Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters
We study the problem of high-speed operation for linear control systems with fuzzy right-hand sides. For this problem, we introduce the notion of optimal solution and establish necessary and sufficient conditions of optimality in the form of the maximum principle.
Gespeichert in:
| Datum: | 2009 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2009
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3026 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509043259342848 |
|---|---|
| author | Molchanyuk, I. V. Plotnikov, A. V. Молчанюк, И. В. Плотников, А. В. Молчанюк, И. В. Плотников, А. В. |
| author_facet | Molchanyuk, I. V. Plotnikov, A. V. Молчанюк, И. В. Плотников, А. В. Молчанюк, И. В. Плотников, А. В. |
| author_sort | Molchanyuk, I. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:43:35Z |
| description | We study the problem of high-speed operation for linear control systems with fuzzy right-hand sides. For this problem, we introduce the notion of optimal solution and establish necessary and sufficient conditions of optimality in the form of the maximum principle. |
| first_indexed | 2026-03-24T02:34:49Z |
| format | Article |
| fulltext |
UDK 517.911.5 + 517.977.5
Y. V. Molçangk, A. V. Plotnykov (Odes. akad. str-va y arxytektur¥)
NEOBXODYMÁE Y DOSTATOÇNÁE USLOVYQ
OPTYMAL|NOSTY V ZADAÇAX UPRAVLENYQ
S NEÇETKYM PARAMETROM
A problem of the high-speed operation is considered for linear control systems with a fuzzy right-hand
side. For this problem, a notion of optimal solution is introduced and necessary and sufficient conditions
of the optimality are established in the form of the maximum principle.
Rozhlqnuto zadaçu ßvydkodi] dlq linijnyx system upravlinnq z neçitkog pravog çastynog.
Dlq ci[] zadaçi vvedeno ponqttq optymal\noho rozv�qzku i vstanovleno neobxidni ta dostatni
umovy optymal\nosti u formi pryncypu maksymumu.
1. Vvedenye. Ponqtye neçetkoho mnoΩestva b¥lo vvedeno v rabote [1]. V ra-
bote [2] vperv¥e rassmatryvalos\ neçetkoe dyfferencyal\noe uravnenye, koto-
roe v dal\nejßem yssledovalos\ v rabotax [3 – 11], a v rabotax [12 – 14] yzuça-
lys\ dyfferencyal\n¥e vklgçenyq s neçetkoj pravoj çast\g, kotor¥e zatem
rassmatryvalys\ v [15, 16].
V dannoj rabote prodolΩen¥ yssledovanyq, naçat¥e v [17]. V nej rassmat-
ryvaetsq odna yz zadaç teoryy optymal\noho upravlenyq � zadaça b¥strodej-
stvyq, kotoraq sformulyrovana dlq upravlqem¥x dyfferencyal\n¥x vklgçe-
nyj s neçetkoj pravoj çast\g. Poluçen¥ neobxodym¥e y dostatoçn¥e uslovyq
optymal\nosty v forme pryncypa maksymuma, kotor¥e obobwagt rezul\tat¥,
poluçenn¥e v [18] dlq ob¥çn¥x upravlqem¥x dyfferencyal\n¥x vklgçenyj.
2. Osnovn¥e opredelenyq y oboznaçenyq. Pust\ Comp( )Rn Conv( )( )Rn
� prostranstvo nepust¥x (v¥pukl¥x) kompaktn¥x podmnoΩestv evklydova
prostranstva Rn s metrykoj Xausdorfa
h A B( , ) = min ( ), ( )r A B S B A Sr r≥ ⊂ + ⊂ +{ }0 0 0 ,
hde A, B � Comp( )Rn (yly Conv( )Rn ), S ar( ) � ßar v Rn radyusa r s cent-
rom v toçke a Rn∈ .
