TIPOEJIEMbI ATIHAMI,TKI{ YIIPABJI,fl EMbIX CIICTEM

vIlK 517.9

JI.A. Btaceuro, A.f. Pymrac, B.B. Cetteue4, A.A. Vuxpui

06 OTTHM AJIbHOM I4MIIYJIbC HOM YII PAB ]IEHI4I4
B,IIECKPI4TITOPHbIX CI4CTEMAX

K'toqeglte o'IoBa: leckprlrnopHat cltcreMa all00epegutarlLHo-zl'Ire6paxqecroe

)?aBHeHIre. KBa.qparH.rllbli t}yuxuroxel KaqecrBa- tgreHc[BHocr]l tMnytLcoB.
MOMcIITH npHJIOxeHttt 

'VnyJILCOB. 
OnTI{MZlrrbHOe tMrryr'IbcHo€ ynpatarteHl{e. COIIpt_

xeHHoe cocrottttte, AByxroqeqHarl (paeBarl 3a.qata. paasorexHnliecKrlli otubrp. ne_

PexoaHot PexrjM.

BBeneHl|e

floHtrHe {MrrynbcHoro y[paBJreHl-tt Bo3Ht{xJIo ,4JUI oflHcallrlt Bo3aetcrBl{ii. Koro-

pbre Bbr3blBaloT MfHoBeHHble r3MeHeHfit B cocTo{HHfi c[creMbl. I4MnyJlr,cHoe y[paB-

,leHue Haxojlr.rr MHoroq[cJIeHHbIe [pllroxeHliq B TexEIrxe. 3KoHoMrlKe, Qlxaucax, Nu-

OopMarHKe, Meariltr{He n r.a. [1. Oautl ll: nepBbIX nocraHoBoK 3a.qaq orITI{Mi}JIbHoro

r{MrryJrbcHoro ynpaBJreHnt np{Bo.[trct B [2, 3]. Meroabl LrccnenoBaHut HfpoBLIx 3aaaq

m [4, 5] ycfleurHo rrpnMeHs;r].Icb e [6 8] nrr A3YrcHHa auHaMI{rIecKI{x Hrp B cl'IcreMax

c yrrpaBJreHUtMlt r{MrrynbcHoro Trna, a B [9] - 
a cncrevax, flpereprleBaroqrt( HM-

rryJ'rr'cHr're Bo3MyrqeHr.rrr B arlcKperHbre MoMeHTr'I BpeMeH[. B [10, l1] peuanach 3aaa'Ia

o6 omr{IraJlbnou ItMrryn6cHoM ylpaBJreHHI-I croxacrlrqecKofi cHcreMon.

Licc,'re,qoBaHte 3aaaq[ onrr,rMaurbHoro HMrry,tbcHoro y[paBJreHnt MLt arjllocrpt-t-

pyeM Ha npllMepe paaHoTexHHqecKon cHcreMr,r - 
3JleKTpr-iqecKoro qerl'IpexrloJrlocHl't'

xa. Ar.{HaM[Ka [oro6Islx crrcrev s o6qeM cJryqae onrchlaaercr 4uQQepenqr.ra,rruo-

&rre6par+recKl{MH )?aBHe}rr{tM{ x rro3ToMy 3TI-t cflcreMhl orHoctrct K le'Iaccy Ae-

clipEnropHhrx cacreu. Tarne cocpeaoroqeHHhle crcreMbl xopouo u3BecrHLI B Teo-

pHH y[paBneHrs [12-15] (cM. rarxe 6H6ruorpaQltrc B 3rl{x pa6orax). Konevnouep-

Hlre ,IecKp[nTopHbre cficreMhr y[paBneHr-rr MoxHo HcI]oJrb3oBaTh aJIt a[npoKCHMa-

uuu 6ecxoneqHorvepHhrx cncreM yrrpaBJreHHr{ TLIIa Co6on€Ba rulli HetBHhIX cucreM,

HanprlMep [16-18].

