Systems of control over set-valued trajectories with terminal quality criterion
We consider the optimal control problem with terminal quality criterion in which the state of a system is described by a set-valued mapping, and an admissible control is a summable function. We describe an algorithm that approximates the admissible control function by a piecewise-constant function a...
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| Date: | 2009 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2009
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3086 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509114377961472 |
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| author | Arsirii, A. V. Plotnikov, A. V. Арсирий, A. В. Плотников, А. В. Арсирий, A. В. Плотников, А. В. |
| author_facet | Arsirii, A. V. Plotnikov, A. V. Арсирий, A. В. Плотников, А. В. Арсирий, A. В. Плотников, А. В. |
| author_sort | Arsirii, A. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:44:57Z |
| description | We consider the optimal control problem with terminal quality criterion in which the state of a system is described by a set-valued mapping, and an admissible control is a summable function. We describe an algorithm that approximates the admissible control function by a piecewise-constant function and prove theorems on the closeness of the corresponding trajectories and the values of quality criteria. |
| first_indexed | 2026-03-24T02:35:57Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDK 517.9
A. V. Arsyryj (Odes. nac. un-t),
A. V. Plotnykov (Odes. akad. str-va y arxytektur¥)
SYSTEMÁ UPRAVLENYQ
MNOHOZNAÇNÁMY TRAEKTORYQMY
S TERMYNAL|NÁM KRYTERYEM KAÇESTVA
We consider the optimal control problem with terminal quality criterion, in which the condition of a
system is described by a set-valued map and an admissible control is a summable function. We describe
the algorithm which approximates the admissible control function by a piecewise constant function and
prove theorems on the proximity of corresponding trajectories and values of quality criteria.
Rozhlqnuto zadaçu optymal\noho keruvannq iz terminal\nym kryteri[m qkosti, u qkij stan sys-
temy opysu[t\sq bahatoznaçnym vidobraΩennqm, a dopustyme keruvannq [ sumovnog funkci[g.
Opysano alhorytm aproksymaci] dopustymo] funkci] keruvannq kuskovo-stalog funkci[g ta
dovedeno teoremy pro blyz\kist\ vidpovidnyx tra[ktorij ta znaçen\ kryteri]v qkosti.
S konca 60-x hodov 20 veka naçalos\ burnoe razvytye teoryy mnohoznaçn¥x oto-
braΩenyj. V rabote [1] M. Xukuxara vvel proyzvodnug y yntehral ot mnoho-
znaçn¥x otobraΩenyj y yssledoval yx svqz\ meΩdu soboj. Vposledstvyy v ra-
bote [2] b¥ly rassmotren¥ dyfferencyal\n¥e uravnenyq s proyzvodnoj Xuku-
xar¥, vveden¥ razlyçn¥e opredelenyq reßenyj y dokazan¥ teorem¥ yx suwest-
vovanyq [3], a v rabotax [4, 5] rassmotrena vozmoΩnost\ prymenenyq nekotor¥x
sxem usrednenyq dlq nyx.
Uravnenyq s proyzvodnoj Xukuxar¥ b¥ly yspol\zovan¥ v rabote [6] pry yzu-
çenyy nekotor¥x svojstv „yntehral\noj voronky” dyfferencyal\noho vklg-
çenyq v banaxovom prostranstve, a v rabotax [7, 8] pry yssledovanyy uravnenyj s
neçetkymy naçal\n¥my uslovyqmy.
V dannoj stat\e rassmatryvaetsq zadaça upravlenyq processom, opys¥vae-
m¥m lynejn¥m dyfferencyal\n¥m uravnenyem s proyzvodnoj Xukuxar¥ s ter-
mynal\n¥m kryteryem kaçestva. Dannug zadaçu moΩno suwestvenno uprostyt\,
esly funkcyg upravlenyq approksymyrovat\ kusoçno-postoqnnoj funkcyej.
Poπtomu pryvodytsq alhorytm postroenyq πtoho pryblyΩennoho kusoçno-po-
stoqnnoho upravlenyq y dokaz¥vaetsq blyzost\ sootvetstvugwyx ym traekto-
ryj y znaçenyj kryteryev kaçestva.
