Problem of impulsive regulator for one dynamical system of the Sobolev type
We establish conditions for the existence of an optimal impulsive control for an implicit operator differential equation with quadratic cost functional. The results obtained are applied to the filtration problem.
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| Date: | 2008 |
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| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2008
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| author | Vlasenko, L. A. Rutkas, A. G. Samoilenko, A. M. Власенко, Л. А. Руткас, А. Г. Самойленко, А. М. Власенко, Л. А. Руткас, А. Г. Самойленко, А. М. |
| author_facet | Vlasenko, L. A. Rutkas, A. G. Samoilenko, A. M. Власенко, Л. А. Руткас, А. Г. Самойленко, А. М. Власенко, Л. А. Руткас, А. Г. Самойленко, А. М. |
| author_sort | Vlasenko, L. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:39Z |
| description | We establish conditions for the existence of an optimal impulsive control for an implicit operator differential equation with quadratic cost functional. The results obtained are applied to the filtration problem. |
| first_indexed | 2026-03-24T02:38:26Z |
| format | Article |
| fulltext |
UDK 517.9
L. A. Vlasenko, A. H. Rutkas (Xar\kov. nac. un-t),
A. M. Samojlenko (Yn-t matematyky NAN Ukrayn¥, Kyev)
PROBLEMA YMPUL|SNOHO REHULQTORA
DLQ ODNOJ DYNAMYÇESKOJ SYSTEMÁ
TYPA SOBOLEVA
Existence conditions of optimal impulse control for an implicit operator differential equation with a
quadratic cost functional are obtained. The results are applied to a filtration problem.
Otrymano umovy isnuvannq optymal\noho impul\snoho upravlinnq dlq neqvnoho dyferencial\-
no-operatornoho rivnqnnq z kvadratyçnym funkcionalom qkosti. Rezul\taty zastosovano do
odni[] zadaçi fil\traci].
Postanovka zadaçy optymal\noho ympul\snoho upravlenyq dlq system, opys¥-
vaem¥x ob¥knovenn¥my dyfferencyal\n¥my uravnenyqmy, pryvedena v [1] (§6,
7, 14). Ympul\snoe upravlenye v kaçestve slahaem¥x soderΩyt ympul\s¥, yn-
tensyvnost\ kotor¥x moΩno rehulyrovat\. Matematyçesky ympul\s¥ opys¥va-
gtsq s pomow\g δ -funkcyy Dyraka. Upravlenyq ympul\snoho typa dlq ras-
predelenn¥x system rassmatryvalys\ v [2] (hl.85). Zadaçy optymal\noho ym-
pul\snoho upravlenyq dlq ob¥knovenn¥x dyfferencyal\n¥x uravnenyj yssle-
dovalys\ v [3] (hl.86), dlq uravnenyj v çastn¥x proyzvodn¥x — v [4].
M¥ budem yzuçat\ zadaçu optymal\noho ympul\snoho upravlenyq dlq syste-
m¥, dynamyka kotoroj na otrezke vremeny [ , ]t T0 opys¥vaetsq dyfferency-
al\no-operatorn¥m uravnenyem typa Soboleva
d
dt
Ay t By t[ ( )] ( )+ = f t Ku( ) + , u = z tk k
k
N
δ τ( )−
=
∑
1
, t0 ≤ t ≤ T, (1)
ne razreßenn¥m otnosytel\no proyzvodnoj. Ynteres k uravnenyg (1) v¥zvan
tem, çto system¥ s raspredelenn¥my parametramy opys¥vagtsq dyfferency-
al\n¥my uravnenyqmy v çastn¥x proyzvodn¥x, kotor¥e dopuskagt abstraktnoe
predstavlenye v vyde dyfferencyal\no-operatorn¥x uravnenyj s dyfferen-
cyal\n¥my operatoramy, dejstvugwymy v prostranstve funkcyj. V obwem
sluçae πty uravnenyq okaz¥vagtsq uravnenyqmy v çastn¥x proyzvodn¥x ne typa
Kovalevskoj yly typa Soboleva, t. e. ne razreßenn¥my otnosytel\no starßej
proyzvodnoj po vremeny [5]. Uçet πtoho fakta obæqsnqet nalyçye operatora A
v uravnenyy (1) v otlyçye ot rabot [1 – 4].