Baeaeu o6orsaqeHw: L2(0,7:IR') fipocrpaHcrBo Bexrop-QyHKrrnfi co 3Ha-

rreHHrMIr B R', cyrauupyevux c KBarparoM HopMbr Ha l0,Tl; w{Q,T:R') npo-

crpaHcrBo Co6o,resa nopaaxa & nerrop-Qynxqxi n: !(0,7; R"). y xoropux o6o6-

ueHHHe npor{3BoaHbre ro nop.{aKa t sxrroqfiren ao npl{Haanexar L2(0, I; Rn).

@lrxuxu n w)to,'r:n') 6y.uev cvrrarr Henpep6rBHhrMri ua l0,Tl, n3MeHrlB r{x,

ec-ra seo6xornuo. Ha MHoxecrae rryJreBofi Mephr. Cnvsoru (.,) 
" ll.ll o6o:Ha'Jarcr

cxalgpHoe flPomBeaenre tl HopMy B coorBercrByrorq[x [pocrpaHcTBax, c]rNraor f
o6o:aa'{aer erHHr.rqHyro Marpnrly HaAJre)rauefi pa3MepHocrh, TpaHcnoHHpoBaHHy}o

. ,-I A B,IACEHKO. A.f pyrKAc. B.B CEMEHEII, A.A. qHKPltfi. 2019
t, I e tcory ap on H b! n Hay\ H o - m er H u\ e c 

^"ui 
)K)p H an

fipo6-E.uu ynpasten8 u unbop*anurur. 2019. Il?.i 5



Malpnrry R Marpqe K o6ostraqaev K', corpnxerrtf',l ollepamp K oneparopy Y
o6o3xaqaeu Y'. Ecrrn 0 - rraAparxar xynerar varprq4 ro noaaraeu 00 = E.

l. IlocruoBrcN 3alaqf, AJrf, 3reRrplsocnof qetrf,

,4n AeMoHcrparEl rqFrJlaraeMoFo MemAa xccne,qoBamr omuMaJILHom trM-
rryJrLcxom ytrpaBJr€Hf,rr paccMorpnM gJrer pnrecxyro qem, rro@axemryro Ha pacyx-
Ke. IIerB nre€r,@a f,cmqEn(a mnprxeHtrr: e(t) n q(t); a(t) - 3.cAassar QyHxrE

or BpeMeHtr, q(f) - ynpaunerrxf, xcrovnn< xanprmenn. IIaporreJBHo c eMKocrr-

un c1,c2 Brjrx)qeux npoBo.u rocT|| gt,gZ. Hz B4rrpemef, rarrf, pactronoxeHhr

tq4trtrf,Bxocrb I n conponrueme r.

Toru I 6 r, I g r, I g r, I g r, I y u uanpDKentrr U 6 r,U g r,(I DIJ 1 ytroBrerBoprror

3aKoraM K4pr<roQa

Ig.,lInr=Igr+Ig2+IL, Ugr+U1+IJ,=e,(JgrrU6r+a1=e (1)

x JpsBHenn'M
ddUL=L;;lL, U,=r17, Is,=Criuri, Ig,=OrAcI j=1,2, (Z)

rye L,r,Cj,gj - floroxmerLnf,re rocrcrHH[re. Coglome areropl{s€cxof, q€ItrI

xapaKr€pByercr B€rcmpoM

faar) [ rz<rr )x(t)=lx2() l=lu.,al I,
l':alJ lu.,1ry.1

cocrctllxM El (3f,€prieTl recKrno) KoMIroEerII, romprle olBernr(}T l|llepqnoHrsfirt 3re_
MeHraM. C troMo4ro (l), (2) norysaeu cncr€My ?pex ,q1ftspe@aJcHo-ame6pa-
tqecxrr( ypaBHeHd qntocEr€Jrbrro rcnepnerlrrecrEx)) KoxnoEeEr:

Lfit r+us, +rIL =4t),

a ft u r, - q !*u 
", 

- I 7 + s{) g, - gzu cz = o,

U6, |Us" =4t)-q(t).
llepemueu cxcr€My )'paBx€rld (3) B Eercropro[ rspue oluocrrernxo Bexropa co-
cmmd r(t):

I

4..4

€

E
4
E
ill
.l

3!
E
EC

rF

Ct

o6d
y?r

(3)