Pust\ Conv( )Rn
— prostranstva nepust¥x kompaktn¥x y v¥pukl¥x pod-
mnoΩestv evklydovoho prostranstva Rn
s metrykoj Xausdorfa h(⋅ ⋅), .
Rassmotrym upravlqemug systemu, kotoraq opys¥vaetsq lynejn¥m dyffe-
rencyal\n¥m uravnenyem s proyzvodnoj Xukuxar¥ vyda
D X t A t X t u t F th ( ) = ( ) ( ) + ( ) + ( ) , X X( ) =0 0 , (1)
hde t T∈[ ]0, ; X T(⋅) [ ]: ,0 → Conv( )Rn
— mnohoznaçnoe otobraΩenye, oprede-
lqgwee sostoqnye system¥; D X th ( ) — proyzvodnaq Xukuxar¥ [1]; A t( ) —
( × )n n -matryca; F T(⋅) [ ]: ,0 → Conv( )Rn
— otklonenye system¥; u(⋅) ∈ U ∈
∈ Conv( )Rn
— upravlqemoe vozdejstvye.
© A. V. ARSYRYJ, A. V. PLOTNYKOV, 2009
1142 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
SYSTEMÁ UPRAVLENYQ MNOHOZNAÇNÁMY TRAEKTORYQMY … 1143
PredpoloΩenye 1. Budem predpolahat\, çto systema (1) udovletvorqet
uslovyqm:
1) matryca A t( ) yzmeryma na [ ]0, T ;
2) suwestvuet konstanta a > 0 takaq, çto A t( ) ≤ a dlq poçty vsex
t T∈[ ]0, ;
3) mnohoznaçnoe otobraΩenye F(⋅) yzmerymo na [ ]0, T ;
4) suwestvuet konstanta f > 0 takaq, çto h F t f( )( ) ≤, 0 dlq poçty
vsex t T∈[ ]0, .
Opredelenye 1 [3]. Reßenyem zadaçy (1), sootvetstvugwym dopustymomu
upravlenyg u(⋅) ∈ U, naz¥vaetsq absolgtno neprer¥vnoe mnohoznaçnoe oto-
braΩenye X u(⋅ ), , udovletvorqgwee (1) poçty vsgdu na [ ]0, T .
Narqdu s zadaçej (1) rassmotrym prysoedynennug k nej zadaçu
X t X A s X s u s F s ds
t
( ) = + ( ) ( ) + ( ) + ( )[ ]∫0
0
. (2)
Yntehral v (2) ponymaetsq v sm¥sle Xukuxar¥ [1].
Opredelenye 2 [3]. Reßenyem zadaçy (2), sootvetstvugwym dopustymomu
upravlenyg u(⋅) ∈ U, naz¥vaetsq absolgtno neprer¥vnoe mnohoznaçnoe oto-
braΩenye X u(⋅ ), , udovletvorqgwee (2) vsgdu na [ ]0, T .
Yz dyfferencyruemosty poçty vsgdu yntehrala s peremenn¥m verxnym pre-
delom sleduet, çto lgboe reßenye zadaçy (2) qvlqetsq reßenyem zadaçy (1)
[2, 3].
Pust\ kaçestvo funkcyonyrovanyq system¥ (1) ocenyvaetsq kryteryem
I u X T u( ) = ( )( )Φ , , (3)
hde Φ(⋅) ( ) →: Conv R Rn 1
.
Lemma 1. Pust\ u t u t u t u tn( ) = ( ) ( ) … ( )( )1 2, , , — yzmerymaq funkcyq na ot-
rezke [ ]0, T takaq, çto u t u uj j j( ) ∈[ ]min max, , j = 1, n , dlq poçty vsex t ∈
∈ [ ]0, T . Razob\em otrezok [ ]0, T na k çastej [ ]( − )i h ih1 , , i = 1, k , h =
T
k
.