Vvedem sledugwye oboznaçenyq: X , Y , Z — kompleksn¥e hyl\bertov¥
prostranstva; L a b Y2( , ; ) — prostranstvo Y -znaçn¥x funkcyj, yntehryruem¥x
s kvadratom na [ , ]a b ; W a b Y2
1( , ; ) — prostranstvo funkcyj yz L a b Y2( , ; ) , u
kotor¥x obobwenn¥e proyzvodn¥e prynadleΩat prostranstvu L a b Y2( , ; ) ;
L ( , )Y X — prostranstvo ohranyçenn¥x lynejn¥x operatorov yz Y v X ;
L ( )Y = L ( , )Y Y ; C I Yp( , ), p = 0, 1, … , — klass Y-znaçn¥x funkcyj, p raz
neprer¥vno dyfferencyruem¥x na I ⊂ R ; C I Y0( , ) = C I Y( , ). Funkcyy
v( ) ( , ; )t W a b Y∈ 2
1
budem sçytat\ neprer¥vn¥my na [ , ]a b , t. e. v( ) ([ , ], )t C a b Y∈ ,
yzmenyv yx, esly πto neobxodymo, na mnoΩestve mer¥ nul\.
Otnosytel\no uravnenyq (1) budem predpolahat\, çto A, B — zamknut¥e ly-
nejn¥e operator¥, dejstvugwye yz Y v X , s oblastqmy opredelenyq DA , DB
sootvetstvenno, D = D DA B∩ ≠ { }0 ; K — ohranyçenn¥j lynejn¥j operator,
dejstvugwyj yz Z v X ; f t L t T X( ) ( , ; )∈ 2 0 ; z Zk ∈ ; τk t T∈[ , ]0 . Ravenstvo y ope-
racyy v (1) budem ponymat\ v sm¥sle teoryy raspredelenyj yly obobwenn¥x
© L. A. VLASENKO, A. H. RUTKAS, A. M. SAMOJLENKO, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8 1027
1028 L. A. VLASENKO, A. H. RUTKAS, A. M. SAMOJLENKO
funkcyj so znaçenyqmy v hyl\bertovom prostranstve (sm., naprymer, [8], hl.81).
Rehulyrovanye dynamyçeskoj systemoj (1) osuwestvlqetsq s pomow\g ym-
pul\snoho upravlenyq u, v kotorom τ τ1, ,… N — moment¥ pryloΩenyq ym-
pul\sov, z zN1, ,… — sootvetstvugwye πtym momentam yntensyvnosty yly vesa;
operator K dejstvuet po pravylu
K u =
k
N
k kKz t
=
∑ −
1
δ τ( ).
Yz çysel t T N0 1, , , ,τ τ… v¥berem vse razlyçn¥e çysla y raspoloΩym yx v po-
rqdke vozrastanyq tak, çto t t t Tn0 1 1< < … < =+ . Tohda ympul\snoe upravle-
nye u v (1) dopuskaet predstavlenye
u =
j
n
j jh t t
=
+
∑ −
0
1
δ( ), hj =
τk jt
kz
=
∑ . (2)
Pod reßenyqmy uravnenyq (1) budem ponymat\ raspredelenyq typa funkcyj
y t L t T Y( ) ( , ; )∈ 2 0 takyx, çto y t D( ) ∈ dlq poçty vsex t t T∈[ , ]0 , Ay t( ) 8∈
∈ W t t Xj j2
1
1( , ; )+ dlq j = 0, … , n, By t L t T X( ) ( , ; )∈ 2 0 , dlq poçty vsex t t T∈[ , ]0
udovletvorqetsq uravnenye
d
dt
Ay t By t( ) ( )[ ] + = f t( ) (3)
y v toçkax mnoΩestva tj v¥polnqgtsq ravenstva (ympul\sn¥e vozdejstvyq)
( )( ) ( )( )Ay t Ay tj j+ − −0 0 = K hj , j = 0, … , n . (4)
Zdes\ znaçenyq ( )( )Ay t0 0+ , ( )( )Ay tn+ −1 0 y ( )( )Ay tj ± 0 dlq j = 1, … , n
ymegt sm¥sl, poskol\ku funkcyq Ay t W t t Xj j( ) ( , ; )∈ +2
1
1 qvlqetsq neprer¥vnoj
na [ ],t tj j+1 posle vozmoΩnoho yzmenenyq na mnoΩestve nulevoj mer¥; znaçe-
nye ( )( )Ay t0 0− zadaetsq:
( )( )Ay t0 0− = q, (5)
znaçenye ( )( )Ay tn+ +1 0 = ( )( )Ay T + 0 opredelqetsq kak
( )( )Ay tn+ +1 0 = ( )( )Ay t Khn n+ +− +1 10 .
Upravlenyg u sootvetstvuet reßenye y ( t ) = y ( t, u ) naçal\noj zadaçy (1), (5).
