Problem of optimal control for a determinate equation with interaction
The problem of optimal control of differential equations with interaction is consider. It is proved that the optimal control satisfies the maximum principle and there exists the generalized optimal control. It is shown that, in the considered problem, new technical aspects arise as compared with t...
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| Date: | 2008 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2008
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3227 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509279382929408 |
|---|---|
| author | Ostapenko, E. V. Остапенко, Е. В. Остапенко, Е. В. |
| author_facet | Ostapenko, E. V. Остапенко, Е. В. Остапенко, Е. В. |
| author_sort | Ostapenko, E. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:39Z |
| description | The problem of optimal control of differential equations with interaction is consider.
It is proved that the optimal control satisfies the maximum principle and there exists the generalized optimal control.
It is shown that, in the considered problem, new technical aspects arise as compared with the usual problem of optimal control. |
| first_indexed | 2026-03-24T02:38:35Z |
| format | Article |
| fulltext |
UDK 519.21
E. V. Ostapenko (Nac. texn. un-t Ukrayn¥ „KPY”, Kyev)
ZADAÇA OPTYMAL|NOHO UPRAVLENYQ
DLQ DETERMYNYROVANNOHO URAVNENYQ
S VZAYMODEJSTVYEM
The problem of optimal control of differential equations with interaction is consider. It is proved that the
optimal control satisfies the maximum principle and there exists the generalized optimal control. It is
shown that, in the considered problem, new technical aspects arise as compared with the usual problem
of optimal control.
Rozhlqda[t\sq zadaça optymal\noho keruvannq dyferencial\nyx rivnqn\ z vza[modi[g. Dovede-
no, wo optymal\ne keruvannq zadovol\nq[ pryncyp maksymumu ta isnu[ uzahal\nene optymal\ne
keruvannq. V zadaçi, wo rozhlqda[t\sq, vynykagt\ novi texniçni momenty u porivnqnni zi zvy-
çajnog zadaçeg optymal\noho keruvannq.
V dannoj stat\e rassmatryvaetsq zadaça optymal\noho upravlenyq potokamy,
voznykagwymy pry reßenyy uravnenyj s vzaymodejstvyem [1]. Takye uravnenyq
ymegt kak koneçnomern¥e, tak y beskoneçnomern¥e svojstva, tak kak, v otly-
çye ot ob¥knovenn¥x dyfferencyal\n¥x uravnenyj, zadagt srazu ves\ potok
na fazovom prostranstve. Poluçen¥ neobxodym¥e uslovyq dlq optymal\noho
reßenyq, analohyçn¥e pryncypu maksymuma Pontrqhyna [2, 3], y pryveden¥ us-
lovyq suwestvovanyq obobwennoho optymal\noho upravlenyq.
Dlq nekotoroj veroqtnostnoj mer¥ µ0 rassmotrym mnoΩestvo funkcyj x :
[ 0, T ] × R → R klassa C1
po pervoj peremennoj y klassa L2 0( )µ po vtoroj.
Pry fyksyrovannoj pervoj peremennoj na prostranstve funkcyj L2 0( )µ zada-
dym neprer¥vn¥j operator F L Lt : ( ) ( )2 0 2 0µ µ→ , zavysqwyj takΩe ot nekoto-
roho parametra u ( t ) ∈ U ⊂ R :
F u x yt( , )( ) = f u t x t y x t z dz
R
( ( ), ( , ), ( , )) ( )µ0∫ ,
hde f : U × R × R → R — ohranyçennaq, neprer¥vnaq po sovokupnosty peremen-
n¥x funkcyq, ymegwaq çastn¥e neprer¥vn¥e y ohranyçenn¥e proyzvodn¥e po
vtoroj y tret\ej peremenn¥m.
Rassmotrym zadaçu optymal\noho upravlenyq
I ( u ) = ϕ µ( ) ( )s dsT
R
∫ → inf,
˙( , )x t x0 = f u t x t x z dzt
R
( ( ), ( , ), ) ( )0 µ∫ , (1)
x x( , )0 0 = x0, x R0 ∈ .
Zdes\ µt = µ0
1� x t− ⋅( , ), ϕ ∈C R1( ), ′ϕ fynytna, u KC T∈ ([ , ])0 ( çerez
KC T([ , ])0 oboznaçen klass vsex kusoçno-neprer¥vn¥x funkcyj na [ , ]0 T ) .
Operator Ft moΩet b¥t\ zapysan v vyde
F u x xt( , )( )0 = f u t x t x z dzt
R
( ( ), ( , ), ) ( )0 µ∫ .
Pod reßenyem zadaçy (1) budem ponymat\ paru ( , )u x — upravlenye, optymy-
zyrugwee funkcyonal, y sootvetstvugwee emu reßenye dyfferencyal\noho
uravnenyq, kotoroe budem naz¥vat\ optymyzyrugwym potokom.
Lemma 1. Pust\ naçal\naq mera µ0 ymeet plotnost\ p0( )⋅ . Tohda dlq
© E. V. OSTAPENKO, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8 1099
1100 E. V. OSTAPENKO
lgboho t T∈[ , ]0 mera µt takΩe ymeet nekotorug plotnost\ pt( )⋅ .
