 3aBHCSII.II.ero OT m.
(7),
,n;o
~HCJIa
"tlT06bI 3TO nOKa3aTb, Heo6xo,n;HM CJIe,n;YIOI.II.HR
npOMe)KYTO~HbIR pe3YJIbTaT, ,n;JISI ~ero BBO,ll;SITCSI BcnOMOraTeJIbHble nOJIHHOMbI
m
q1(>'), q2(>'), q3(>'), q4(>'): q1(>') +Q2(>') +Q3(>') +Q4(>')
= LPj(>'),
j=l
npH'leM nOJIHHOMbI
Q1 (>.), Q2 (>.)
co,n;ep)KaT TOJIbKO 'leTHble CTeneHH, nOJIHHOMbI
Q3 (>.), Q4 (>.) ­
He~eTHble, Q1(iw) ~ 0, Q2(iw) ~ 0, LQ3(iw) ~ 0, i~Q4(iw) ~ 0, w > 0.
JIeMMa 3. M'HoatCecm6a 'Ha 1COMn.ae1CC'HOU n.aOC1Cocmu
C06naaarom.
,n;OCTaTO~O nOKa3aTb, ~TO ,n;JISI nOJIO)KHTeJIbHbIX ~HCeJI
,n; 0 K a 3 aTe JI b C T B O.
aI, ... ,a m
MHO)KeCTBa
hI =
H
{t, l'
{(t
e-;"TU dgj(u)
aj
h2
=
aj )
!
~
h2
J
J
0
0
hI
+ ... + am)
l'
Idgj(u)1 = 1, j = 1, ... ,m }
dg(u)
=
J
Idg(u) I =
!
1
e-
iWTU
1}
0
CJIe,n;yeT H3 paBeHcTBa
1
(a1
dgj(u) =
e-;"'TUdg(u)
]-1
COBna,n;aIOT. BKJIIO'leHHe
l'
d9(U)
o
= a1
!
1
e-
iWTU
d9(U)
+ ... + am
0
e-
iWTU
d9(U).
0
06paTHoe BKJIIO~eHHe ­
!
. )!
1 1 1
a1
!
e-iWTUd91 ( U )
+ ... + am
0
r,n;e
<PYHK~SI g(u) =
f
j=1
e-iWTUd9m (U)
0
= ( a1 + ... + am
e-iwTUd-(
9 U) ,
0
aj:!!j(u) npHHa,D,JIe)KllT MHO)KeCTBY
E aj
<PYHK~R
pacnpe,n;eJIeHHSI KaK
j=l
BbmYKJIaSI KOM6HHaU;HSI <PYHK~R 3Toro MHO)KeCTBa.
153
Tenepb B03MO)KHO C<p0pMYJmpOBaTb 06ll1,UH pe3YJIbTaT 0 ,n;OCTaTO~HbIX yCJIOBlUIX YCTOH­
~HBOCTH KBa3HnOJIHHOMa
(2).
TeopeMa 5. IIycm'b no nO.4U'HOMaM PI (..\), ... , Pm (..\) nOCmpOe'H'bt nO.4U'HOM'bt ql (..\), q2("\),
Q3("\), Q4("\), no 1WmOp'btM,
8 C80ro o"tepea'b, onpeae.4e'H'bt 8e.4U"tU'H'bt
Ma'KCUMYM cpeau 'HUX. Tozaa 'K8aaUn0.4U'HOM (2) ycmou"tu6 npu
2
<
m9JC{7j}
3
M 1, M 2 , M 3, M4 U M* ­
-arcctgM*. w*
,l:t; 0 K a 3 aTe JI b C T B O. CnpaBe,n;JIHBbI
BKJIIO~eHHH
m
+ 2: Pj(..\)e- ATi
Po(..\)
E
i=1
= {Po(A) + 'fP;(A)
3=1
r,n;e 7
= m9JC{7j}
j
j
e-,rqd9;(U)
0
d9;(U) =
0
. ITo JIeMMe
j I 9;(u)1
d
= I,j = I, """' m}
,
0
MHO)KeCTBa 3Ha~eHHM Ha KOMnJIeKCHOM nJIOCKOCTH CeMeMCTB
3
3
HI
H
H2 = {Po(A)
+
t
j
IJi(A)
3=1
COBna,n;aroT, CJIe,n;oBaTeJIbHO, no
yCTOH~HBOCTH
74
d9; (U) =
0
j
Idg;(u) I = I, j = I, 2, 3, 4}
0
npHH~HTIY HCKJIIO~eHHH HYJIH
3THX ceMeMCTB. TaK KaK
= 7, TO ,n;JIH ceMeMCTBa H2
3aKJIIO~eHHe
j
e-,rq d9;(U)
0
d:
arg (Qj
(iw))
[8]
COBna,n,aroT H YCJIOBHH
= 0, j = 1,
BbITIOJIHeHbI yCJIOBHH TeopeMbI
2, 3, 4, H 71 = 72 = 73 =
4 npH 70 = 0, oTKy,n;a CJIe,n;yeT
TeOpeMbI.