Pust\ povedenye upravlqemoj system¥ opys¥vaetsq lynejn¥m dyfferen-
cyal\n¥m uravnenyem
ẋ = A t x( ) + B t u( ) + C t( )v, x(0) = x0, (1)
hde x Rn∈ � fazov¥j vektor; A t( ), B t( ), C t( ) � matryc¥ sootvetstvugwyx
razmernostej (n × n), (n × m), (n × k); u t( ) ∈ U t( ) � vektor upravlenyq; U( )⋅ :
R+
1 → Conv( )Rm � mnohoznaçnoe otobraΩenye; v( )t ∈ V � neçetkoe vneßnee
vozdejstvye (pomexa); V � neçetkoe mnoΩestvo s xarakterystyçeskoj funk-
cyej µ( )x , µ( )⋅ : Rk → 0 1,[ ], kotor¥e udovletvorqgt sledugwym uslovyqm.
PredpoloΩenye 1. 1. Matryc¥ A t( ), B t( ), C t( ) yzmerym¥ na R+
1 .
2. Suwestvugt summyruem¥e funkcyy a t( ) > 0, b t( ) > 0, c t( ) > 0 takye,
çto A t( ) ≤ a t( ), B t( ) ≤ b t( ), C t( ) ≤ c t( ) dlq poçty vsex t ∈ R+
1 .
3. Mnohoznaçnoe otobraΩenye U t( ) yzmerymo na R+
1 .
4. Suwestvuet summyruemaq funkcyq g t( ) > 0 takaq, çto h U t( )( , 0{ }) ≤
≤ g t( ) dlq poçty vsex t ∈ R+
1 .
5. Xarakterystyçeskaq funkcyq µ( )⋅ : Rk → 0 1,[ ] udovletvorqet uslo-
vyqm:
a) funkcyq µ( )⋅ modal\naq, t . e. suwestvuet xotq b¥ odno y Rk
0 ∈ t a -
koe, çto µ( )y0 = 1;
© Y. V. MOLÇANGK, A. V. PLOTNYKOV, 2009
384 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
NEOBXODYMÁE Y DOSTATOÇNÁE USLOVYQ OPTYMAL|NOSTY V ZADAÇAX … 385
b) funkcyq µ( )y neprer¥vna po y na Rk ;
v) dlq lgboho ε > 0 y y ∈ y Rk∈{ µ( )y ∈ ( , )0 1 } suwestvugt y1, y2 ∈
∈ Rk takye, çto y y– 1 < ε, y y– 2 < ε y µ( )y1 < µ( )y < µ( )y2 ;
h) mnoΩestvo V[ ]0 = cl y{ µ( )y > 0} kompaktno.
Opredelenye 1. MnoΩestvo vsex yzmerym¥x selektorov U( )⋅ n a 0, ∞[ ) bu-
dem naz¥vat\ mnoΩestvom dopustym¥x upravlenyj y oboznaçat\ U.
Vvedem neçetkoe upravlqemoe dyfferencyal\noe vklgçenye
˙ ( )x A t x∈ + B t u( ) + C t V( ) , x(0) = x0, (2)
kotoroe poluçaetsq yz system¥ (1) v rezul\tate zamen¥ parametra v( )t na ne-
çetkoe mnoΩestvo V.
Oboznaçym çerez X u( ) neçetkyj puçok traektoryj system¥ (2), sootvet-
stvugwyx dopustymomu upravlenyg u( )⋅ , a çerez X t u( , ) seçenye puçka X u( )
v moment vremeny t > 0, kotoroe qvlqetsq nekotor¥m neçetkym mnoΩestvom s
xarakterystyçeskoj funkcyej χ(x , t, u).
Zameçanye 1. Yz [17] yzvestno, çto pry v¥polnenyy uslovyj predpoloΩe-
nyq 1 neçetkoe mnohoznaçnoe otobraΩenye X u( , )⋅ ymeet vyd
X t u( , ) = Φ( )t x0 + Φ Φ( ) ( ) ( ) ( )–t s B s u s ds
t
0
1∫ + Φ Φ( ) ( ) ( )–t s C s V ds
t
0
1∫ (3)
y qvlqetsq neçetko v¥pukl¥m y neçetko kompaktn¥m v kaΩd¥j moment vreme-
ny t > 0 y absolgtno neprer¥vn¥m na R+
1 , xarakterystyçeskaq funkcyq χ(x ,
t, u) udovletvorqet uslovyg 5 predpoloΩenyq 1 po x dlq vsex t ≥ 0 y vsex do-
pustym¥x upravlenyj u( )⋅ , Φ( )t � matryca Koßy dyfferencyal\noho urav-
nenyq ẋ = A t x( ) , a yntehral v poslednem slahaemom ponymaetsq v sm¥sle [19].