Dokazatel\stvo. Yz neprer¥vnosty f y ′f2 sleduet [4] suwestvovanye
∂
∂
x t y
y
( , )
, pryçem
∂
∂
x t y
y
( , ) = e
t
t
R
f u t x t y z dz dt
0
2∫ ′∫
( ( ), ( , ), ) ( )µ
> 0.
Yz teorem¥ ob obratnoj funkcyy sleduet suwestvovanye obratnoho otobraΩe-
nyq x t− ⋅1( , ) y proyzvodnoj
∂
∂
−x t y
y
1( , )
.
Sohlasno opredelenyg
µt B( ) = µ0 0 0{ }( , )x x t x B∈ = p z dz
x t B
0
1
( )
( , )( )− ⋅
∫ =
= p x t y
x t y
y
dy
B
0
1
1
( )( , ) ( , )−
−∂
∂∫ .
Sledovatel\no,
p yt( ) = p x t y
x t y
y0
1
1
( )( , ) ( , )−
−∂
∂
.
Lemma dokazana.
Lemma 2. Pust\ µ0 — haussova mera. Tohda dlq lgboho t T∈[ , ]0 reße-
nye uravnenyq x t( , )⋅ prynadleΩyt klassu L2 0( )µ .
Dokazatel\stvo. Yz neprer¥vnosty f y ′f2 sleduet suwestvovanye
∂
∂
x t y
y
( , )
, pryçem
∂
∂
x t y
y
( , ) = e
t
t
R
f u t x t y z dz dt
0
2∫ ′∫
( ( ), ( , ), ) ( )µ
.
A tak kak ′f2 ohranyçena, to
∂
∂
x t y
y
( , )
ohranyçena y, sledovatel\no,
x t( , )⋅ @∈ L2 0( )µ .
Lemma dokazana.
Pust\ ℵ — mnoΩestvo vsex veroqtnostn¥x mer na borelevoj σ-alhebre
B ( R ) . Dlq dvux mer µ, ν@∈ ℵ opredelym mnoΩestvo C ( µ, ν ) vsex veroqtnost-
n¥x mer na borelevoj σ-alhebre B ( R
2
) , ymegwyx µ y ν svoymy proekcyqmy.
Opredelym rasstoqnye meΩdu meramy µ y ν sledugwym obrazom:
γ1 ( µ, ν ) =
inf ( , )
( , )κ µ ν
κ
∈ ∫∫ −
C
R
u du d
2
v v .
Lemma 3. Pust\ x t1( , )⋅ y x t2( , )⋅ — potoky, sootvetstvugwye razlyç-
n¥m upravlenyqm, µt
x1
y µt
x2
— mer¥, perenosym¥e πtymy potokamy. Tohda
γ µ µ( ),t
x
t
x1 2 ≤ x t x t1 2( , ) ( , )⋅ − ⋅ .
Dokazatel\stvo. Ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
ZADAÇA OPTYMAL|NOHO UPRAVLENYQ DLQ DETERMYNYROVANNOHO … 1101
γ µ µ( ),t
x
t
x1 2 =
inf ( , )
( , )κ µ µ
κ
t t
x
t
xC
R
tu du d
∈ ∫∫ −
1 2
2
v v =
=
inf ( , ) ( , ) ( , )
( , )κ µ µ
κ
∈ ∫∫ −
C
R
x t u x t du d
0 0 2
1 2 v v ≤
≤
R
x t u x t u du∫ −1 2 0( , ) ( , ) ( )µ ≤
≤
R
x t u x t u du∫ −
( )( , ) ( , ) ( )
/
1 2
2
0
1 2
µ = x t x t1 2( , ) ( , )⋅ − ⋅ .
Lemma dokazana.
Pust\ ψ : [ 0, T ] × R → R — nekotoraq funkcyq klassa C1
po pervoj pere-
mennoj y klassa L2 0( )µ po vtoroj. Na prostranstve L2 0( )µ rassmotrym ne-
prer¥vn¥j funkcyonal H ( u, x, ψ ) . Dlq kaΩdoho x t L( , ) ( )⋅ ∈ 2 0µ
H ( u, x, ψ ) = 〈 ⋅ ⋅ 〉ψ( , ), ( , )( )t F u xt =
=
R R
tt y f u t x t y z dz dy∫ ∫
ψ µ µ( , ) ( ), ( , ), ( ) ( )( ) 0 .
V kaçestve ψ( , )t ⋅ yspol\zuem reßenye soprqΩennoj system¥ [2, 3]
˙ ( , )ψ t ⋅ = – ′H u xx( , , )ψ ,
udovletvorqgwee uslovyg transversal\nosty
ψ( , )T ⋅ = – ′ ⋅ϕ ( ( , ))x T ,
hde pod ′Hx ponymaetsq proyzvodnaq Freße funkcyonala.