Osnovnoe predpoloΩenye na operatorn¥e koπffycyent¥ v (1) zaklgçaetsq
v sledugwem: v nekotoroj zamknutoj okrestnosty beskoneçno udalennoj toçky
λ ≥ C2 suwestvuet rezol\venta ( ) ( , )λA B X Y+ ∈−1 L , kotoraq udovletvorqet
ohranyçenyg
( )λ A B+ −1 ≤ C r
1 λ , λ ≥ C2
, (6)
hde r — neotrycatel\noe celoe çyslo. Poπtomu (sm.8[6], lemmu84.1, a takΩe [7],
lemm¥82.1, 2.2) spravedlyv¥ prqm¥e razloΩenyq lyneala D = D D1 2+̇ v Y y
prostranstva X = X X1 2+̇ takye, çto operator¥ A, B otobraΩagt Dj v Xj ,
j = 1, 2, D2 est\ lyneal sobstvenn¥x y prysoedynenn¥x vektorov puçka
µB A+ v toçke µ = 0, X2 = BD2 , X1 = AD1, Ker A D∩ 1 = { }0 , Ker B D∩ 2 =
= { }0 . Pust\ P1, P2 = E P− 1, y Q1, Q2 = E Q− 1 — par¥ vzaymno dopolny-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
PROBLEMA YMPUL|SNOHO REHULQTORA DLQ ODNOJ DYNAMYÇESKOJ … 1029
tel\n¥x proektorov sootvetstvenno na D1, D2 y X1, X2:
P1 = 1
2
1
2
π
λ λ
λ
i
A B d A
C
( )+ −
=
∫ , Q1 = 1
2
1
2
π
λ λ
λ
i
A A B d
C
( )+ −
=
∫ .
Zamknut¥j lynejn¥j operator
G = AP BP1 2+ = Q A Q B1 2+ : D → X, DG = D,
otobraΩaet Dj v Xj y ymeet ohranyçenn¥j obratn¥j G X Y− ∈1 L ( , ) , pryçem
G AP−1
1 = P1, G BP−1
2 = P2 , AG Q−1
1 = Q1,
BG Q−1
2 = Q2 , ( )AG Qr− +1 1
2 = 0. (7)
Oboznaçym
W = – Q BG1
1−
∈ L ( )X , F = AG Q−1
2 ∈ L ( )X .
V sluçae r ≥ 1 predpolahaem, çto F f t W t T Xl l( ) ( , ; )∈ 2 0 dlq l = 1, … , r. Esly
v¥polnqgtsq uslovyq
Q q2 =
l
r
l
l
l
l
t t
d
dt
F f t
=
+
=∑ − [ ]
0
11
0
( ) ( ) , l = 0, … , r , (8)
to suwestvuet edynstvennaq funkcyq y t L t T Y( ) ( , ; )∈ 2 0 takaq, çto y ( t ) ∈ D
dlq poçty vsex t t T∈[ , ]0 , Ay t W t T X( ) ( , ; )∈ 2
1
0 , By t L t T X( ) ( , ; )∈ 2 0 , y ( t ) udovlet-
vorqet uravnenyg (3) dlq poçty vsex t t T∈[ , ]0 y naçal\nomu uslovyg
( )( )Ay t0 0+ = q . Spravedlyvo predstavlenye
y ( t ) = ϕ ( t ) � G e Q q e Q f s ds d
dt
F Q f tW t t
t
t
W t s
l
r
l
l
l
l− − −
=
+ + − [ ]
∫ ∑1
1 1
0
2
0
0
1( ) ( ) ( ) ( ) ( ) (9)
( polahaem 00 = E ) . Podrobnoe dokazatel\stvo formul¥ (9) sm. v [7] (lem-
ma83.1). Esly
Q Khj2 = Q K z
k jt
k2
τ =
∑ = 0, j = 0, 1, … , n + 1 , (10)
to, posledovatel\no prymenqq πtot rezul\tat k uravnenyg (3) na otrezkax
[ ],t tj j+1 s naçal\n¥my uslovyqmy ( )( )Ay tj + 0 = ( )( )Ay t hj j− +0 , poluçaem su-
westvovanye y edynstvennost\ reßenyq zadaçy (1), (5) y eho predstavlenye v vy-
de
y ( t ) = ϕ χ( ) ( )
( )
t G t t e h
j
n
j
W t t
j
j+ −−
=
+
−∑1
0
1
, (11)
hde ϕ ( t ) opredelqetsq v¥raΩenyem (9), χ ( t ) — funkcyq Xevysajda. V çastno-
sty, sootnoßenyq (10) v¥polnqgtsq, esly
Im K ⊂ X1 . (12)
Zdes\ Im K — obraz operatora K . V πtom sluçae reßenye (11) dopuskaet pred-
stavlenye
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1030 L. A. VLASENKO, A. H. RUTKAS, A. M. SAMOJLENKO
y ( t ) = ϕ χ τ τ( ) ( ) ( )t G t e Kz
k
N
k
W t
k
k+ −−
=
−∑1
1
. (13)
Oboznaçym çerez z = { z1 , … , zN } vektor prostranstva Z
N
, çerez h =
= { h0 , h1 , … , hn + 1 } vektor prostranstva Zn+2
y çerez τ = { τ1 , … , τN } vek-
tor prostranstva R
N . Vvedem mnoΩestvo vektorov Θ = { }: [ , ]τ τ∈ ∈R
N
k t T0 .