Tohda suwestvuet kusoçno-postoqnnaq funkcyq u t*( ) , udovletvorqgwaq
sledugwym uslovyqm:
1) u t*( ) postoqnna na kaΩdom yz otrezkov [ ]( − )i h ih1 , , i = 1, k ;
2) u si
*( ) = {( )( ) ( ) … ( ) ( ) ∈{u s u s u s u s ui i i
n T
i
j j* * * *
min, , , :1 2 ,, maxu j } , i = 1, k , j = 1, n }
dlq vsex s ∈ [ ]0, T ;
3) dlq lgboho t ∈ [ ]0, T v¥polnqetsq neravenstvo
u s ds u x ds u u h
t t
( ) − ( ) ≤ −∫ ∫
0 0
1
2
*
max min . (4)
Dokazatel\stvo. Oboznaçym U t U U Ui i i i
n T( ) = ( … )1 2, , , , hde Ui
j =
= u s dsj
i h
ih
( )
( − )
∫
1
, j = 1, n , i = 1, k y u t*( ) = ui = ( … )u u ui i i
n T1 2, , , : [ u ui
j j∈{ min ,
u j
max} , j = 1, n ], t ∈ [ ]( − )i h ih1 , , i = 1, k .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
1144 A. V. ARSYRYJ, A. V. PLOTNYKOV
Skonstruyruem funkcyg u t*( ) , t. e. opredelym u u un1 2, , ,… sledugwym
obrazom:
1) u u u un
1 1
1
1
2
1= ( … ), , , :
u
u U u u h
u
j
j j j j
j
1
1
1
2=
≥ ( + )max max min
min
,
,
esly
eslyy U u u hj j j
1
1
2
< ( + )
max min
, j = 1, n ;
2) predpoloΩym, çto m¥ uΩe opredelyly u u ui1 2, , ,… ; tohda u ui i+ += (1 1
1 ,
u ui i
n
+ +… )1
2
1, , :
u
u U u h u u
i
j
j
i
j
k
j
k
i
j
+
+
==
− ≥ ( +∑
1
1
1
1
2max max m, esly iin
min max m,
j
j
i
j
k
j
k
i
j
h
u U u h u u
)
− < ( ++
=
∑esly 1
1
1
2 iin
j h)
, j = 1, n .
Yz opredelenyq Ui sleduet, çto u h U U u hj
i
j
i
j j
min max≤ − ≤−1 , j = 1, n , yly
u h U U u hi imin max≤ − ≤−1 .
Poπtomu esly m¥ konstruyruem ui
, i = 1, k , po pryvedennoj v¥ße proce-
dure, to ymegt mesto sledugwye sootnoßenyq:
dlq i = 1 y j = 1, n
esly u uj j
1 = max , to 0
1
21 1≥ − ≥ ( − )U u h u u hj j j j
max min ,
esly u uj j
1 = min , to
1
2
01 1( − ) > − ≥u u h U u hj j j j
max min ,
t. e. U u h u u hj j j j
1 1
1
2
− ≥ ( − )max min , j = 1, n , yly
U u h u u h1 1
1
2
− ≥ −max min . (5)
Analohyçno, dlq i = 2, k y j = 1, n ymeem:
esly u ui
j j
+ =1 max , to
U u h U u h u h ui
j
k
j
k
i
i
j
i
j
k
j
k
i
− ≥ − − ≥ − (
=
+ +
=
∑ ∑
1
1 1
1
1
2 maax min
j ju h− ) ,
esly u ui
j j
+ =1 min , to
1
2 1 1
1
( − ) > − − ≥+ +
=
∑u u h U u h u h Uj j
i
j
i
j
k
j
k
i
i
j
max min −−
=
∑ u hk
j
k
i
1
,
t. e.
U u h u u h U ui
j
k
j
k
i
j j
i
j
+
=
+
− ≤ ( − ) −∑1
1
1 1
2
max ,max min kk
j
k
i
h
=
∑
1
, j = 1, n ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
SYSTEMÁ UPRAVLENYQ MNOHOZNAÇNÁMY TRAEKTORYQMY … 1145
yly
U u h u u h U u hi j
j
i
i j
j
i
+
=
+
=
− ≤ − −∑ ∑1
1
1
1
1
2
max ,max min
. (6)
Sootnoßenyq (5) y (6) dokaz¥vagt, çto neravenstvo (4) v¥polnqetsq dlq
vsex t = i h, i = 1, k , t. e. dlq hranyçn¥x toçek otrezkov [ ]( − )i h ih1 , , i = 1, k .