V pokoordynatnoj forme soprqΩennaq systema ymeet vyd
˙ ( , )ψ t x0 =
= – ψ µ ψ µ( , ) ( ), ( , ), ( ) ( , ) ( ), ( , ), ( , ) ( )( ) ( )t x f u t x t x z dz t y f u t x t y x t x dy
R
t
R
0 2 0 3 0 0∫ ∫′ + ′
,
ψ( , )T x0 = – ′ϕ ( ( , ))x T x0 , x R0 ∈ .
Lemma 4. Pust\ µ0 — haussova mera. Tohda dlq lgboho t T∈[ , ]0 reße-
nye soprqΩennoho uravnenyq ψ( , )t ⋅ prynadleΩyt klassu L2 0( )µ .
Dokazatel\stvo. Pravug çast\ soprqΩennoj system¥ moΩno rassmatry-
vat\ kak lynejn¥j neprer¥vn¥j operator. Oboznaçym çerez Ωt
T u( , )⋅ , t T∈[ , ]0 ,
matrycant soprqΩennoj system¥. Tohda, s uçetom uslovyq transversal\nosty
na pravom konce,
ψ( , )t ⋅ = Ωt
T u T( , ) ( , )⋅ ⋅ψ = – Ωt
T u x T( , ) ( ( , ))⋅ ′ ⋅ϕ .
Poskol\ku ′ϕ fynytna, to
R
t
T u y x T y dy∫ ′( )Ω ( , ) ( ( , )) ( )ϕ µ
2
0 =
=
R
t
T
tu x T z z dz∫ − ′( )Ω ( ), ( , ) ( ) ( )1 2
ϕ µ < + ∞ .
Lemma dokazana.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1102 E. V. OSTAPENKO
Dopolnytel\no predpoloΩym, çto suwestvuet konstanta M takaq, çto
′ + − ′ϕ ϕ( ) ( )x r x ≤ M r ,
f u h x r z f u x z( ) ( ), , , ,+ + − ≤ M h r( )+ ,
′ + + − ′f u h x r z f u x z2 2( ) ( ), , , , ≤ M h r( )+ ,
f u x z f u x z( , , ) ( , , )1 2− ≤ M z z( )1 2− ,
′ − ′f u x z f u x z2 1 2 2( , , ) ( , , ) ≤ M z z( )1 2− ,
′ + + − ′f u h x z r f u x z3 3( ) ( ), , , , ≤ M h r( )+ .
Tohda dlq haussovoj mer¥ µ0 spravedlyva sledugwaq teorema.
Teorema 1. Esly ( , )u x — reßenye zadaçy (1), to dlq lgboho t T∈[ , ]0
v¥polnqetsq uslovye maksymuma
H ( u, x, ψ ) =
max ( , , )
v
v
∈U
H x ψ .
Dokazatel\stvo. Prydadym upravlenyg u pryrawenye h takoe, çto u + h
qvlqetsq dopustym¥m, t. e. ( ( ) ( ))u t h t U+ ∈ , t T∈[ , ]0 . Pust\ x y x + r — po-
toky, sootvetstvugwye upravlenyqm u y u + h , µt
x , µt
x r+
— mer¥, perenosy-
m¥e potokamy x y x + r sootvetstvenno. Tohda
˙( , )r t x0 =
R
t
x rf u t h t x t x r t x z dz∫ + + +( )( ) ( ), ( , ) ( , ), ( )0 0 µ –
–
R
t
xf u t x t x z dz∫ ( )( ), ( , ), ( )0 µ ,
r x( , )0 0 = 0, x R0 ∈ .
Proyntehryruem ot 0 do t :
r t x( , )0 =
=
0
0 0 0
t
R
x r
R
xf u h x x r x z dz f u x x z dz d∫ ∫ ∫+ + −
+( ) ( )( ) ( ), ( , ) ( , ), ( ) ( ), ( , ), ( )τ τ τ τ µ τ τ µ ττ τ .
Oboznaçym çerez κτ meru na R2
, ymegwug svoymy proekcyqmy mer¥ µτ
x r+
y µτ
x
. Tohda
r t( , )⋅ ≤
≤
0
1 2 1 2
2
t
R
f u h x r z f u x z dz dz d∫ ∫ + ⋅ + ⋅ − ⋅
( ) ( )( ) ( ), ( , ) ( , ), ( ), ( , ), ( , )τ τ τ τ τ τ κ ττ ≤
=
0
1 1 1 2
2
t
R
f u h x r z f u x z dz dz∫ ∫ + ⋅ + ⋅ − ⋅
( ) ( )( ) ( ), ( , ) ( , ), ( ), ( , ), ( , )τ τ τ τ τ τ κτ +
+
R
f u x z f u x z dz dz d
2
1 2 1 2∫ ⋅ − ⋅
( ) ( )( ), ( , ), ( ), ( , ), ( , )τ τ τ τ κ ττ ≤
≤
0
1 2 1 2
2
t
R
M h r z z dz dz d∫ ∫+ ⋅ + −
( ) ( , ) ( , )τ τ κ ττ ≤
≤ M h d r d d
t t t
x x r
0 0 0
1∫ ∫ ∫+ ⋅ + ( )
+( ) ( , ) ,τ τ τ τ γ µ µ ττ τ ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
ZADAÇA OPTYMAL|NOHO UPRAVLENYQ DLQ DETERMYNYROVANNOHO … 1103
≤ M h d r d
t t
0 0
2∫ ∫+ ⋅
( ) ( , )τ τ τ τ .