Opredelenye 2. Neçetkym mnoΩestvom dostyΩymosty Y T( ) system¥ (2)
nazovem mnoΩestvo vsex neçetkyx mnoΩestv X(T, u), t. e.
Y T( ) = X T u u U( , ) ( )⋅ ∈{ }.
Zameçanye 2. Yz [17] yzvestno, çto pry v¥polnenyy uslovyj predpoloΩe-
nyq 1 neçetkoe mnoΩestvo dostyΩymosty Y T( ) qvlqetsq v¥pukl¥m y kom-
paktn¥m mnoΩestvom neçetkyx mnoΩestv.
3. Zadaçy b¥strodejstvyq neçetkymy puçkamy traektoryj. Rassmotrym
sledugwug zadaçu optymal\noho upravlenyq: opredelyt\ mynymal\nug vely-
çynu T > 0 y dopustymoe upravlenye u∗ ⋅( ) ∈ U takye, çto dlq sootvetstvug-
weho seçenyq puçka X T u( , )∗ system¥ (2) v¥polnqetsq odno yz uslovyj
X T u( , )∗ I Sk ≠ ∅, (4)
X T u( , )∗ � Sk , (5)
X T u( , )∗ � Sk , (6)
hde Sk � neçetkoe celevoe mnoΩestvo s xarakterystyçeskoj funkcyej ς( )x ,
udovletvorqgwej uslovyg 5 predpoloΩenyq 1.
Opredelenye 3. Budem hovoryt\, çto para u∗ ⋅( ( ) , X(T, u∗ )) udovletvorq-
et uslovyg maksymuma na otrezke 0, T[ ], esly suwestvuet dlq vsex t ∈ 0, T[ ]
vektornaq funkcyq ψ( )⋅ � reßenye soprqΩennoj system¥
ψ̇ = – ( ) ( )A t tT ψ , ψ( ) ( )0 01∈S , (7)
y v¥polnen¥ uslovyq:
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
386 Y. V. MOLÇANGK, A. V. PLOTNYKOV
1) uslovye maksymuma: dlq poçty vsex t ∈ 0, T[ ]
u t t∗( )( ), ( )ψ = C B t U t t( ) ( ), ( )ψ( ); (8)
2) uslovye transversal\nosty na Sk :
a) dlq sluçaq (4)
C X T u t( , ) , ( )∗[ ]( )1
ψ = − [ ]( )C S Tk
1, – ( )ψ ; (9)
b) dlq sluçaq (5): dlq lgboho α ∈ 0 1,[ ]
C X T u t( , ) , ( )∗[ ]( )α
ψ ≤ C S Tk[ ]( )α ψ, ( )
y suwestvuet xotq b¥ odno ′α ∈ 0 1,[ ] takoe, çto
C X T u t( , ) , ( )∗[ ]( )α
ψ = C S Tk[ ]( )α ψ, ( ) ;
v) dlq sluçaq (6): dlq lgboho α ∈ 0 1,[ ]
C X T u t( , ) , – ( )∗[ ]( )α
ψ ≤ C S Tk[ ]( )α ψ, – ( )
y suwestvuet xotq b¥ odno ′α ∈ 0 1,[ ] takoe, çto
C X T u t( , ) , – ( )∗[ ]( )α
ψ = C S Tk[ ]( )α ψ, – ( ) ,
hde G[ ]α � α-srezka neçetkoho mnoΩestva [1], t. e.
G[ ]α =
y y
y y
β α α
β α
( ) , , ,
( ) , ,
≥{ } ∈( ]
>{ } =
0 1
0 0cl
β( )⋅ � xarakterystyçeskaq funkcyq neçetkoho mnoΩestva G, C X( , )ψ =
= max( , )
x X
x
∈
ψ � opornaq funkcyq mnoΩestva X ∈ Comp ( )Rn po vektoru ψ ∈
∈ Rn [20].