E
c[
eoi

I

=&
re
gl

ynF
ETT

cu
h,4

IL

I
uzl

I
U,I

ISSN 0572-259r



L;*<,>l * o<,1 = /(t) + &,' (t),

u2=lw14t -r*)-
k=l

(4)

rAe /' r o) f'o) fol

^=l', 
:, tl'=l l' n' --n'[rar=l o.li*'=l o I' u'

loo o) \;;;) [*rJ [-tJ

Ot 
" " 

U'r, rctrycraeM' tIIo B MoAeIE gJtelcrpnsecaofi cxgI€MrI por'MErpnBaerE

*d';"il;**'y,ffiffiffiffiffi#
E lfxtxo,xAeEEf, eJlerlpFl€cKoro 'Iolsl ru lIl--i,*.""'",*r""ED( Ep3ge[crBd.

tr"T"-t *" I'r*',*;o r{r*oro rma tErterct f,"'oqHm r try&cl

,Ft rpm' ampotoon * pt"yo"' te"roo't"'"to' vro mr 1;(l) p€rympycrc'r n\r-

Eysamora BosAeilclilt B MoMeBrbr Ep€ridem t1' t2"'' t'v e [0' Il:

N {t)le,r= I t&*+ 0) - 11(t1 - 0) = w1 ' k=l'--N' (6)

3trecb lq,.'- w,v - nmeEcgBxocru nnu D€ca' oroeqaloqrt€ MoMeHTaM riMfiylbctlsrx

EGAeisrBrd. MatreMamq€crs IofiyJrrchr omcLrBar(/rct c lououo-S-'0Vmow AAem

6(r) (pocryegoremre rnr o6obqenna Qynaur)' Pery4oBauf,c orcreMor (4) c roMo-

il. ffi*". *'otn",s,d (6) p€a,*ye* E Bxae f,Mtryrrcsorc ytrpBBtemit

(7)

c uonoqrc nm)arcsoro ypaBlremt (?) f, Msrprer K2 rpx nmyJscsoM ynp@Ileltrm

(L\
*., =lo I (8)- 

loj

cncreua(4)cnmymcmo.rnrosAeficTBnnm(6)rrpeo6pasyercnrrnry

d' 
1nQ)1+ Bx(t) = f (t) + Ku,t(t) + K2u2' (9)

dt

PaBeFcDa E oltep64|It B (9) noE alorct B oMLIGJte Teopm'pacnp€lererd rulr

oSo6nemo< i[yuud' gro osua;; qm BMesro ypaBsel t (9) Mrr paccMarploaeM

yparxeme (4) .{Jn f * t* ' a r rowa:< '* 
p*"ttp*" rn'rtr)aEcEre Bo3treficTEllr:

(A)(tt + 0) - ('{'r)(r* -0) = K2*1' k =l'"" N '

MareMarus€cxaf, Mo,Ilstb stel(rplt.recro[ [eEt " 1*]1::Y:tt 
cocpeAororreE-

Errun 3JreMe'ra*x' Ko"oa mrur ;^J;;; no,*eprarorct^nxnyrbcs*r sogael-

crBf,ru, ormcasa E lzol' n rror f6# rayqarlacb nepexoFrre flpouecchl B 3[enpr-

"**f 
** ^"w*Eoe 

ytrpasJrerme He paccMarpl|EaJlocb'

PerytupoEastr€ cucrenx 19) o"yu"t-""t"" c EoMoqbrc- yupaEJtesat u=

= lu{t),u2\ ' B xoropoM nr(rl I <jrIqsoe )mpaBJresf,e)' E aarbneltueM Ear6na€-

Moe BMepxrinlM yllpaMegrcM' a 
'l2 - 

ilxcrc l|MnyllDclroe ynpaBJreEf,e c B@aMx rtt

f, MoMelImMf, npruIoxeg|it *T ',*o" 
lr' 3a!8qa "T1T-:"t 

uMtryrrlfloro

vnDaBleHf,t B crrsr€Me (r) cocronr " ""'6op" 
BxoIIEom (FlpaBrnoqero) Falprlne-

;ffi,'d;ffi;'",o v"p;;; zz (7)' peamrToun M'I'BMvM n,Hep."rFre-

cllono) rctarpcrf,sHoro Oyrtrg$oHara o nrnyJrf,cHHMn trnreHcliBEocrtMn:

uAlneoaaaw,tno-nanvcnw%1, *, 
1

* nfrniu l'twletut n utttioptut'nu<



flpaaaa vacru Qopr'rylu (29) co4epxar rrl+2 ner:secrublx BerwtnH AQJ),

j =0,1,....n, r p2(0). Hanoruuur,l, rrro 3Aec6 r'lor'aernH {lr}f=o - pa3nHqHhte Mo-