Teper\, sohlasno lemme Hronuolla – Bellmana,
r t( , )⋅ ≤ C h t dt
T
1
0
∫ ( ) , t T∈[ , ]0 , hde C1 = Me MT2 .
Vvedem oboznaçenye
I u( ) =
R
Ts ds∫ ϕ µ( ) ( ) =
R
x T y dy∫ ϕ µ( ( , )) ( )0 = Φ( ( , ))x T ⋅ .
Tohda Φ moΩno rassmatryvat\ kak funkcyonal na L2( )µ , a pod ′Φ =
= ′Φ ( ( , ))x T x0 = ′ϕ ( ( , ))x T x0 , x R0 ∈ , budem ponymat\ eho proyzvodnug Freße.
Rassmotrym pryrawenye funkcyonala
∆ I = I u h I u( ) ( )+ − .
Tohda suwestvuet θ1 0 1∈[ , ] takoe, çto
∆ I = Φ Φ( ( , ) ( , )) ( ( , ))x T r T x T⋅ + ⋅ − ⋅ = ′ ⋅ + ⋅ ⋅Φ ( ( , ) ( , )); ( , )x T r T r Tθ1 =
= ′ ⋅ ⋅ + ′ ⋅ + ⋅ − ′ ⋅ ⋅Φ Φ Φ( ( , )); ( , ) ( ( , ) ( , )) ( ( , )); ( , )x T r T x T r T x T r Tθ1 .
Oboznaçym
R1 = ′ ⋅ + ⋅ − ′ ⋅ ⋅Φ Φ( ( , ) ( , )) ( ( , )); ( , )x T r T x T r Tθ1 .
Tohda
R1 ≤ r T x T r T x T( , ) ( ( , ) ( , )) ( ( , ))⋅ ′ ⋅ + ⋅ − ′ ⋅Φ Φθ1 ≤
≤ r T x T r T x T( , ) ( ( , ) ( , )) ( ( , ))⋅ ′ ⋅ + ⋅ − ′ ⋅ϕ θ ϕ1 ≤
≤ r T M r T( , ) ( , )⋅ ⋅θ1 ≤ M r T( , )⋅ 2
.
Poskol\ku ′ϕ ( ( , ))x T x0 = – ψ( , )T x0 , to, rassmatryvaq pervoe slahaemoe, polu-
çaem
′ ⋅ ⋅Φ ( ( , )); ( , )x T r T = ′ ⋅ ⋅ϕ ( ( , )); ( , )x T r T =
=
R
x T x r T x dx∫ ′ϕ µ( ( , )) ( , ) ( )0 0 0 0 =
= –
R
T x r T x dx∫ ψ( , ) ( , )0 0 0 = – ψ( , ); ( , )T r T⋅ ⋅ =
= –
0
T
d
dt
t r t dt∫ ⋅ ⋅ψ( , ); ( , ) = –
0
T
t r t t r t dt∫ ⋅ ⋅ + ⋅ ⋅[ ]ψ ψ( , ); ˙( , ) ˙ ( , ); ( , ) =
= –
0
T
R
t
x r
R
t
xt f u t h t x t r t z dz f u t x t z dz dt∫ ∫ ∫⋅ + ⋅ + ⋅( ) − ⋅( )+ψ µ µ( , ); ( ) ( ), ( , ) ( , ), ( ) ( ), ( , ), ( ) +
+
0
T
xH u x r t dt∫ ′ ⋅( , , ); ( , )ψ =
= –
0 0
T T
xH u h x r H u x dt H u x r t dt∫ ∫+ + −[ ] + ′ ⋅( , , ) ( , , ) ( , , ); ( , )ψ ψ ψ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1104 E. V. OSTAPENKO
Tak kak
H u h x r( , , )+ + ψ = H u h x H u h x r rx( , , ) ( , , );+ + ′ + +ψ θ ψ2 , θ2 0 1∈[ , ],
to
∆ I = –
0
1 2
T
H u h x H u x dt R R∫ + −[ ] + +( , , ) ( , , )ψ ψ ,
hde
R2 = –
0
2
T
x xH u h x r H u x r dt∫ ′ + + − ′( , , ) ( , , );θ ψ ψ .
Tohda
R2 ≤
0
T
r t∫ ⋅( , ) ×
× ′ + ⋅ + ⋅ ⋅( ) − ′ ⋅ ⋅( )H u t h t x t r t t H u t x t t dtx x( ) ( ), ( , ) ( , ), ( , ) ( ), ( , ), ( , )θ ψ ψ2 .