Oboznaçym
Z T( ) = Φ Φ( ) ( ) ( )–T s C s V ds
T
0
1∫ = Φ Φ( ) ( ) ( )–T s C s dsV
T
0
1∫ .
Teorema 1 (neobxodymoe uslovye optymal\nosty). Pust\ spravedlyv¥ uslo-
vyq predpoloΩenyq 1, u∗ ⋅( ) � optymal\noe upravlenye, a X u( , )⋅ ∗ � soot-
vetstvugwyj emu puçok system¥ (2).
Tohda para u∗ ⋅( ( ) , X( , )⋅ )∗u udovletvorqet sledugwym uslovyqm:
1) pryncypu maksymuma na 0, T[ ];
2) suwestvuet mnoΩestvo Κ ∈ Comp ( )Rn takoe, çto:
dlq sluçaq (5)
Κ = k R Z T k Sn
k∈ + ⊂{ }( ) , (10)
dlq sluçaq (6)
Κ = k R S Z T kn
k∈ ⊂ +{ }( ) (11)
y upravlenye u∗ ⋅( ) udovletvorqet uslovyg
Φ( )T x0 + Φ Φ Κ( ) ( ) ( ) ( )–T s B s u s ds
T
0
1∫ ∗ ∈ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
NEOBXODYMÁE Y DOSTATOÇNÁE USLOVYQ OPTYMAL|NOSTY V ZADAÇAX … 387
Dokazatel\stvo. Rassmotrym podrobno zadaçu (2), (4).
Dlq dokazatel\stva prymenym sxemu rassuΩdenyj, analohyçnug yspol\zue-
moj pry dokazatel\stve pryncypa maksymuma dlq lynejn¥x dyfferencyal\-
n¥x uravnenyj [20].
Pust\ u∗ ⋅( ) � optymal\noe upravlenye, X u( , )⋅ ∗ � sootvetstvugwyj emu
puçok system¥ (1), kotor¥j udovletvorqet hranyçn¥m uslovyqm:
1) X T u( , )∗ ∈ Y T( );
2) X T u( , )∗ I Sk ≠ ∅.
Oçevydno, çto yz predpoloΩenyq 1 uslovye X T u( , )∗[ ]1 I Sk[ ]1 ≠ ∅ haranty-
ruet v¥polnenye uslovyq 2. ∏to oznaçaet, çto dlq lgboho ψ ∈ S1 0( )
max ( , )
( )X Y T
C X
∈[ ]1
ψ ≥ – , –C Sk[ ]( )1 ψ . (12)
Sledovatel\no,
p ≡ max min ( , )
( ) ( )X Y T S
C X
∈[ ] ∈1
1 0ψ
ψ + C Sk[ ]( )1, –ψ ≥ 0. (13)
Dejstvytel\no, esly p < 0, to ne suwestvuet takoho mnoΩestva X ∈ Y T( )[ ]1,
çtob¥ dlq vsex ψ ∈ S1 0( ) v¥polnqlos\ neravenstvo
C X( , )ψ + C Sk[ ]( )1, –ψ ≥ 0,
çto protyvoreçyt neravenstvu (12).
PokaΩem, çto suwestvugt ψ = ψ ∈ S1 0( ) y X = X T u( , )∗[ ]1, pry kotor¥x
p < 0. Yz sootnoßenyq X T u( , )∗[ ]1 I Sk[ ]1 ≠ ∅ sleduet, çto dlq lgboho ψ ∈
∈ S1 0( )
q T( , )ψ = C X T u( , ) ,∗[ ]( )1 ψ + C Sk[ ]( )1, –ψ ≥ 0. (14)
Poskol\ku X T u( , )∗[ ]1 neprer¥vno zavysyt ot vremeny [17], funkcyq q T( , )ψ
neprer¥vna po ψ y T.
Esly predpoloΩyt\, çto q T( , )ψ > 0 dlq lgboho ψ ∈ S1 0( ), a sledovatel\-
no, y p > 0, to poluçym
q T0( ) = min ( , )
( )ψ
ψ
∈S
q T
1 0
≥ γ > 0,
hde funkcyq q T0( ) neprer¥vna. Sledovatel\no, najdetsq τ < T takoe, çto
q0( )τ ≥ 0. ∏to oznaçaet, çto dlq lgboho ψ ∈ S1 0( )
C X u( , ) ,τ ψ∗[ ]( )1 + C Sk[ ]( )1, –ψ ≥ 0,
t. e. X T u( , )∗[ ]1 I Sk[ ]1 ≠ ∅. ∏to protyvoreçyt optymal\nosty vremeny T.