MeHTbr lrMrryJrbcoB a: {r1};'=1, pacno;roxeHHbre B Bo3pacrilroqeM nopr.qxe. yxa3aH-

Hhre Hen3BecrHhre naoxuo ualrrr rl: crrcreMbt u+2 nurcfrHbtx alre6panvecxlrx ypal-
HeH[trJ xoropar floryqqrcr. ecJru B rperbeM coorHorxeHrrj a (29) noloxmr | = t J Ms
j=1,...,n, B qerBeproM rlojloxlrrb t=tray=7, a MoMeHTbr nMrryrbcoB Tt BLIpa3HTb

rrepe3 MoMeFrLr t r. ?Ta cvcTeMa lwrefnux alre6patvecKr-tx )paBHeH[r xMeeT eIxH-

cTBeHHOe perueHHe B cruTy cyqecTBoBaHrri { ear,rHcTBeHHocTx peueHHr aB),xToqeqHofi
KpaeBofi,3a(a'{Il (27).

3ar.nro,lexne

PaccMorpeHa 3aAarra oIITIjMaJrbHoro r{MnyJltcHoro ynpaBneH{r,qecxplrnrop-
HoF c[creMofi, 3axrroqarouarcr B HaxoxaeHr. t MaBHMyMa rra.uparavnoro Qynx-
rlHoHana xaqecrBa c flMrryJrr,cHbtMI, rrHTeHc[BHocrrMu. ,{nnaur,ra noAo6gux cr.t_

creM onncbrBaercr lnQQepexuxa,ruro-alre6patvecnrun ypaBHeHurMrr! He pa3pe-
UIeHH;IMH orHocr{Ter;Ho rrpox3aoaHotr. OcnosHoe npeanoJroxeHte npn HccJre-

AoBaHI-.tlt [ocraaJleHHofi 3aAa.lr cocrorrr B peryr.{pHocrx xapaxreprrcrllqecxoro
nyqKa MaTpfiu, oTBeqabruero ypaBHeHHlo. ?ro npeano,roxenue rro3BoJrreT IIpe.{-
craBarb peueHfle EMrryJrLcHoro 

^u{QQepexqnarrxo-a,rre6patvecroro ypaBHeHr.rr B

BLIAe 0opMyn6r Bap ar+rq nocTotHHLrx H rapaHTr-tpoBarb cylIecrBoaaHr_re a eAHH_

crBeHHocr; perueHl-it nocraBJrenHoi 3ana,ta onrHMaJIbHoro t MnyJrbcHoro yflpaB-
neHxr. qepe3 perneHre conp.sxeHHoi aByxro.teqHoi KpaeBofi 3aAaqx crporrc, on-
TnMiurbHoe IlMrrynbcHoe y[paBJreHae u onT[MiulbHar TpaeKTopxr cHcTeMLI.

flpeAnoxeHrcrfi Merolt uccJreAoBaHxr rlpoHjurrocrpr{poBaH Ha npr,lMepe orrrlr_
MIIIbHOTO XMnyIbCHOfO ynpaBJreHHr nepexoaHblMI_r pexlrMaMll oaHofo paal_ioTexHa_
,rec(oro (DtrBTpa c fHeprerrrrrecKr.rM Qyurur.ronalou KaqecrBa. C noMotrlbro r[opuyln
Bapnaulav nocrornHr,Ix omjcaHht nepexoaHLle pexlMbr Qrurrpa npll l_lMrrylrbcHhrx
Bo3MyueHrrx Toxoa r.r HanprxeHntr. flpelnoxeHHhrfi B pa6oTe MeroA {ccreAoBaHrj,
3aaaqH omr{MaJILHoTO HMrryJrr,CHoIo yIIpaBJIeHr_rr rro3Bojlr,rjr fiocTporIlb orrTr,tMarbHoe
HMnyrbcHoe ynpaBneHne r.i orBeqanrqrii euy onmua,rluufi nepexoquoft pexnM pa_

AuorexHt{.recKorc (DIlnbrpa.