Poskol\ku ohranyçenyq na ympul\s¥ (10) qvlqgtsq ne tol\ko dostatoçn¥my
dlq razreßymosty zadaçy (1), (5), no y neobxodym¥my, mnoΩestvo dopustym¥x
upravlenyj sostoyt yz upravlenyj u = u ( τ, z ) (1), dlq kotor¥x τk t T∈[ , ]0 y
zk udovletvorqgt sootnoßenyqm (10). V çastnosty, dostatoçno predpoloΩyt\
v¥polnenye ohranyçenyq (12). Pust\ snaçala upravlenye systemoj (1), (5) osu-
westvlqetsq putem yzmenenyq yntensyvnostej z1 , … , zN v fyksyrovann¥e mo-
ment¥ vremeny τ1 , … , τN . Zadaça zaklgçaetsq v naxoΩdenyy mynymuma
min ( , )
z Z N
J z
∈
τ funkcyonala kaçestva
J z( , )τ = J ( u ) =
t
T
Y ZRy t y t dt Sz z N
0
∫ +( ), ( ) , (14)
na reßenyqx y ( t ) = y ( t, u ) system¥ (1), (5). V (14) çerez 〈⋅ ⋅〉, Y oboznaçeno ska-
lqrnoe proyzvedenye v prostranstve Y. Otnosytel\no operatorov R Y∈L ( ) y
S Z N∈L ( ) predpolahaem, çto ony qvlqgtsq neotrycatel\no opredelenn¥my y
S E≥ δ , δ > 0. Upravlenye u∗τ, sootvetstvugwee πlementu z Z N
∗ ∈τ , na ko-
torom dostyhaetsq mynymum funkcyonala (14), budem naz¥vat\ τ -optymal\-
n¥m upravlenyem, a sootvetstvugwee reßenye y t∗τ ( ) = y t u( , )∗τ system¥ (1),
(5) — τ -optymal\n¥m reßenyem. V sledugwej teoreme ustanavlyvagtsq su-
westvovanye y edynstvennost\ τ -optymal\noho upravlenyq zadaçy (1), (5), (14).
Teorema&1. Pust\ v¥polnen¥ ohranyçenyq (6), (12); f t L t T X( ) ( , ; )∈ 2 0 y es-
ly r ≥ 1, to F f t W t T Xl l( ) ( , ; )∈ 2 0 , l = 1, … , r ; vektor q v naçal\nom us-
lovyy (5) udovletvorqet ohranyçenyg (8). Tohda dlq lgboho τ ∈ Θ suwest-
vugt edynstvenn¥j vektor z Z N
∗ ∈τ y sootvetstvugwee emu upravlenye
u∗τ = u z( , )τ τ∗ , na kotorom dostyhaetsq mynymum min ( , )
z Z N
J z
∈
τ funkcyonala
kaçestva (14).
Dokazatel\stvo. Predstavym funkcyonal J ( u ) (14) kak kvadratyçnug
formu, opredelennug na Z N . Oboznaçym HX = L t T X2 0( , ; ) , H Y = L t T Y2 0( , ; ).
PoloΩym
w ( t ) = G e Q q e Q f s ds d
dt
F Q f tW t t
t
t
W t s
l
r
l
l
l
l− − −
=
+ + − [ ]
∫ ∑1
1 1
0
2
0
0
1( ) ( ) ( ) ( ) ( ) .
Ponqtno, çto w ( t ) ∈ HY
. Opredelym ohranyçenn¥j lynejn¥j operator L yz
Z N
v HY
:
L v =
G t e Q K
k
N
k
W t
k
k−
=
−∑ −1
1
1χ τ τ( ) ( ) v , v = { v1 , … , vN } ∈ Z
N
. (15)
Dlq lgb¥x τ ∈ Θ , z ∈ Z
N
suwestvuet edynstvennoe reßenye y ( t, u ( t, z )) ∈ HY
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
PROBLEMA YMPUL|SNOHO REHULQTORA DLQ ODNOJ DYNAMYÇESKOJ … 1031
zadaçy (1), (5):
y ( t, u ( t, z )) = ( L z ) ( t ) + w ( t ) . (16)
Tohda funkcyonal J ( u ) v (14) moΩno predstavyt\ v vyde
J ( u ) = R Lz w Lz w Sz zH ZY
N( ), ,+ + + .
Operator
M = S L RL+ ∗
∈ L ( )Z N
(17)
qvlqetsq samosoprqΩenn¥m y ymeet ohranyçenn¥j obratn¥j M Z N− ∈1 L ( ) ,
tak kak
M Z Nv v, ≥ δ v
Z N
2 , M−1 ≤ 1
δ
. (18)
PokaΩem, çto τ -optymal\noe upravlenye u∗τ est\
u∗τ =
k
N
k kz t
=
∗∑ −
1
τ δ τ( ) , z∗τ = { }, ,z z N∗ ∗…τ τ1 = – M L Rw− ∗1 . (19)
Dejstvytel\no, preΩde vseho zametym, çto
J ( u ) = Mz z L Rw z Rw wZ Z HN N Y
, Re , ,+ +∗2 . (20)
Otsgda poluçaem
J u J u( ) ( )− ∗τ = M z z z z
Z N( ),− −∗ ∗τ τ ≥ δ τz z
Z N− ∗
2
.