Esly predpoloΩyt\, çto p > 0 y maksymum v (13) dostyhaetsq pry X ≠
≠ X T u( , )∗[ ]1, to analohyçno pred¥duwemu pryxodym k protyvoreçyg.
Takym obrazom, suwestvuet vektor ψ ∈ S1 0( ) takoj, çto
C X T u( , ) ,∗[ ]( )1 ψ = max ( , )
( )X Y T
C X
∈[ ]1
ψ , (15)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
388 Y. V. MOLÇANGK, A. V. PLOTNYKOV
C X T u( , ) ,∗[ ]( )1 ψ = – , –C Sk[ ]( )1 ψ . (16)
Yz sootnoßenyj (3) y (15) poluçaem
0
1
T
T s B s u s ds∫ ∗
Φ Φ( ) ( ) ( ) ( ) ,– ψ = max ( ) ( ) ( ) ( ) ,
( )
–
u U
T
T s B s u s ds
⋅ ∈ ∫
0
1Φ Φ ψ . (17)
Yz (17), yspol\zuq svojstva yntehrala y skalqrnoj funkcyy, poluçaem
Φ Φ( ) ( ) ( ) ( ),–T t B t u t1 ∗( )ψ = max ( ) ( ) ( ) ( ),
( )
–
u U
T t B t u t
⋅ ∈
( )Φ Φ 1 ψ (18)
dlq poçty vsex t ∈ 0, T[ ].
Uçyt¥vaq, çto ψ( )t = Φ Φ Φ Φ( ) ( ) ( ) ( )– –T t T t
T T1 1( ) ( )ψ ψ qvlqetsq reße-
nyem soprqΩennoj system¥ ψ̇ = – ( )A tT ψ( )t s naçal\n¥m uslovyem ψ (0) ∈
∈ S1 0( ) dlq poçty vsex t ∈ 0, T[ ], sootnoßenyq (16) y (18) moΩno zapysat\ v
forme sootvetstvugwyx sootnoßenyj pryncypa maksymuma.
Sluçay (5) y (6) dokaz¥vagtsq analohyçno s nebol\ßymy yzmenenyqmy uslo-
vyq (12):
dlq sluçaq (5): dlq lgboho α ∈ 0 1,[ ]
max – ( , – )
( )X Y T
C X
∈[ ]α
ψ + C Sk[ ]( )α ψ, – ≥ 0
y suwestvuet ′α ∈ 0 1,[ ] takoe, çto
max – ( , – )
( )X Y T
C X
∈[ ] ′α
ψ + C Sk[ ]( )′α ψ, – = 0;
dlq sluçaq (6): dlq lgboho α ∈ 0 1,[ ]
max ( , )
( )X Y T
C X
∈[ ]α
ψ + C Sk[ ]( )α ψ, ≥ 0
y suwestvuet ′α ∈ 0 1,[ ] takoe, çto
max ( , )
( )X Y T
C X
∈[ ] ′α
ψ + C Sk[ ]( )′α ψ, = 0.
Tem sam¥m uslovye 1 dokazano. Netrudno vydet\, çto uslovye 2 � harantyq to-
ho, çto X T u( , )∗ � Sk yly X T u( , )∗ � Sk .
Teorema dokazana.