Rassmotrym normu raznosty proyzvodn¥x:
′ + + − ′H u h x r H u xx x( , , ) ( , , )θ ψ ψ2 =
= ψ θ µ θ( , ) ( ) ( ), ( , ) ( , ), ( )t f u t h t x t r t z dz
R
t
x r⋅ ′ + ⋅ + ⋅( )
∫ +
2 2 1 1
2 –
–
R
t
xf u t x t z dz∫ ′ ⋅( )
2 2 2( ), ( , ), ( )µ +
+
R
t y f u t h t x t y r t y x t r t∫ ′ + + ⋅ + ⋅( )(ψ θ θ( , ) ( ) ( ), ( , ) ( , ), ( , ) ( , )3 2 2 –
– ′ ⋅( ))f u t x t y x t dy3 0( ), ( , ), ( , ) ( )µ ≤
≤ ψ θ µ θ( , ) ( ) ( ), ( , ) ( , ), ( )t f u t h t x t r t z dz
R
t
x s⋅ ′ + ⋅ + ⋅( )∫ +
2 2 1 1
2 –
–
R
t
xf u t x t z dz∫ ′ ⋅( )2 2 2( ), ( , ), ( )µ +
+
R
t y f u t h t x t y r t y x t r t∫ ′ + + ⋅ + ⋅( )ψ θ θ( , ) ( ) ( ), ( , ) ( , ), ( , ) ( , )3 2 2 –
– ′ ⋅( )f u t x t y x t dy3 0( ), ( , ), ( , ) ( )µ .
Ocenym pervoe slahaemoe, predpoloΩyv, çto κt — mera na R2
, ymegwaq
svoymy proekcyqmy mer¥ µ θ
t
x r+ 2
y µt
x
:
ψ θ µ θ( , ) ( ) ( ), ( , ) ( , ), ( )t f u t h t x t r t z dz
R
t
x s⋅ ′ + ⋅ + ⋅( )∫ +
2 2 1 1
2 –
–
R
t
xf u t x t z dz∫ ′ ⋅( )2 2 2( ), ( , ), ( )µ ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
ZADAÇA OPTYMAL|NOHO UPRAVLENYQ DLQ DETERMYNYROVANNOHO … 1105
≤ ψ τ τ τ θ τ( , ) ( ) ( ), ( , ) ( , ),t f u h x r z
R
⋅ ′ + ⋅ + ⋅( )∫
2
2 2 1 –
– ′ ⋅( )f u x z dz dzt2 2 1 2( ), ( , ), ( , )τ τ κ ≤
≤ ψ τ τ τ θ τ( , ) ( ) ( ), ( , ) ( , ),t f u h x r z
R
⋅ ′ + ⋅ + ⋅( )
∫
2
2 2 1 –
– ′ ⋅( )f u x z dz dzt2 1 1 2( ), ( , ), ( , )τ τ κ +
+
R
tf u x z f u x z dz dz
2
2 1 2 2 1 2∫ ′ ⋅( ) − ′ ⋅( )
( ), ( , ), ( ), ( , ), ( , )τ τ τ τ κ ≤
≤ M t h t r t z z dz dz
R
tψ θ κ( , ) ( ) ( , ) ( , )⋅ + ⋅ + −
∫2 1 2 1 2
2
≤
≤ M t h t r t t
x
t
x rψ θ γ µ µ θ( , ) ( ) ( , ) ,⋅ + ⋅ + ( )( )+
2
2 ≤
≤ M t h t r t
t T
max ( , ) ( ) ( , )
0
2
≤ ≤
⋅ + ⋅( )ψ .
Ocenym vtoroe slahaemoe:
R
t y f u t h t x t y x t r t∫ ′ + ⋅ + ⋅( )ψ θ( , ) ( ) ( ), ( , ), ( , ) ( , )3 2 –
– ′ ⋅( )f u t x t y x t dy3 0( ), ( , ), ( , ) ( )µ ≤
≤ M t y h t r t dy
R
∫ + ⋅( )ψ θ µ( , ) ( ) ( , ) ( )2 0 ≤
≤ M t h t r t
t T
max ( , ) ( ) ( , )
0≤ ≤
⋅ + ⋅( )ψ .
Tohda
R2 ≤ M t r t h t r t dt
t T
T
max ( , ) ( , ) ( ) ( , )
0
0
2 3
≤ ≤
⋅ ⋅ + ⋅( )∫ψ ≤
≤ M t C C T h t dt
t T
T
max ( , ) ( )
0
1 1
2
0
2
2 3
≤ ≤
⋅ +( )
∫ψ .
Dlq pryrawenyq funkcyonala poluçaem predstavlenye
∆ I = –
0
T
H u h x H u x dt R∫ + −[ ] +( , , ) ( , , )ψ ψ
y
R ≤ C h t dt
T
0
2
∫
( ) ,
hde
C = M C t C C T
t T
1
2
0
1 1
22 3+ ⋅ +( )
≤ ≤
max ( , )ψ .