JI.A. Bnqcenro, A.f. Pymrcac, B.B. Ceueueqa, A.O. guxpii

T]PO OTITT4MAJIbHE IMNYJIbCHE KEPYBAH}UI
B AECKPI4I]TOPHHX CI4CTEMAX

Baotaevo taravy onrrMitlllxoro ilttryllonoro (epysaHHr 3 KBaapaar.rHr.rM
$yuxuioxaaov rrocri 4lr lecxpulropHoi cI,|crcMH. EBonroqi, 

"r"""", onn-
cycrLcr AraoepeHrlifiHo-a,rre6paiqHUM pinanxnenr, u1o ae po:r'l:re nintocno
tloxiasoi craBy. KepyBaHllr clcrevoro tAificurceucr ru,trxoM :uinl axuipto_
ro KepyBaEHr ra 'rncro irunyarcroro kepyBaHHr. gtcro iuny,rtclre (epyBaHH,
xapaloepr3y€Tbcr ilteHcuruoctlllr iuny:ucie ra MoMet{TaMrr npyrcJtararJlx
iuny,rtcia. Ocsogge o6uexeaxl norrrar s roMy. uo xaparrepacra,rnri xvy_
ToK Mar![q6, uro BianoBiaa€ piBH*urro cranis. e perynlpxnM_ y repvixax xa-
pakrepucrllqttoro MarpnqBofo xMyrl(a acral{oBnrorcTrc, yMoB, ictyoannr i
eauHocri orrtManhHoro KepyBaHH, Ta BilnosiaHoro onrlMtur6Hofo craHy. OII-
TuMaJIr,Fe l(epyBaHlfi Ta olrtManbH{i craH 6y.0yrorscr 3a AonoMorolo cDprxe-
Horo craDy. ulo € po3B'r3KoM cnplxeHoi Inorouroloi Kpafiosoi 3a!a.ri. pe3ynh-
Tarri rltrocrpyrorr,ct Ha npqftnaai ,leckprrnropHoi ctcreMx, tqo onhcy€ nepex;_

l

{

I

ISSN 0572-269t



3.

4.

l.

7.

Aqi p€xnMr y pa,qior€xrisEoMy lDirIETpi. AJI' qid crcr€xR po3rffAae$ct
eHeprqrusrn4f oynxqior.r rf,osri 3 ixlryn6csro|It iuTesclosocrrr.n, qo xapax'
r€pBye eBepriro isepuitE{x ete[enrir ra rxiA[oi ssnpynr oinbrpy, a rsKox
inreEcrtDnocri Ta MoMe[Tq upxxts,[aEru ixrryarcia. flepexiali perulnr npn
ixnyarcrux s6l4rennex orpyxie i mnpyr onsgypttct 38 aoloxororo tlopnynn
rapiauii cra:rax,qrr ir.nytrbclroi aeoxpsllopHoi clrsreMtr

trrroqori crou: a€c[pl|InlpHE auqr€Ma, &it€p€IuiajEEo-ajtr€6pairye piBHrEI&

roa4)amqux oyuoionar rlocri, irntsqosocri irfiyJDcir, MoreFrtl rD|rjraras'
sr innyarcir, onrnxatrrttr ixnynLcre xepyBslfit, cnpax€rrtr cras, aBorocroEa
rpalola 3aAss4 pa.qiorexd{d Qfurrp, nepo(i,ld pexor.