3. IIpHMep.
3, 4
ITpHMeHHM TeopeMbI
,n;JIH nOJIy~eHHH ,n;OCTaTO~HbIX YCJIOBHM po6aCT­
HOM yCTOM~HBOCTH ceMeMCTBa KBa3HnOJIHHOMOB
HJIH ,n;OCTaTO~HbIX YCJIOBHM yCTOM~HBOCTH KBa3HnOJIHHOMa
Nl
f(..\)
= ..\2 + ..\ 2: alje-
N2
AT1j
+ 2: a2je- AT2j ,
j=1
B KOTOPOM a11
> 0, ... , a1Nl > 0, a21 > 0, ... , a2N2 > 0.
raTeJIbHOe ceMeMCTBO KBa3HnOJIHHOMOB
r,n;e
Nl
al
=
2:
j=1
154
N2
a lj,
a2
= 2: a 2j,
j=1
(9)
j=1
,l:t;JIH 3Toro paCCMaTpHBaeTCH BcnOMO­
,lI,JUI onpe,n;eJIeHlUI Kpl1Tl1'"leCKHX 3ana3,n;hIBaHHH CTPOflTCfl <PYHKI.I,l1H
hew, UI, U2)
= -w 2(ui + l)(u~ + 1) + 2a1w(u~ + 1)U1 + a2(ui + l)(u~ +i(a1w(u~
ii...(
) -_
'±' U1 , U2
1 - (U1
4 U2
4 -2
a 1a2
- 4U1U~
r,n;e
k
1) + + l)(ui -1) - 2a2(ui + l)u2), 4k U14 U22 + 4U31U23 - 2U2U24
1
+ U~ + 2ui - 4U1U2 -
+ 4U13 U2 -
8kU12U2­
2
4ku~ - 1),
= -3
a .
1
HeTpy,n;Ho npOBepHTh paBeHCTBO
<I>(M, M)
= 0,
r,n;e
M
J2k
+ "';4k 2 + 1,
a TaK)Ke
HepaaeHCTBa
_
8 i1. +h . CP(U1'
I
U2)
> 0,
8u t21 8u 2t2
Uj=M
i1
= 0,
... , 4; i2
= 0,
+ i2 =f. 0) .
... , 4(i1
'YpaBHeHl1e ,n;JIfl onpe,n;eJIeHl1fl BeJIl1'"lHHhI
w· =
y(1 + J4k + 1)
TO
1
T.
(9)
e.
npH T1
<
.
cJIe,n;ylOrn;ee:
2
<
4.
arcctg
a1(1+J4k+1)
TeJIhHhI, CJIe,n;yeT, COrJIaCHO TeopeMe
4.
> 0,
i, j
3aKJIIOQeHHe.
'feHHfl YCJIOBHH
J
2k +
TO Bce KBa3HnOJIHHOMhI ceMeHCTBa
YCTOH'"lHBhI. ,lI,JIfl CJIY'fafl, Kor,n;a B KBa3HnOJIl1HOMe
B KOTOPOM aij
w2 - a1W - a2
0,
oTKy,n;a
Tor,n;a
= 7<0 = TO =
TO, T2
w·
= 1,
5,
(9)
J 4k2 + 1,
(8),
a TaK)Ke KBa3HnOJIHHOM
He Bce K03<P<PHlI,HeHThI nOJIO)KH­
paCCMaTpHBaTh ceMeHCTBO
2.
TaKHM 06Pa30M, B HacTOflrn;eH CTaThe npe,n;JIO)KeH aJIrOpHTM nOJIy­
YCTOH'fHBOCTH JIHHeHHhIX CTalI,HOHapHhIX CHCTeM
<pepeHlI,HaJIhHhIX ypaBHeHHH,
<PYHKI.I,l10HaJIhHO-,n;H<p­
oCHoBaHHhIH Ha npHHlI,Hne 3arpy6JIeHHfl, T. e.
Ha 3aMeHe
Hcxo,n;Horo xapaKTepHCTH'"leCKOrO KBa3HnOJIHHOMa ceMeHCTBOM KBa3HnOJIl1HOMOB.