∏to oznaçaet, çto u∗τ (19) qvlqetsq edynstvenn¥m τ -optymal\n¥m upravleny-
em.
Teorema dokazana.
Yz (16), (17), (19) poluçaem, çto sootnoßenye
Sz L Ry u+ ∗ ( ) = 0 (21)
v¥polnqetsq tohda y tol\ko tohda, kohda u = u∗τ qvlqetsq τ -optymal\n¥m
upravlenyem, a y = y u( )∗τ — τ -optymal\n¥m reßenyem. Dlq operatora L (15)
naxodym soprqΩenn¥j operator L H ZY
N∗ ∈L ( ), :
L x∗ = K Q e G x t dt
k
k
T
W t
k
N
∗ ∗ − − ∗
=
∫
∗
1
1
1τ
τ( )[ ] ( ) . (22)
Tohda sootnoßenye (21) moΩno zapysat\ v vyde
Sz K p k k
N+ ∗
={ }( )τ 1 = 0, p t( ) =
t
T
W s te Q G Ry s ds∫
∗ − ∗ − ∗( ) [ ] ( )1
1 . (23)
Zametym, çto p t W t T X( ) ( , ; )∈ 2
1
0 qvlqetsq edynstvenn¥m reßenyem zadaçy
′p t( ) = – W p t Q G Ry t∗ ∗ − ∗−( ) [ ] ( )1
1 p.v. t0 ≤ t ≤ T, p T( ) = 0. (24)
S pomow\g (23) vektor z∗τ dopuskaet predstavlenye
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1032 L. A. VLASENKO, A. H. RUTKAS, A. M. SAMOJLENKO
z∗τ = – S K p k k
N− ∗
=
1
1{ }( )τ . (25)
Takym obrazom, m¥ poluçyly sledugwyj rezul\tat.
Teorema&2. Pust\ v¥polnen¥ ohranyçenyq (6), (12); f t L t T X( ) ( , ; )∈ 2 0 y es-
ly r ≥ 1, to F f t W t T Xl l( ) ( , ; )∈ 2 0 , l = 1, … , r ; vektor q v naçal\nom us-
lovyy (5) udovletvorqet ohranyçenyg (8). Tohda zadaça (1), (5), (23), (24)
ymeet edynstvennoe reßenye y ( t ) = y t∗τ( ) ∈ L t T Y2 0( , ; ) , p ( t ) ∈ W t T X2
1
0( , ; ) ,
z = z∗τ . Ympul\snaq funkcyq u∗τ = z tk kk
N
∗= −∑ τ δ τ( )
1
qvlqetsq τ -opty-
mal\n¥m upravlenyem, a funkcyq y t∗τ( ) — sootvetstvugwym τ -optymal\-
n¥m reßenyem.
Teper\ upravlenye systemoj (1), (5) budem osuwestvlqt\ putem yzmenenyq
momentov ympul\sn¥x vozdejstvyj τ1 , … , τN y sootvetstvugwyx yntensyvnos-
tej ympul\sov z1 , … , zN
. Ywem mynymum min ( , )
,τ
τ
∈ ∈Θ z Z N
J z funkcyonala kaçe-
stva (14). Upravlenye u∗ = u z∗ ∗ ∗( , )τ , sootvetstvugwee πlementam τ∗ ∈Θ ,
z Z N
∗ ∈ , na kotorom dostyhaetsq πtot mynymum, budem naz¥vat\ optymal\n¥m
upravlenyem, a sootvetstvugwee reßenye y t∗( ) = y t u( , )∗ system¥ (1), (5) —
optymal\n¥m reßenyem.
Operator L = L( )τ (15) syl\no neprer¥ven po τ ∈Θ. Eho soprqΩenn¥j
L∗ = L∗( )τ (22) takΩe syl\no neprer¥ven po τ ∈Θ. Poπtomu syl\no nepre-
r¥vn¥m po τ ∈Θ qvlqetsq operator M = M( )τ (17). V sylu ocenky (18)
obratn¥j operator M−1( )τ ravnomerno ohranyçen y, sledovatel\no, syl\no ne-
prer¥ven po τ ∈Θ. Otsgda poluçaem, çto z∗τ (19), kak funkcyq τ ∈Θ so
znaçenyqmy v Z N , neprer¥vna. Yspol\zuq predstavlenye (20), ustanavlyvaem,
çto min ( , )
z Z N
J z
∈
τ = J u( )∗τ qvlqetsq funkcyej, neprer¥vnoj po τ ∈Θ. Esly
τ∗ = { }, ,τ τ∗ ∗…1 N ∈ Θ — vektor, na kotorom dostyhaetsq mynymum funkcyy
J u( )∗τ , to optymal\noe upravlenye u∗ est\
u u z t
k
N
k k∗ ∗
=
∗ ∗= = −
∗ ∑τ δ τ
1
( ), z∗ = z z N∗ ∗…{ }1, , = – M L Rw−
∗
∗
∗
1( ) ( )τ τ . (26)
Takym obrazom, spravedlyva sledugwaq teorema.