Teorema 2 (dostatoçnoe uslovye optymal\nosty). Pust\ spravedlyv¥ uslo-
vyq predpoloΩenyq 1, u∗ ⋅( ) � dopustymoe upravlenye y para u∗ ⋅( ( ) , X T u( , )∗ )
udovletvorqet uslovyqm teorem¥ 1 na 0, T[ ]. Krome toho, pust\ X t u( , )∗
udovletvorqet usylennomu uslovyg transversal\nosty na mnoΩestve Sk s
funkcyej ψ( )⋅ , t. e. dlq vsex t ∈ 0, T[ ) v¥polnqgtsq neravenstva:
a) dlq sluçaq (4)
C X t u t( , ) , ( )∗[ ]( )1 ψ < – , – ( )C S tk[ ]( )1 ψ ; (19)
b) dlq sluçaq (5)
C X t u t( , ) , ( )∗[ ]( )α
ψ > C S tk[ ]( )α ψ, ( ) (20)
dlq lgboho α ∈ 0 1,[ ];
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
NEOBXODYMÁE Y DOSTATOÇNÁE USLOVYQ OPTYMAL|NOSTY V ZADAÇAX … 389
v) dlq sluçaq (6)
C X t u t( , ) , ( )∗[ ]( )α
ψ < C S tk[ ]( )α ψ, ( ) (21)
dlq lgboho α ∈ 0 1,[ ].
Tohda u∗ ⋅( ) � optymal\noe upravlenye.
Dokazatel\stvo provedem dlq sluçaev (2) y (4) (v sluçaqx (5) y (6) dokaza-
tel\stvo provodytsq analohyçno s nebol\ßymy yzmenenyqmy). V¥polnymost\
neobxodym¥x uslovyj optymal\nosty oçevydna. DokaΩem dopolnytel\noe us-
lovye optymal\nosty.
Zametym, çto esly uslovye (19) v¥polnqetsq dlq nekotoroho t ∈ 0, T[ ) , to
uslovye X t u( , )∗ I Sk ≠ ∅ ne ymeet mesta.
Voz\mem proyzvol\noe dopustymoe upravlenye u∗ ⋅( ) ∈ U na otrezke 0 1, t[ ],
t T1 < , y pust\ X u( , )⋅ � sootvetstvugwaq emu mnohoznaçnaq traektoryq.
Uçyt¥vaq v¥polnenye uslovyq maksymuma, poluçaem neravenstvo
C X t u t( , ) , ( )[ ]( )1 ψ ≤ C X t u t( , ) , ( )∗[ ]( )1 ψ , (22)
kotoroe spravedlyvo dlq vsex t ∈ 0 1, t[ ].
Pust\ teper\ zadan nekotor¥j moment vremeny τ ≤ t1 < T. Tohda yz (19) y
(22) sleduet, çto
C X u( , ) , ( )τ ψ τ[ ]( )1 < – , – ( )C Sk[ ]( )1 ψ τ ,
t. e. uslovye X u( , )τ I Sk ≠ ∅ ne v¥polnqetsq.
V sylu proyzvol\nosty v¥bora dopustymoho upravlenyq u( )⋅ ∈ U moΩno
utverΩdat\, çto ne suwestvuet ny odnoj mnohoznaçnosty traektoryy, udovlet-
vorqgwej pry τ < T uslovyg X u( , )τ I Sk ≠ ∅. Tem sam¥m upravlenye u∗ ⋅( )
optymal\no.
Teorema 2 dokazana.
Prymer. Pust\ povedenye obæekta opys¥vaetsq systemoj
ẋ1 = x2 + u1 + v1, x1(0) = 0,
ẋ2 = – x1 + u2 + v2 , x2(0) = 0,
hde (x1, x2) � fazovoe prostranstvo; u ∈ U = S1(0) � vektor upravlenyq;
v ∈R2 � vektor pomexy, kotor¥j prynadleΩyt neçetkomu mnoΩestvu V s xa-
rakterystyçeskoj funkcyej
ν( )v =
1 4 9 4 9 1
0 4 9 1
1
2
2
2
1
2
2
2
1
2
2
2
– – , ,
, .
v v v v
v v
+ ≤
+ >
Rassmotrym zadaçu b¥strodejstvyq typa (4), t. e. neobxodymo najty takye
T∗ y u∗ ⋅( ) , çto T∗ = min T(u), X T u( , )∗ ∗ I Sk ≠ ∅, hde Sk � neçetkoe mno-
Ωestvo s xarakterystyçeskoj funkcyej
ς( )x =
1 2 1 1
1 2 1 1
1 2 1 1
0
1
2
2
2
2
1
2
2
1
2
2
2
2
– ( – ) – ( – ) , , ,
– ( – ) , , – ,
– ( – ) – ( ) , , – ,
, ,
x x x Q x
x x Q x
x x x Q x
x Q
π
π
π
∈ ≥
∈ < <
+ ∈ ≤
∉
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
390 Y. V. MOLÇANGK, A. V. PLOTNYKOV
Q = ( , )
– ,
– ( – ) – – ( – )
x x R
x
x x x
1 2
2
1
1
2
2 1
2
2 1 2 1
1 2 1 1 2 1
∈
≤ ≤ +
≤ ≤ +
π π
π π
.