Pust\ u — optymal\noe upravlenye. V¥berem proyzvol\noe v ∈U , t ,
t T+ ∈ε [ , )0 , ε > 0, y rassmotrym pryrawenye vyda
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1106 E. V. OSTAPENKO
h ( t ) =
v( ) ( ), ,
, [ , ] [ , ).\
τ τ τ ε
τ ε
− ≤ < +
∈ +
u t t
T t t0 0
Tohda
∆ I = –
t
t
H x H u x d R
+
∫ −[ ] +
ε
ψ ψ τ( , , ) ( , , )v .
Vospol\zuemsq neravenstvom Koßy – Bunqkovskoho
R ≤ C h d
t
t+
∫
ε
τ τ( )
2
≤ ε τ τ
ε
C h d
t
t+
∫ ( ) 2 .
Oboznaçym
g ( τ ) = H x H u xv( ), ( , ), ( , ) ( ), ( , ), ( , )τ τ ψ τ τ τ ψ τ⋅ ⋅( ) − ⋅ ⋅( ).
Tohda
t
t
g d
+
∫
ε
τ τ( ) = ε θ εg t( )+ 3 , hde θ3 0 1∈[ , ].
Poskol\ku u — optymal\noe upravlenye, to ∆ I ≥ 0. Poπtomu
0 ≤ ∆ I = – ε θ εg t R( )+ +3 ≤ – ε θ ε ε τ τ
ε
g t C h d
t
t
( ) ( )+ +
+
∫3
2 .
Razdelym na ε :
g t( )+ θ ε3 ≤ C h d
t
t+
∫
ε
τ τ( ) 2 .
Pry ε → 0 poluçaem g ( t ) ≤ 0, t. e. H x H u x( , , ) ( , , )v ψ ψ− ≤ 0.
Teorema dokazana.
Prymer 1. Rassmotrym zadaçu optymal\noho upravlenyq bez vzaymodejst-
vyq:
R
Ts ds∫ ϕ µ( ) ( ) → inf,
ẋ = f ( u, x ) ,
x ( 0 ) = x0 , x0 ∈ R ,
xt = µ0
1� x t( , )⋅ − .
Pust\ µ0 ymeet plotnost\ p0 . Dlq takoj zadaçy
H u x( , , )ψ =
R
t y f u t x t y dy∫ ( )ψ µ( , ) ( ), ( , ) ( )0 ,
hde ψ( , )t ⋅ — reßenye soprqΩennoj system¥
˙ ( , )ψ t x0 = – ψ( , ) ( ), ( , )t x f u t x t x0 2 0′( ), x0 ∈ R ,
udovletvorqgwee uslovyg transversal\nosty
ψ( , )T x = – ′ϕ ( )( , )x T x0 , x0 ∈ R ,
a ymenno
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
ZADAÇA OPTYMAL|NOHO UPRAVLENYQ DLQ DETERMYNYROVANNOHO … 1107
ψ( , )t x0 = – ′ ∫ ′( )
ϕ
τ τ τ
( )( , )
( ), ( , )
x T x e
f u x x d
t
T
0
2 0 ,
H u x( , , )ψ = –
R
f u x y d
x T y e f u t x t y dyt
T
∫ ′ ∫ ( )
′( )
ϕ µ
τ τ τ
( )( , ) ( ), ( , ) ( )
( ), ( , )2
0 .
Poluçaem uravnenye dlq opredelenyq optymal\noho upravlenyq u :
R
f u x y d
x T y e t
T
∫ ′ ∫ ′( )
ϕ
τ τ τ
( )( , )
( ), ( , )2 ×
× ′( ) + ′′ ( )
∫f u t x t y f u x y d dy
t
T
1 1 2 0( ), ( , ) ( ), ( , ) ( ), τ τ τ µ = 0.
Obratymsq k voprosu suwestvovanyq reßenyq zadaçy (1). Sledugwyj pry-
mer pokaz¥vaet, çto pry neohranyçennom upravlenyy daΩe v prostejßem ly-
nejnom sluçae reßenye moΩet ne suwestvovat\.
Prymer 2. Rassmotrym sluçaj naçal\noj mer¥, sosredotoçennoj v toçke y .
Pust\ dynamyka tqΩeloj çastyc¥ opys¥vaetsq uravnenyem ẋ = ux , a ϕ — lg-
baq funkcyq klassa C R1( ), ymegwaq strohyj mynymum v toçke 0. Tohda v za-
daçe optymal\noho upravlenyq
I = ϕ( ( ))x T → min,
ẋ = ux ,
x ( 0 ) = y , u ∈ R ,
mynymum ne dostyhaetsq, tak kak x ( T ) = yeuT > 0 dlq lgboho u ∈ R , no
x ( T ) = yeuT → 0, u → – ∞ .
Lemma 5. Dlq lgboho x0 ∈ R mnoΩestvo traektoryj { }( , )x x⋅ 0 , soot-
vetstvugwyx razlyçn¥m upravlenyqm y naçynagwyxsq v x0
, predkompaktno v
C ( [ 0, T ] ) .
Dokazatel\stvo. Poskol\ku funkcyq f ohranyçena, suwestvuet C > 0
takoe, çto
f u x z( , , ) ≤ C, u, x, z ∈ R .