3a,u,a'fa
YCTOH'fHBOCTH KBa3HnOJIHHOMOB nOJIY'feHHOrO ceMeHcTBa CBO,1J;l1TCfl K nOCTpoeHHIO npflMoro
KOHyca, lI,eJIHKOM pacnOJIO)KeHHOrO B MHO)KeCTBe, rpaHHlI,a KOToporo - H3BeCTHafl aJIre6pa­
H'feCKafl nOBepxHocTh.
I1apaMeTphI 3Toro KOHyca MO)KHO nOJIy'fHTh KaK peIIIeHHe 3a,u,a'fH
YCJIOBHOH onTHMH3alI,HH. Mx MO)KHO ou,eHHTb 'fepe3 MaKCl1MaJIhHhle KOpHl1 (HJIH HX OlI,eHKY)
(2n+ l)m
BcnOMoraTeJIhHhIX aJIre6paH'feCKHX ypaBHeHHH OTHOCl1TeJIhHO O,1J;HOH nepeMeHHoH.
,lI,JIfl nOJIY'feHl1fl 60JIee rpy6hIx OlI,eHOK MO)KHO OrpaHH'"lHThCfl
ypaaHeHHH, T.
e.
(2n
+ 1)4
BcnOMOraTeJIhHhIX
'fHCJIO ypaBHeHHH He 6y,n;eT 3aBHCeTh OT KOJIH'feCTBa 3ana3,n;hIBaHHH.
AJIrOpHTM n03BOJIfleT YCTaHOBHTh TaK)Ke ,n;OCTaTO'fHhle YCJIOBHfl po6aCTHOH YCTOH'"lHBOCTH
ceMeHCTB KBa3l1nOJIl1HOMOB KaK C napaMeTpH'feCKOH, TaK H C HenapaMeTpl1'"leCKOH Heonpe,n;e­
JIeHHOCThlO. 9TH yCJIOBHfl ,n;OBO,1J;flT ,n;o aHaJIHTH'feCKOH <pOPMhI, HanpHMep, B CJIy'"lae CHCTeMhI
BToporo nopfl,n;Ka.
155
Summary
Dmitrishin D. V. The problem of stability of quasypolynomials and quasypolynomial sets deci­
sion methods.
The effective practical methods for determining conditions of stability of time delay systems
and robust stability of the systems with non-parametric uncertainty are suggested.
JIHTepaTypa
1. 3y606 B. H. K Teopilli JlliHeHHhIX CTa~QHapHhIX CHCTeM c 3ana3~BaIO~M aprYMeHTOM / /
113B. By30B. Cep. MaT. 1958. T. 6. C. 86-95.
2. lIoHmp.1/,zuH JI. C. 0 HYJIHX HeKoTophlX :meMeHTapHhlx TpaHCu;eHAeHTHhIX <l>yHK~H / / 113B.
AH CCCP. Cep. MaT. 1942. T. 6. C. 115-134.
3. MeilMaH H. H., TJe6omape6 H.
IIpo6JIeMa Payca-rypBHu;a AJIH nOJlliHOMOB H u;eJIhIX
<l>YHKu;HH / / Tpy,LJ;h1 MaTeM. HH-Ta AH CCCP. 1949. T. 26. 332 c.
4. Stepan F. Stability charts for linear functional differential equations / / Diff. Equations.
Szeged. 1984. Vol. 47. P. 1049-1057.
5. XapumoHo6 B. JI. 06 onpeAeJIeHHH MaKCHMaJIhHO AonYCTHMoro 3ana3,LJ;hlBaHHH B 3aAaqax
cTa6HJlli3~ / / ,lJ;H<l><l>. YPaBHeHHH. 1982. T. 17. C. 723-724.
6. JKa6'1(;o A. II., XapumoHo6 B. JI. MeTO,LJ;h1 JIHHeOOOH aJIre6phl B 3a,n;aQax ynpaBJIeHHH. CII6.:
113,n;-Bo C.-IIeTep6. YH-Ta, 1993. 320 c.
7. YC06 A. B., ,lI,y6p06 A. H., ,lI,.A-mmpuwuH,lI,. B. MOAeJIHpOBaHHe CHCTeM C pacnpeAeJIeHHblMH
napaMeTPaMH. OAecca: ACTponpHHT, 2002. 664 c.
8. 1I0000.1/,'I(; B. T., I{wn'l(;UH H. 3. Po6aCTHaH yCTOH'IHBOCTh JIHHeHHhIX CHCTeM (0630p) / / 11TorH
HayKH H TeXHHKH. TeXHHQeCKaH KH6epHeTHKa. M.: BIIHIITII, 1991. T.32. C. 3-31.
r.
CTaTbH nocTynHJIa
B pe~aK~HIO
24 HOH6pH 2005 r.
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