Lehko proveryt\, çto optymal\naq para T∗ = 2π y u t∗( ) = (cos t, – sin t) udov-
letvorqet uslovyqm teorem¥ 1:
1) u t∗( ( ), ψ( )t ) = C U( , ψ( )t ) dlq poçty vsex t ∈ 0 2, π[ ];
2) C X T u( , )∗ ∗[ ]( 1
, ψ( )T∗ ) = –C Sk[ ]( 1
, – ( )ψ T∗ ),
hde ψ( )t = (cos t, – sin t) dlq poçty vsex t ∈ 0 2, π[ ], X T u( , )∗ ∗[ ]1 = T T∗ ∗( cos ,
– sinT T∗ ∗) = (2π, 0), Sk[ ]1 = ( , )x x1 2{ x1 = 2π, – 1 ≤ x2 ≤ 1} .
1. Zadeh L. A. Fuzzy sets // Inf. Control. – 1965. – 8. – P. 338 – 353.
2. Kaleva O. Fuzzy differential equations // Fuzzy Sets and Systems. – 1987. – 24, # 3. – P. 301 –
317.
3. Komleva T. A., Plotnykov A. V., Plotnykova L. Y. Usrednenye neçetkyx dyfferencyal\-
n¥x uravnenyj // Tr. Odes. polytexn. un-ta. � 2007. � V¥p. 1 (27). � S. 185 � 190.
4. Komleva T. A., Plotnykov A. V., Skrypnyk N. V. Ω-prostranstvo y eho svqz\ s teoryej ne-
çetkyx mnoΩestv // Tam Ωe. � 2007. � V¥p. 2 (28). � S. 182 � 191.
5. Kaleva O. The Cauchy problem for fuzzy differential equations // Fuzzy Sets and Systems. – 1990.
– 35. – P. 389 – 396.
6. Kaleva O. The Peano theorem for fuzzy differential equations revisited // Ibid. – 1998. – # 98. –
P. 147 – 148.
7. Kaleva O. O notes on fuzzy differential equations // Nonlinear Anal. – 2006. – # 64. – P. 895 –
900.
8. Lakshmikantham V., Granna Bhaskar T., Vasundhara Devi J. Theory of set differential equations
in metric spaces. – Cambridge Sci. Publ., 2006. – 204 p.
9. Park J. Y., Han H. K. Existence and uniqueness theorem for a solution of fuzzy differential equa-
tions // Int. J. Math. and Math. Sci. – 1999. – 22, # 2. – P. 271 – 279.
10. Park J. Y., Han H. K. Fuzzy differential equations // Fuzzy Sets and Systems. – 2000. – # 110. –
P. 69 – 77.
11. Vorobiev D., Seikkala S. Towards the theory of fuzzy differential equations // Ibid. – 2002. –
# 125. – P. 231 – 237.
12. Aubin J. P. Fuzzy differential inclusions // Probl. Control and Inform. Theory. – 1990. – 19, # 1. –
P. 55 – 67.
13. Baidosov V. A. Differential inclusions with fuzzy right-hand side // Sov. Math. – 1990. – 40, # 3.
– P. 567 – 569.
14. Baidosov V. A. Fuzzy differential inclusions // J. Appl. Math. and Mech. – 1990. – 54, # 1. –
P. 8 – 13.
15. Hullermeier E. An approach to modeling and simulation of uncertain dynamical systems // Int. J.
Uncertainty Fuzziness Knowledge Based Systems. – 1997. – # 5. – P. 117 – 137.
16. Lakshmikantham V., Mohapatra R. Theory of fuzzy differential equations and inclusions // Ser.
Math. Anal. and Appl. – London: Taylor and Francis, Ltd., 2003. – 143 p.