Teorema&3. Pust\ v¥polnen¥ ohranyçenyq (6), (12); f t L t T X( ) ( , ; )∈ 2 0 y
esly r ≥ 1, to F f t W t T Xl l( ) ( , ; )∈ 2 0 , l = 1, … , r ; vektor q v naçal\nom
uslovyy (5) udovletvorqet ohranyçenyg (8). Tohda suwestvugt vektor¥
τ∗ ∈Θ , z Z N
∗ ∈ y sootvetstvugwee ym optymal\noe upravlenye u∗ (26), na
kotorom dostyhaetsq mynymum min ( , )
,τ
τ
∈ ∈Θ z Z N
J z funkcyonala kaçestva (14).
Zameçanye. Teorem¥8818–83 uprowagtsq v sledugwyx çastn¥x sluçaqx:
1)88suwestvuet A X Y− ∈1 L ( , ) , 2) A = E, B Y X∈L ( , ), 3) dim X = dim Y < ∞ .
PokaΩem, kak poluçenn¥e abstraktn¥e rezul\tat¥ prymenqgtsq k upravle-
nyg systemamy, opys¥vaem¥my dyfferencyal\n¥my uravnenyqmy v çastn¥x
proyzvodn¥x. Pry yssledovanyy processa fyl\tracyy Ωydkosty v trewynova-
to-poryst¥x porodax voznykaet uravnenye v çastn¥x proyzvodn¥x ne typa Kova-
levskoj dlq davlenyq Ωydkosty v trewynax [9]. Predpolahaq nalyçye nesvo-
bodnoho raspredelennoho vneßneho ystoçnyka [7] (razdel81.5), v sluçae uprav-
lenyq s pomow\g ympul\sn¥x vozdejstvyj pryxodym k sledugwej naçal\no-
kraevoj zadaçe v oblasty 0 ≤ t ≤ T, 0 ≤ x ≤ π :
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
PROBLEMA YMPUL|SNOHO REHULQTORA DLQ ODNOJ DYNAMYÇESKOJ … 1033
∂
∂
+ ∂
∂
+ − ∂
∂t
y t x
y t x
x
y t x
y t x
x
( , ) ( , ) ( , ) ( , )2
2 1 2
2
2δ δ = Ku f t x+ ( , ) ,
(27)
u =
k
N
k kz x t
=
∑ −
1
( ) ( )δ τ , y x
y x
x
( , ) ( , )− + ∂ −
∂
0 02
2 = q ( x ) , y ( t, 0 ) = y ( t, π ) = 0,
hde δ1 ≥ 0, δ2 > 0, q ( x ) , z x Lk ( ) ( , )∈ 2 0 π , f t x L t T L( , ) ( , ; ( , ))∈ 2 0 2 0 π =
= L T2 0 0([ , ] [ , ])× π , K L∈L ( ( , ))2 0 π .
Zadaça optymal\noho ympul\snoho upravlenyq zaklgçaetsq v naxoΩdenyy
momentov pryloΩenyq ympul\sov τ = τ τ1, ,…{ }N ∈ Θ = τ τ: [ , ]k T∈{ }0 y
sootvetstvugwyx yntensyvnostej z ( x ) = z x z xN1( ), , ( )…{ } ∈ [ ]( , )L N
2 0 π , myny-
myzyrugwyx kryteryj kaçestva
J ( u ) =
0 0
2
1 0
2
T
k
N
ky t x dx dt z x dx∫ ∫ ∑ ∫+
=
π π
( , ) ( ) (28)
na reßenyqx y ( t, x ) zadaçy (27). Lgbug funkcyg y : ( t, x ) → y ( t, x ) m¥ budem
tak Ωe rassmatryvat\ kak funkcyg ot t so znaçenyqmy v prostranstve funk-
cyj ot x y zapys¥vat\ kak y ( t ) ( x ) . V prostranstve X = Y = L2 0( , )π smeßan-
naq zadaça (27) zapys¥vaetsq v abstraktnoj forme (1), (5) s dyfferencyal\n¥-
my operatoramy
A g = g x
d g x
dx
( ) ( )+
2
2 , B g = δ δ1 2
2
2g x
d g x
dx
( ) ( )− ,
D = DA = DB =
�
W2
2 0( , )π = g x W g g( ) ( , ), ( ) ( )∈ = ={ }2
2 0 0 0π π ,
hde W2
2 0( , )π — prostranstvo Soboleva porqdka82 funkcyj yz L2 0( , )π .