MnoΩestvo { }( , )x x⋅ 0 ohranyçeno, tak kak
x t x( , )0 = x f u s x s x z dz ds
t
R
s0
0
0+ ( )∫ ∫ ( ), ( , ), ( )µ ≤ x CT0 + ,
y ravnostepenno neprer¥vno, tak kak
x t x x t x( , ) ( , )1 0 2 0− ≤
t
t
R
sf u s x s x z dz ds
1
2
0∫ ∫ ( )( ), ( , ), ( )µ ≤ C t t1 2− .
Sledovatel\no, sohlasno teoreme Askoly – Arcela, mnoΩestvo { }( , )x x⋅ 0 pred-
kompaktno v C T([ , ])0 .
Lemma dokazana.
Dlq proyzvol\noho potoka x opredelym operator, dejstvugwyj na kusoç-
no-neprer¥vn¥x funkcyqx na [ , ]0 T : dlq lgboho u KC T∈ ([ , ])0
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
1108 E. V. OSTAPENKO
G u x t( , )( ) = f u t x t x x t z dz
R
( ( ), ( , ), ( , )) ( )0 0µ∫ .
Rasßyrym mnoΩestvo prav¥x çastej uravnenyq dynamyky vsevozmoΩn¥my sla-
b¥my predelamy operatorov takoho vyda.
Zameçanye. V sluçae, kohda ′f u x z1( , , ) ≥ ε > 0, u, x, z ∈ R , takym slab¥m
predelam sootvetstvugt slab¥e predel¥ nekotor¥x posledovatel\nostej up-
ravlenyj.
Teorema 2. Suwestvuet optymyzyrugwyj potok zadaçy (1).
Dokazatel\stvo. Pust\ posledovatel\nost\ { },u nn ≥ 1 takova, çto
I un( ) → inf ( )
([ , ])u KC T
I u
∈ 0
, n → ∞ .
KaΩdomu upravlenyg un sootvetstvuet potok x tn( , )⋅ , t T∈[ , ]0 .
Dlq lgboho racyonal\noho x0 yz { }( , ),x x nn ⋅ ≥0 1 moΩno v¥brat\ ravno-
merno sxodqwugsq na [ 0, T ] podposledovatel\nost\ { }( , ),x x knk
⋅ ≥0 1 takug,
çto x x x xnk
( , ) ˜( , )⋅ ⇒ ⋅0 0 , k → ∞ , na [ , ]0 T .
Sohlasno dyahonal\nomu metodu Kantora, moΩno v¥brat\ takug podposle-
dovatel\nost\ { }( , ),x x lnl
⋅ ≥0 1 , çto dlq lgboho racyonal\noho x0 v¥polneno
x x x xnl
( , ) ˜( , )⋅ ⇒ ⋅0 0 , l → ∞ , na [ , ]0 T . Pust\ k = nl . Tohda dlq lgboho A > 0
v¥polneno x xk ⇒ ˜ , k → ∞ , na [ , ] [ , ]0 T A A× − y
˜( , )x t x0 = x f u s x s x x s z dz ds
k
t
k k k
R
0
0
0 0+
→∞ ∫ ∫lim ( ( ), ( , ), ( , )) ( )µ .
Ocenym raznost\ G u x sk k( , )( ) y G u x sk( , ˜)( ):
G u x s G u x sk k k( , )( ) ( , ˜)( )− ≤
≤ f u s x s x x s z f u s x s x x s z dzk k k k
R
( ( ), ( , ), ( , )) ( ( ), ˜( , ), ˜( , )) ( )0 0 0−∫ µ ≤
≤
R
k k k k kf u s x s x x s z f u s x s x x s z∫ −[ ( ( ), ( , ), ( , )) ( ( ), ˜( , ), ( , ))0 0 +
+ f u s x s x x s z f u s x s x x s z dzk k k( ( ), ˜( , ), ( , )) ( ( ), ˜( , ), ˜( , )) ( )0 0 0− ]µ ≤
≤ M x s x x s x dz M x s z x s z dzk
R
k
R
( , ) ˜( , ) ( ) ( , ) ˜( , ) ( )0 0 0 0− + −∫ ∫µ µ =
= M x s x x s x x s z x s z dz x s z x s z dzk k
z A
k
z A
( , ) ˜( , ) ( , ) ˜( , ) ( ) ( , ) ˜( , ) ( )0 0 0 0− + − + −
≤ >
∫ ∫µ µ ≤
≤ M x s x x s x A A x s z x s zk
z A
k( , ) ˜( , ) ([ , ]) sup ( , ) ˜( , )0 0 0− + − −
≤
µ +
+ x s z x s z dzk
z A
( , ) ˜( , ) ( )−
>
∫ µ0 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
ZADAÇA OPTYMAL|NOHO UPRAVLENYQ DLQ DETERMYNYROVANNOHO … 1109
Poskol\ku
∂
∂
x t y
y
( , ) = e
t
t
R
f u t x t y z dz dt
0
2∫ ′∫
( ( ), ( , ), ) ( )µ
, suwestvuet C > 0 takoe, çto
∂
∂
x s y
y
k ( , )
≤ C, k ≥ 1. Sledovatel\no, x s zk ( , ) ≤ C z x sk+ ( , )0 , k ≥ 1. Pe-
rejdem k predelu pry k → ∞ : ˜( , )x s z ≤ C z x s+ ˜( , )0 .