L.A- Vlasenko, A.G. Rutkas, Y.V. Semenets, A.A. Chifuii

ON THE OPTIMAL IMPULSE CONTROL
IN DESCRIPTORSYSTEMS

we study the optimal impulse control problem with quadratic perfoma[ce func'
tional for a descriptot system. The system evolution is described by a linear
differ€ntial-algebraic equation not solved with respect to the derivative of the
state. The system is controlled by changing the measurable coottol and the pure

impulse control. The pure impulse coltrol is characterized by impulse intensi-

ties and moments of impulse applications. The main restdction is thst the char'
act€ristic mstrix pencil corresponding to the state equatiott is regular' ln terms
ofthc characteristic matrix pencil, ws establish the conditions for the existence
and uniqueness ofthe optimal control and the corresponding optimal stEte. The
optimal control and thc optimal state 8le constructed by using the adjoiot state,

which is a solution to the adjoint two-point boundary value problem. Th€ re-
sults are illustrated on an example of descriptor system that dcsc.ibes transient
states ill a.adio tech cal filter. Fo. this system, we consider ao energetic per-
fomance functional with impulse intensities chanclerizing the energy of i[er-
tial elemcots and input voltag€ ofthe lilter and also moments ofimpulse appli-
cations. Transient states under impulsive pertubations of currents attd voltages
are described by using the formula of variation of constsnts for the impulsive
descriptor syst€m.

Keyrord!: descriptor sysenr, differential algebraia equation, quadratic performance

functional, impulse intensities, momenls of impulse applications, optimal impulse

contol, adjoint statg two.point boutrdary value prcblsm, radio technical filt€r, tran-
sient state.

EcEcyoas A., Jkortc x.-JI. firfiyrrcHo€ ytDaue&s n xB!3xBapsauoEE6re 8@aBenqrEa. M. :

Haylca, 1987. 600 c.

IQamcrod ILIL TeQnr )i[paBlcrnrr.BrDN(er{|€ra. Jt |efHEI€ cnqterGl. M. I Hayr4 1968. 476 o.

SylrolcF|t A,f. CTyrrlTssr ftopx,r pscqeaeressLrx cucleic M. : ltayo, I 9?7. 320 c.

Chilaii A.A. Conflict-controlled proc€sses, Dordrecht : Springer Science and Business Medi4
20f 3. 424 p. htp://doi.otgll0.l007/918-94-017 -1135-7.

Chilaii A.A. An analytical method in a dynamic pusuit games. Proce?dings of the SleHot Intli-
tate of Ma&ematiat. 201 0. 27t, N l. P. 69-85. htF://doi.orgl10.1 I 34lS008 15438 10040073

I(prDoHoc IO.f., Mfls.n|tI Ir-I4, tlDerd A.A. ,4qavrt'r€€iore rtIpEI c palpEBtrlwr rpaeKropn-

irs. K. : Ila),& Aa|rca, 2(n5.220 c
Chifqii A.A., Mdichh t.l. Differcotial grmes with impulse control ofplaye',f' Prcceedings of he
Steklov Institrre of Marhendtics (Suwl.). 2005. Suppl. l. P. 568-581. htlps://mathscinelams.-

orglmathscinet-gditerntnr2l 56250.

Chilsii A.A., Matycbyn I.1,, Chikii K.A. Differ€ntial games with impulse control. Advanc€s in

dynamic gane trcry. Vol. 9. ofAnnals ofthe Intemational Society ofDynamic Gam€s. Boston :

Birt<hluser, 2007. P. 37-55. http://doi.org/10. 100'l 1978-0-817f.4.553-3-2

Matychyn I., Chilaii A., Onyshchenko V. Conflict-controlled process€s involving ftactional dif-
f€r€ntial equations with impulses. Mathemotica Ealkonica. 2012.26, N l-2. P. 159-168, http://-

www.malhbalkadc&info/

Meed$qodnb,, t@yo.o-rrct Becna, *wra,
(n$Evhtyrq6pr8atl$q @'It ur, 2019, $e 3



ll.