Ostalos\ ustanovyt\ neravenstvo (4) dlq vnutrennyx toçek πtyx otrezkov.
Pust\ t ∈ (( − ) )i h ih1 , , i = 1, k . Tohda
u s ds u s ds U u h
t t
i j
j
i
( ) − ( ) = −
+∫ ∫ ∑−
=
−
0 0
1
1
1
* (( )( ) −
( − )
∫ u s u dsi
i h
t
1
.
Pry ocenke vtoroho slahaemoho v pravoj çasty vozmoΩn¥ dva sluçaq: 1) ui
j =
= u j
max , j = 1, n ; 2) u ui
j j= min , j = 1, n .
Oçevydno, çto v pervom sluçae dlq lgboho j = 1, … , n moΩno zapysat\
ocenku
0 ≥ ( ) ( )( ) − ≥ ( ) −
( − ) ( − )
∫ u s u ds u s u dsj
i
j
i h
t
j
i
j
i h
ih
1 1
∫∫ = − −−U U u hi
j
i
j
i
j
1 .
Posle preobrazovanyj poluçym
U u h U u h u s u dsi
j
k
j
k
i
i
j
k
j
k
i
j
i
j− ≤ − + ( ) −
=
−
=
∑ ∑ ( )
1
1
1 (( − )
−
=
−
∫ ∑≤ −
i h
t
i
j
k
j
k
i
U u h
1
1
1
1
, j = 1, n .
Analohyçno, vo vtorom sluçae dlq lgboho j = 1, n
U u h U u h u s u dsi
j
k
j
k
i
i
j
k
j
k
i
j
i
j− ≥ − + ( ) −
=
−
=
∑ ∑ ( )
1
1
1 (( − )
−
=
−
∫ ∑≥ −
i h
t
i
j
k
j
k
i
U u h
1
1
1
1
.
Sledovatel\no, dlq vsex j = 1, n y t ∈ (( − ) )i h ih1 ,
u s ds u s ds U u h Uj
t
i
j
t
i
j
k
j
k
i
i( ) − ( ) ≤ −∫ ∫ ∑
=
−
0 0 1
max , 11
1
1
j
k
j
k
i
u h−
=
−
∑ ,
t. e.
u s ds u s ds U u h U u
t t
i j
j
i
i j( ) − ( ) ≤ − −∫ ∫ ∑
=
−
0 0 1
1
* max , hh
j
i
=
−
∑
1
1
.
Takym obrazom, neravenstvo (4) v¥polnqetsq dlq lgboho t T∈[ ]0, .
Lemma dokazana.
Teper\ dokaΩem blyzost\ reßenyj system¥ (1) y znaçenyj kryteryev ka-
çestva, sootvetstvugwyx ysxodnomu yzmerymomu upravlenyg y postroennomu
kusoçno-postoqnnomu.
Teorema 1. Pust\ u(⋅) — proyzvol\noe dopustymoe upravlenye, X t u( ), —
sootvetstvugwaq emu traektoryq system¥ (1) s naçal\n¥m sostoqnyem
X t u( )0 , = X0 , a otobraΩenye Φ(⋅) neprer¥vno y udovletvorqet uslovyg
Lypßyca s postoqnnoj λ . Razob\em otrezok [ ]0, T n a k çastej y skonst-
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
1146 A. V. ARSYRYJ, A. V. PLOTNYKOV
ruyruem upravlenye u*(⋅) sohlasno lemme 1, y pust\ X t u( ), *
— sootvet-
stvugwaq traektoryq system¥ (1) s naçal\n¥m uslovyem X t u( )0 , * = X0 .