Pust\ d = sup ( , )
, [ , ]k s T
kx s
≥ ∈1 0
0 < ∞ . Tohda
x s zk ( , ) ≤ C z d+ , k ≥ 1,
˜( , )x s z ≤ C z d+ .
V πtom sluçae poluçaem
G u x s G u x sk k k( , )( ) ( , ˜)( )− ≤ M x s x x s xk ( , ) ˜( , )0 0−
+
+ µ µ0 02([ , ]) sup ( , ) ˜( , ) ( )( )− − + +
≤ >
∫A A x s z x s z C z d dz
z A
k
z A
.
Tohda dlq lgboho ε > 0 suwestvugt A > 0 takoe, çto
( ) ( )C z d dz
z A
+
>
∫ µ0 < ε
6M
,
y k ≥ 1 takoe, çto
sup ( , ) ˜( , )
z A
kx s z x s z
≤
− < ε
µ3 0M A A([ , ])−
,
x s x x s xk ( , ) ˜( , )0 0− < ε
3M
,
t. e. G u x s G u x sk k k( , )( ) ( , ˜)( )− ≤ ε . Sledovatel\no,
˜( , )x t x0 = x f u s x s x x s z dz ds
k
t
k
R
0
0
0 0+
→∞ ∫ ∫lim ( ( ), ˜( , ), ˜( , )) ( )µ
y qvlqetsq optymyzyrugwym potokom dlq zadaçy (1).
Teorema dokazana.
1. Dorogovtsev A. A. Stochastic flows with interactions and measure-valued processes // Int. J. Math.
and Math. Sci. – 2003. – 63. – P. 3963 – 3977.
2. Yoffe A. D., Tyxomyrov V. M. Teoryq πkstremal\n¥x zadaç. – M.: Nauka, 1974. – 480 s.
3. Alekseev V. M., Tyxomyrov V. M., Fomyn S. V. Optymal\noe upravlenye. – M.: Nauka, 1979. –
432 s.
4. Eruhyn N. P., Ítokalo Y. Z. Kurs ob¥knovenn¥x dyfferencyal\n¥x uravnenyj. – Kyev:
Vywa ßk., 1974. – 472 s.
Poluçeno 04.07.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 8
|
| id | umjimathkievua-article-3227 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:35Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2a/86024f9c4105d4bd8ef786f6da75212a.pdf |
| spelling | umjimathkievua-article-32272020-03-18T19:48:39Z Problem of optimal control for a determinate equation with interaction Задача оптимального управления для детерминированного уравнения с взаимодействием Ostapenko, E. V. Остапенко, Е. В. Остапенко, Е. В. The problem of optimal control of differential equations with interaction is consider. It is proved that the optimal control satisfies the maximum principle and there exists the generalized optimal control. It is shown that, in the considered problem, new technical aspects arise as compared with the usual problem of optimal control. Розглядається задача оптимального керування диференціальних рівнянь з взаємодією. Доведено, що оптимальне керування задовольняє принцип максимуму та існує узагальнене оптимальне керування. В задачі, що розглядається, виникають нові технічні моменти у порівнянні зі звичайною задачею оптимального керування. Institute of Mathematics, NAS of Ukraine 2008-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3227 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 8 (2008); 1099–1109 Український математичний журнал; Том 60 № 8 (2008); 1099–1109 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3227/3200 https://umj.imath.kiev.ua/index.php/umj/article/view/3227/3201 Copyright (c) 2008 Ostapenko E. V. |
| spellingShingle | Ostapenko, E. V. Остапенко, Е. В. Остапенко, Е. В. Problem of optimal control for a determinate equation with interaction |
| title | Problem of optimal control for a determinate equation with interaction |
| title_alt | Задача оптимального управления для детерминированного уравнения с взаимодействием |
| title_full | Problem of optimal control for a determinate equation with interaction |
| title_fullStr | Problem of optimal control for a determinate equation with interaction |
| title_full_unstemmed | Problem of optimal control for a determinate equation with interaction |
| title_short | Problem of optimal control for a determinate equation with interaction |
| title_sort | problem of optimal control for a determinate equation with interaction |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3227 |
| work_keys_str_mv | AT ostapenkoev problemofoptimalcontrolforadeterminateequationwithinteraction AT ostapenkoev problemofoptimalcontrolforadeterminateequationwithinteraction AT ostapenkoev problemofoptimalcontrolforadeterminateequationwithinteraction AT ostapenkoev zadačaoptimalʹnogoupravleniâdlâdeterminirovannogouravneniâsvzaimodejstviem AT ostapenkoev zadačaoptimalʹnogoupravleniâdlâdeterminirovannogouravneniâsvzaimodejstviem AT ostapenkoev zadačaoptimalʹnogoupravleniâdlâdeterminirovannogouravneniâsvzaimodejstviem |