17. Plotnykov A. V., Molçangk Y. V. Lynejn¥e system¥ upravlenyq s neçetkym parametrom //
Nelinijni kolyvannq. � 2006. � 9, # 1. � S. 63 � 73.
18. Plotnykov A. V. Lynejn¥e system¥ upravlenyq s mnohoznaçn¥my traektoryqmy // Kyber-
netyka. � 1987. � # 4. � S. 130 � 131.
19. Puri M. L., Ralescu D. A. Differential of fuzzy functions // J. Math. Anal. and Appl. – 1983. –
# 91. – P. 552 – 558.
20. Blahodatskyx V. Y. Lynejnaq teoryq optymal\noho upravlenyq. � M.: Yzd-vo Mosk. un-ta,
1978. � 95 s.
Poluçeno 17.09.07,
posle dorabotky � 28.05.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 3
|
| id | umjimathkievua-article-3026 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:34:49Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ee/ae4625c5706ef1652524b52e7a0c0bee.pdf |
| spelling | umjimathkievua-article-30262020-03-18T19:43:35Z Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters Необходимые и достаточные условия оптимальности в задачах управления с нечетким параметром Molchanyuk, I. V. Plotnikov, A. V. Молчанюк, И. В. Плотников, А. В. Молчанюк, И. В. Плотников, А. В. We study the problem of high-speed operation for linear control systems with fuzzy right-hand sides. For this problem, we introduce the notion of optimal solution and establish necessary and sufficient conditions of optimality in the form of the maximum principle. Розглянуто задачу швидкодії для лінійних систем управління з нечіткою правою частиною. Для цієї задачі введено поняття оптимального розв'язку і встановлено необхідні та достатні умови оптимальності у формі принципу максимуму. Institute of Mathematics, NAS of Ukraine 2009-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3026 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 3 (2009); 384-390 Український математичний журнал; Том 61 № 3 (2009); 384-390 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3026/2801 https://umj.imath.kiev.ua/index.php/umj/article/view/3026/2802 Copyright (c) 2009 Molchanyuk I. V.; Plotnikov A. V. |
| spellingShingle | Molchanyuk, I. V. Plotnikov, A. V. Молчанюк, И. В. Плотников, А. В. Молчанюк, И. В. Плотников, А. В. Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters |
| title | Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters |
| title_alt | Необходимые и достаточные условия оптимальности в задачах управления с нечетким параметром |
| title_full | Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters |
| title_fullStr | Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters |
| title_full_unstemmed | Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters |
| title_short | Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters |
| title_sort | necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3026 |
| work_keys_str_mv | AT molchanyukiv necessaryandsufficientconditionsofoptimalityintheproblemsofcontrolwithfuzzyparameters AT plotnikovav necessaryandsufficientconditionsofoptimalityintheproblemsofcontrolwithfuzzyparameters AT molčanûkiv necessaryandsufficientconditionsofoptimalityintheproblemsofcontrolwithfuzzyparameters AT plotnikovav necessaryandsufficientconditionsofoptimalityintheproblemsofcontrolwithfuzzyparameters AT molčanûkiv necessaryandsufficientconditionsofoptimalityintheproblemsofcontrolwithfuzzyparameters AT plotnikovav necessaryandsufficientconditionsofoptimalityintheproblemsofcontrolwithfuzzyparameters AT molchanyukiv neobhodimyeidostatočnyeusloviâoptimalʹnostivzadačahupravleniâsnečetkimparametrom AT plotnikovav neobhodimyeidostatočnyeusloviâoptimalʹnostivzadačahupravleniâsnečetkimparametrom AT molčanûkiv neobhodimyeidostatočnyeusloviâoptimalʹnostivzadačahupravleniâsnečetkimparametrom AT plotnikovav neobhodimyeidostatočnyeusloviâoptimalʹnostivzadačahupravleniâsnečetkimparametrom AT molčanûkiv neobhodimyeidostatočnyeusloviâoptimalʹnostivzadačahupravleniâsnečetkimparametrom AT plotnikovav neobhodimyeidostatočnyeusloviâoptimalʹnostivzadačahupravleniâsnečetkimparametrom |