Tohda:
1) suwestvuet konstanta C eaT
1 = takaq, çto dlq vsex t T∈[ ]0, v¥ -
polnqetsq neravenstvo
h X t u X t u C u u
h( )( ) ( ) ≤ −, , , *
max min1 2
; (7)
2) suwestvuet konstanta C2 0> takaq, çto dlq vsex t T∈[ ]0, v¥pol-
nqetsq neravenstvo
I u I u C u u
h( ) − ( ) ≤ −*
max min2 2
. (8)
Dokazatel\stvo. Perexodq ot uravnenyq (1) k sootvetstvugwemu ynteh-
ral\nomu uravnenyg, dlq dopustym¥x upravlenyj u t( ) y u t*( ) y sootvetstvu-
gwyx ym traektoryj X t u( ), y X t u( ), *
poluçaem
h X t u X t u( )( ) ( ), , , * =
= h A s X s u u s F s ds A s X s u u
t
[ ] [( ) ( ) + ( ) + ( ) ( ) ( ) +∫ , , , *
0
**( ) + ( )
]∫ s F s ds
t
0
≤
≤ h A s X s u u s F s ds A s X s u u
t
[ ] [( ) ( ) + ( ) + ( ) ( ) ( ) +∫ , , , *
0
(( ) + ( )
]∫ s F s ds
t
0
+
+ h A s X s u u s F s ds A s X s u
t
[ ] [( ) ( ) + ( ) + ( ) ( ) ( ) +∫ , , ,* *
0
uu s F s ds
t
*( ) + ( )
]∫
0
≤
≤ u s ds u s ds h A s X s u ds A
t t t
( ) − ( ) + ( ) ( ) (∫ ∫ ∫ [ ] [
0 0 0
* , , ss X s u ds
t
) ( )
]∫ , *
0
≤
≤ u s ds u s ds a h X s u X s u ds
t t t
( ) − ( ) + ( ) ( )∫ ∫ ( )
0 0 0
* *, , ,∫∫ .
Na osnovanyy lemm¥ Hronuolla – Bellmana moΩno zapysat\
h X t u X t u u s ds u s ds e e
t t
aT( )( ) ( ) ≤ ( ) − ( ) ≤∫ ∫, , , * *
0 0
aaT u u
h
max min−
2
.
Oboznaçaq C eaT
1 = , poluçaem (7). Takym obrazom, blyzost\ reßenyj dokazana.
Poskol\ku otobraΩenye Φ(⋅) neprer¥vno y udovletvorqet uslovyg Lyp-
ßyca s postoqnnoj λ, to
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
SYSTEMÁ UPRAVLENYQ MNOHOZNAÇNÁMY TRAEKTORYQMY … 1147
I u I u X t u X t u( ) − ( ) = ( ) − ( )( ) ( )* *, ,Φ Φ ≤
≤ λ λh X t u X t u C u u
h( )( ) ( ) ≤ −, , , *
max min1 2
.
Oboznaçaq C C2 1= λ , poluçaem (8). Sledovatel\no, dokazana blyzost\ kryte-
ryev kaçestva.
Teorema dokazana.
Zameçanye 1. V rabote [9] poluçen analohyçn¥j rezul\tat dlq ob¥knoven-
noho bylynejnoho dyfferencyal\noho uravnenyq so skalqrn¥m upravlenyem.
Pust\ teper\ kryteryj kaçestva funkcyonyrovanyq system¥ (1) budet mno-
hoznaçn¥m, t. e.
J u X T u( ) = ( )( )Ψ , , (9)
hde Ψ(⋅) ( ) → ( ): Conv ConvR Rn 1
.
Opredelenye 3 [4]. Upravlenye u U* ∈ nazovem maksymynn¥m (maksy-
maksn¥m) dlq zadaçy (1), (9), esly dlq lgboho upravlenyq u U∈ v¥polnqet-
sq neravenstvo
mJ u mJ u( ) ≤ ( )*
( MJ u MJ u( ) ≤ ( )* ),
hde mA a a A A R= ∈ ∈ ( ){ | }min , Conv 1
, MA a a A A R= ∈ ∈ ( ){ | }max , Conv 1
.
Tohda, kak y pry dokazatel\stve teorem¥ 1, moΩno dokazat\ blyzost\ maksy-
mynn¥x y maksymaksn¥x znaçenyj kryteryev kaçestva, sootvetstvugwyx ysxod-
nomu yzmerymomu y postroennomu kusoçno-postoqnnomu upravlenyqm, t. e.
suwestvuet takaq konstanta C3 > 0, çto dlq vsex t T∈[ ]0, v¥polnqgtsq ne-
ravenstva
mJ u mJ u C u u
h( ) − ( ) ≤ −*
max min3 2
,
MJ u MJ u C u u
h( ) − ( ) ≤ −*
max min3 2
.