Reßenye zadaçy (27) ponymaetsq v sm¥sle reßenyq abstraktnoj zadaçy (1), (5).
Dlq funkcyy g x L( ) ( , )∈ 2 0 π budem yspol\zovat\ oboznaçenye gm dlq ee
koπffycyentov v razloΩenyy
g ( x ) = g mxm
m
sin
=
∞
∑
1
, gm = 2
0
π
π
∫ g x mx dx( )sin , m = 1, 2, … . (29)
Suwestvuet rezol\venta
( ) ( )λ A B g x+ −1 =
g mx
m m
m
m
sin
λ δ λ δ+ − +=
∞
∑
1
2 2
21
, λ ≠ αm =
δ δ1
2
2
2 1
+
−
m
m
, m = 2, 3, … ,
dlq kotoroj v¥polnena ocenka (6) s r = 0. Pust\ funkcyq q ( x ) y operator K
v (27) takye, çto q x xdx( )sin
0
π
∫ = 0 y K x∗ sin = 0. Dlq reßenyq zadaçy myny-
myzacyy funkcyonala (28) na reßenyqx y ( t, x ) system¥ (27) moΩno prymenyt\
teorem¥81 – 3. Predstavlenyq dlq y ( t, x ) (13) y p ( t, x ) (23) prynymagt vyd
y ( t, x ) =
f t x
y t mxm
m
1
1 2 2
( )sin
( ) sin
δ δ+
+
=
∞
∑ , p ( t, x ) = sin ( )( )mx
m
e y s ds
m t
T
s t
m
m
1 2
2 −=
∞
−∑ ∫ α ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1034 L. A. VLASENKO, A. H. RUTKAS, A. M. SAMOJLENKO
ym ( t ) = 1
1 2
01−
+ − +
− −
=
∫∑
m
e q t e Kz e f s dsm m k mt
m k
t
k m
t
t s
m
k
N
α α τ αχ τ( ) ( ) ( )( ) ( ) ,
m = 2, 3, … .
M¥ yspol\zuem oboznaçenye (29) dlq funkcyj q ( x ) , f ( t, x ) , zk ( x ) , y ( t, x ) ,
p ( t, x ) . Zapyßem sootnoßenye (25) dlq τ - optymal\n¥x yntensyvnostej ym-
pul\sov zk ( x ) :
zk ( x ) = – ( )( ) ( )K p xk
∗ τ ,
pm ( t ) = β β τ βm m m j j m
j
N
m m
T
t q t Kz t s f s ds( , ) ( , )( ) ( , ) ( )0
1 0
+ +
=
∑ ∫ ,
βm ( t, s ) = e e
m
m mT t s s t
m
α α
α
( )
( )
2
2 22 1
− − −−
−
, m = 2, 3, … .
V çastnosty, esly N = 1, to τ - optymal\naq yntensyvnost\ opredelqetsq kak
z x∗τ( ) = E K K a x+[ ]∗ −
Φ( ) ( )( )τ τ
1
, Φ( ) ( )τ g x = β τ τm m
m
g mx( , ) sin
=
∞
∑
2
,
a ( τ, x ) = – K q s f s ds mx
m
m m
T
m m
∗
=
∞
∑ ∫+
2 0
0β τ β τ( , ) ( , ) ( ) sin .
1. Krasovskyj N. N. Teoryq optymal\noho upravlenyq dvyΩenyem. Lynejn¥e system¥. – M.:
Nauka, 1968. – 476 s.
2. Butkovskyj A. H. Strukturnaq teoryq raspredelenn¥x system. – M.: Nauka, 1977. – 320 s.
3. Samojlenko A. M., Perestgk N. A. Dyfferencyal\n¥e uravnenyq s ympul\sn¥m vozdejst-
vyem. – Kyev: Vywa ßk., 1987. – 288 s.
4. Lqßko S. Y. Ympul\snoe optymal\noe upravlenye lynejn¥my systemamy s raspredelen-
n¥my paremetramy // Dokl. AN SSSR. – 1984. – 276, # 2. – S.8285 – 287.
5. Sobolev S. L. Zadaça Koßy dlq çastnoho sluçaq system, ne prynadleΩawyx typu
Kovalevskoj // Tam Ωe. – 1952. – 82, # 2. – S.8205 – 208.
6. Rutkas A. H. Zadaça Koßy dlq uravnenyq Ax t Bx t f t′ + =( ) ( ) ( ) // Dyfferenc. uravne-
nyq. – 1975. – 11, # 11. – S.81996 – 2010.