10. Vlasenko L.A., Rutkas A.G. Stochasiic

Differcntiol Equations. 20ll' 47' N

t4.

l).

irnprdsc cqntol of p66bolic systems of Sobolev qpe

10. P. t498-1507. http://doi.orgl10.1134/500122661

r l100132.
Viur.Jo f-,,q , nutku" A.G Optimal contsol of a class of ratrdom distributed SoboleY tvpe sys'

tems with after€ffect. Jo umal of Auto'atiott a'd Informaior kiences 2013 45'N9-P 6G76'

htg://doi.orgl10. l6l 5AAutomattofscien-vil5'i9'60'

;;i;. ;i"$i. *"*l svsterns. Berliq Heidelbcrg' Nsrv YoL' t'ddor'" Pads' Tokvo : Springer-

Verlag, 1989. 332 p. http://doi.org/10.1007/Bft0m2475'

fJJaC.,l., Mez n On linear luadr*ic optinal corol frobl@s q tine-variables descriPtor

.rl*^t siit r controt opiin- 2004' 42' N 6' P' 20r2:2077' hltp://doi ors/lo'l137/

s0363012900380991.
i"- in e""iytr" -d design of rtestriprc lirg ryslcos' l'lcrt Yod'' Dordrech! Heidelberg'

London: Springer, 2010. 494 p. hq/&i'ae/t 0'lm7m&-l4l91Bn4'. 
.

vf^"*" L.e- nutkas 1.G., Sernenas V-v' Scqreric c{q6itidt lrd decomposition of de'

s"'rip,o, *nooi tr*"." . Jotnal of Atmtab od @abt s.:ie/,es 2018 50' N 9 P 60-

?5. http://doi.ore/ I 0. I 6 I S/JAutomarlnf Scinv50 j9J0'

vtasentol.e.,-nutkasA.G.ontl|coglilDa|cffiolof$s*ccrib€dbtimPlicitevo|ution

"iti"i. oin"**.l Equotiont. 2009' 't5' l{ 3' P' 129438 htp://doi ors/l0 1134/

s0012266109030124.
vi""o*" I_.,1., cr,*rli A.A. TIle merhod ofrGolvirg fE ri* fq I dynamic garne in a Sobo-

fev system. Joumal of Automation ond Lforotit' Sci'rces 2Ol4' 46' N ?' P' l-l l '

hnp://doi.org/10. l6l 5dAutornatlnfScietv{5'i7'10-
vr""r"r." li., t"*^ A.G. On a differarid gre i! . sts.o G.rib€d by an- implicir differcn-

;;;;"; equeLior.. Dfurential Epalio'6 m15' 5l- N 5' P' 79H07' http://doi org/

l0.l 134/500122661 150601 17.

PoceE{erba A.C., fts{Icotr E.}L IIeqoEE 4@r r o6fuie oyngl M : Ha]'l(a.

1966.440 e
VtasenLo I-.4., Perestyuk N.A' On $e sohdiliry of ildirc diff€t€ntial-algebraic equations'

Ilkrairian Marhematical Joumo!. 2005- 57,\ 1' P' 551-fl'@2ll&ri'org/t 0 1007/sl 1253-005-

0209.4.
Vlasenko L.A., Rutkas A.G., Samoileoko r{.lr'{- Prdrb dinbive regulator for one dynamical

system of the Sobolev tvpe. UkrainiDn Yffi 'bzf 2008 60' N 8 P 1200-1209'

I
:

C

C
t
I
I
I
a
I

16.

18.

19.

20.

-
-rtt-
--1

--
Gr
G rs)tfi
-lrclEabF
---I
44l|

=341-

--l
(r. l|-|
r+
crE-l
tfr-tl
Etl-
rl--(

-4

-I

tltrtlts
rlICrF
a.l-tJ

'ia--
tGIE
d--oltt-ccrEol,t+
.=;

21.

bttp://doi.orgl10.1007/sl 1253409{l3rv'
22. Vlasenko L.A., Samoilenko A.M. Optioat cmol ril -FlsiE cornponent for systems de-

scaibed by op€rator diffetpntial q],II1o6' IrtbaLbt tfut'l Joa al 2009 61' N I
P. 1250-1263 http://doi.org/ 10.1007/s l l2530lO|n|/Cl'

23. JhoEa )K.-JI. OnMMaJlEIroc JmpaalteE c GrcttrE crEm lDorsesnxM c qasmuMu

rDorr3Eo,ryllMlr' M. : MI'P ' 
1972. 415 c'

IIa'OYexo 22'02 20 I 9

l8 rssN 0572-2691