1. Hukuhara M. Integration des applications mesurables dont la valeur est un compact convexe //
Funkc. ekvacioj. – 1967. – # 10. – P. 205 – 223.
2. de Blasi F. S., Iervolino F. Equazioni differentiali con soluzioni a valore compatto convesso //
Boll. Unione mat. ital. – 1969. – 2, # 4 – 5. – P. 491 – 501.
3. Brandao Lopes Pinto A. J., de Blast F. S., Iervolino F. Uniqueness and existence theorems for
differential equations with compact convex valued solutions // Ibid. – 1970. – # 4. – P. 534 – 538.
4. Plotnykov V. A., Plotnykov A. V., Vytgk A. N. Dyfferencyal\n¥e uravnenyq s mnoho-
znaçnoj pravoj çast\g. Asymptotyçeskye metod¥. – Odessa: AstroPrynt, 1999. – 354 s.
5. Kisielewicz M. Method of averaging for differential equations with compact convex valued
solutions // Rend. mat. – 1976. – 9, # 3. – P. 397 – 408.
6. Tolstonohov A. L. Dyfferencyal\n¥e vklgçenyq v banaxovom prostranstve. – Novosy-
byrsk: Nauka, 1986. – 296 s.
7. Kaleva O. Fuzzy differential equations // Fuzzy Sets and Systems. – 1987. – 24, # 3. – P. 301 –
317.
8. Kaleva O. The Cauchy problem for fuzzy differential equations // Fuzzy Sets and Systems. – 1990.
– 35. – P. 389 – 396.
9. Celikovsky S. On the representation of trajectories of bilinear systems and its applications //
Kybernetika. – 1987. – 23, # 3. – P. 198 – 213.
Poluçeno 03.04.07,
posle dorabotky — 15.05.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
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| id | umjimathkievua-article-3086 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:35:57Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2c/767e54e956667db26d42030160545d2c.pdf |
| spelling | umjimathkievua-article-30862020-03-18T19:44:57Z Systems of control over set-valued trajectories with terminal quality criterion Системы управления многозначными траекториями с терминальным критерием качества Arsirii, A. V. Plotnikov, A. V. Арсирий, A. В. Плотников, А. В. Арсирий, A. В. Плотников, А. В. We consider the optimal control problem with terminal quality criterion in which the state of a system is described by a set-valued mapping, and an admissible control is a summable function. We describe an algorithm that approximates the admissible control function by a piecewise-constant function and prove theorems on the closeness of the corresponding trajectories and the values of quality criteria. Розглянуто задачу оптимального керування із термінальним критерієм якості, у якій стан системи описується багатозначним відображенням, а допустиме керування є сумовною функцією. Описано алгоритм апроксимації допустимої функції керування кусково-сталою функцією та доведено теореми про близькість відповідних траєкторій та значень критеріїв якості. Institute of Mathematics, NAS of Ukraine 2009-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3086 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 8 (2009); 1142-1147 Український математичний журнал; Том 61 № 8 (2009); 1142-1147 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3086/2921 https://umj.imath.kiev.ua/index.php/umj/article/view/3086/2922 Copyright (c) 2009 Arsirii A. V.; Plotnikov A. V. |
| spellingShingle | Arsirii, A. V. Plotnikov, A. V. Арсирий, A. В. Плотников, А. В. Арсирий, A. В. Плотников, А. В. Systems of control over set-valued trajectories with terminal quality criterion |
| title | Systems of control over set-valued trajectories with terminal quality criterion |
| title_alt | Системы управления многозначными траекториями с
терминальным критерием качества |
| title_full | Systems of control over set-valued trajectories with terminal quality criterion |
| title_fullStr | Systems of control over set-valued trajectories with terminal quality criterion |
| title_full_unstemmed | Systems of control over set-valued trajectories with terminal quality criterion |
| title_short | Systems of control over set-valued trajectories with terminal quality criterion |
| title_sort | systems of control over set-valued trajectories with terminal quality criterion |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3086 |
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