7. Vlasenko L. A. ∏volgcyonn¥e modely s neqvn¥my y v¥roΩdenn¥my dyfferencyal\n¥my
uravnenyqmy. – Dnepropetrovsk: System. texnolohyy, 2006. – 273 s.
8. Lyons Û.-L., MadΩenes ∏. Neodnorodn¥e hranyçn¥e zadaçy y yx pryloΩenyq. – M.: Myr,
1971. – 372 s.
9. Barenblatt H. Y., Ûeltov G. P., Koçyna Y. N. Ob osnovn¥x predstavlenyqx teoryy
fyl\tracyy odnorodn¥x Ωydkostej v trewynovat¥x porodax // Prykl. matematyka y
mexanyka. – 1960. – 24. – S.8852 – 864.
Poluçeno 07.05.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
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| id | umjimathkievua-article-3220 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:26Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ff/5b81469250fac2b643084febf32569ff.pdf |
| spelling | umjimathkievua-article-32202020-03-18T19:48:39Z Problem of impulsive regulator for one dynamical system of the Sobolev type Проблема импульсного регулятора для одной динамической системы типа Соболева Vlasenko, L. A. Rutkas, A. G. Samoilenko, A. M. Власенко, Л. А. Руткас, А. Г. Самойленко, А. М. Власенко, Л. А. Руткас, А. Г. Самойленко, А. М. We establish conditions for the existence of an optimal impulsive control for an implicit operator differential equation with quadratic cost functional. The results obtained are applied to the filtration problem. Отримано умови існування оптимального імпульсного управління для неявного диференціально-операторного рівняння з квадратичним функціоналом якості. Результати застосовано до однієї задачі фільтрації. Institute of Mathematics, NAS of Ukraine 2008-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3220 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 8 (2008); 1027–1034 Український математичний журнал; Том 60 № 8 (2008); 1027–1034 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3220/3186 https://umj.imath.kiev.ua/index.php/umj/article/view/3220/3187 Copyright (c) 2008 Vlasenko L. A.; Rutkas A. G.; Samoilenko A. M. |
| spellingShingle | Vlasenko, L. A. Rutkas, A. G. Samoilenko, A. M. Власенко, Л. А. Руткас, А. Г. Самойленко, А. М. Власенко, Л. А. Руткас, А. Г. Самойленко, А. М. Problem of impulsive regulator for one dynamical system of the Sobolev type |
| title | Problem of impulsive regulator for one dynamical system of the Sobolev type |
| title_alt | Проблема импульсного регулятора для одной динамической системы типа Соболева |
| title_full | Problem of impulsive regulator for one dynamical system of the Sobolev type |
| title_fullStr | Problem of impulsive regulator for one dynamical system of the Sobolev type |
| title_full_unstemmed | Problem of impulsive regulator for one dynamical system of the Sobolev type |
| title_short | Problem of impulsive regulator for one dynamical system of the Sobolev type |
| title_sort | problem of impulsive regulator for one dynamical system of the sobolev type |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3220 |
| work_keys_str_mv | AT vlasenkola problemofimpulsiveregulatorforonedynamicalsystemofthesobolevtype AT rutkasag problemofimpulsiveregulatorforonedynamicalsystemofthesobolevtype AT samoilenkoam problemofimpulsiveregulatorforonedynamicalsystemofthesobolevtype AT vlasenkola problemofimpulsiveregulatorforonedynamicalsystemofthesobolevtype AT rutkasag problemofimpulsiveregulatorforonedynamicalsystemofthesobolevtype AT samojlenkoam problemofimpulsiveregulatorforonedynamicalsystemofthesobolevtype AT vlasenkola problemofimpulsiveregulatorforonedynamicalsystemofthesobolevtype AT rutkasag problemofimpulsiveregulatorforonedynamicalsystemofthesobolevtype AT samojlenkoam problemofimpulsiveregulatorforonedynamicalsystemofthesobolevtype AT vlasenkola problemaimpulʹsnogoregulâtoradlâodnojdinamičeskojsistemytipasoboleva AT rutkasag problemaimpulʹsnogoregulâtoradlâodnojdinamičeskojsistemytipasoboleva AT samoilenkoam problemaimpulʹsnogoregulâtoradlâodnojdinamičeskojsistemytipasoboleva AT vlasenkola problemaimpulʹsnogoregulâtoradlâodnojdinamičeskojsistemytipasoboleva AT rutkasag problemaimpulʹsnogoregulâtoradlâodnojdinamičeskojsistemytipasoboleva AT samojlenkoam problemaimpulʹsnogoregulâtoradlâodnojdinamičeskojsistemytipasoboleva AT vlasenkola problemaimpulʹsnogoregulâtoradlâodnojdinamičeskojsistemytipasoboleva AT rutkasag problemaimpulʹsnogoregulâtoradlâodnojdinamičeskojsistemytipasoboleva AT samojlenkoam problemaimpulʹsnogoregulâtoradlâodnojdinamičeskojsistemytipasoboleva |