Необхідні умови оптимального керування об’єктами з розподіленими параметрами
Problems of optimal control over objects with distributed constants described by nonlinear differential equations with partial derivatives of elliptic, parabolic and hyperbolic types have been considered.
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2019
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System research and information technologies| _version_ | 1866302297699516416 |
|---|---|
| author | Iovane, G. Mizernyy, V. M. |
| author_facet | Iovane, G. Mizernyy, V. M. |
| author_sort | Iovane, G. |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2019-01-18T15:10:28Z |
| description | Problems of optimal control over objects with distributed constants described by nonlinear differential equations with partial derivatives of elliptic, parabolic and hyperbolic types have been considered. |
| first_indexed | 2025-07-17T10:24:24Z |
| format | Article |
| fulltext |
© G. Iovane, V.M. Mizernyy, 2006
Системні дослідження та інформаційні технології, 2006, № 4 77
TIДC
МЕТОДИ ОПТИМІЗАЦІЇ, ОПТИМАЛЬНЕ
УПРАВЛІННЯ І ТЕОРІЯ ІГОР
UDC 517.9
NECESSARY CONDITIONS FOR CONTROL OF OBJECTS WITH
DISTRIBUTED CONSTANTS
G. IOVANE, V.M. MIZERNYY
Problems of optimal control over objects with distributed constants described by
nonlinear differential equations with partial derivatives of elliptic, parabolic and hy-
perbolic types have been considered.
INTRODUCTION
The significant progress in the development of non-linear functional analysis
methods [1, 2, 3], which have become widely adopted in different sections of
mathematics, favours the research of applied non-linear tasks, which are in natural
environment and used for many industrial technologies. Bringing them to corre-
sponding operators or differential-operator equations in functional spaces allows
to reveal general regularities and connections for entire tasks classes, which are
different according to their specific content [4].
With the help of some methods given by the non-linear analyses, we can re-
search on the question of extreme of functionals with restrictions, which appear
during solution of great number of important manufacturing and technical tasks.
The restrictions in the kind of functional equations and inequalities allow to form
up mathematical models of objects functioning, considering the physical essence
of the task.
The present work considers the task of optimal controlling for objects with
distributed parameters, which are described by non-linear differential equations
with partial derivatives of elliptical parabolic and hyperbolic types.
TASK SETTING
Let UYX ,, be Banach spaces, the functional J be determined in UX × and
the operator G reflect the space UX × on Y , that is RUXJ →×: ,
YUXG →×: .
Let us consider an extremal task:
inf),( →uxJ , (1)
YyUuXxyuxG ∈∈∈= , , ,),( , (2)
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 4 78
where the functional J and the operator G are non-linear.
Let us mark through *** ,, UYX the conjugated spaces to UYX ,, respec-
tively;
),( ),;( YULYXL are spaces of non-linear continuous operators, which act
on X and U on Y respectively;
GDGDJDJD uxux , , , are partial derivatives according to Gato [1] in the
point UXux ×∈);( of the reflection J and G , that is
x
uxJJDx ∂
∂
=
),( ,
u
uxJJDu ∂
∂
=
),( ,
x
uxGJDx ∂
∂
=
),( ,
u
uxGJDu ∂
∂
=
),( .
Theorem. Let us consider
1) the functional J and the operator G have partial derivatives according to
Gato in some interval UXW ux ×⊂),( 00
of the element UXux ×∈);( 00 and the
reflection
);(:,:,: );(
*
);(
*
);( 000000000
YXLWGDUWJDXWJD uxxuxuuxx →→→
and
);(: );( 000
YULWGD uxu → are continuous;
2) the space patterns X , U with the reflections GDx0
and GDu0
are
closed in Y .
At the same time, if the element (pair) ),( 00 ux is the solution of the tasks
(1), (2), then such correlations occur:
yuxG =);( 00 , (3)
UXyxxyuxGDxuxJD XxXx ×∈∀=〉〈+〉〈 ),( 0,)];([),;( *
1
*
00001 00
λ , (4)
0,)];([),;( *
2
*
00002 00
=〉〈+〉〈 UuUx uyuxGDuuxJDλ , (5)
where **
2
*
121 , ,, YyyR ∈∈λλ and 0**
*
2
*
121 ≠+++
YY
yyλλ .
Proof. According to the conditions of the multitude theorem
{ } { } YUuuuxGDPYXxxuxGDP xx ⊆∈∀=⊆∈∀= ;);( , ;);( 002001 00
form closed spaces in Y , that is subspaces 1P and 2P hold all their border points.
If YP ≠1 and YP ≠2 , that is 1P and 2P are proper subspaces of the Banach
space Y , then according to lemma about annihilator [5], non-zero functionals
**
2
*
1 , Yyy ∈ can be found; they are equals to zero at 1P and 2P correspondingly.
For linear continuous functionals *
1y and *
2y with Xx∈∀ and Uu∈∀ we ob-
tain
0);(,,)];([ 00
*
1
*
1
*
00 00
=〈=〉
YxXx xuxGDyxyuxGD
and
Necessary conditions for control of objects with distributed constants
Системні дослідження та інформаційні технології, 2006, № 4 79
0);(,,)];([ 00
*
2
*
2
*
00 00
=〈=〉
YuUu uuxGDyuyuxGD ,
as elements xuxGDx );( 000
and uuxGDu );( 000
belongs correspondingly to sub-
spaces YP ⊂1 , YP ⊂2 .
Let us assume that 01 =λ and 02 =λ if we take into account the last corre-
lations, we obtain equations (4), (5).
Let us now consider the case YP =1 and YP =2 . If we apply to the reflec-
tion G the theorem of Lustenik [5, 6], we will have that Xh∈∀ and Uu∈∀ ,
which satisfy the conditions
0),;( 000
=〉〈 Xx huxGD and 0),;( 000
=〉〈 Uu uxGD ν , (6)
at rather small numbers t and τ there exist such elements
)(),( 10 trhtxhtx ++= and )(),( 10 τντντ ruu ++= ,
that 0
)(
,0
)(
,0)),(),,(( 21 →→=−
τ
τ
ντ UX r
t
tr
yuhtxG with 0→t and
0→τ .
Let us consider the function )),(),,((),( νττϕ uhtxLt = . Its partial derivatives
according to Gato become
100
0
0
),;(
0
ChuxJD
t Xxt
=〉〈=
∂
∂
=
=
τ
ϕ , 200
0
0
),;(
0
CuxJD Uut
=〉〈=
∂
∂
=
=
ν
τ
ϕ
τ
and should be equal to zero. Indeed, if
0),;( 1000
≠=〉〈 ChuxJD Xx and 0),;( 2000
≠=〉〈 CuxJD Uu ν ,
then the signs of expressions
=+〉−〈=− )(),(),;();()),,(( 000000 0
toxhtxuxJDuxJuhtxJ Xx
+〉〈=+〉+〈= XxXx huxJDttotrthuxJD ),;()()(),;( 00100 00
)()(),;()()(),;( 1001100 00
totruxJDtCtotruxJD XxXx +〉〈+=+〉〈+
and
=+〉−〈=− )(),(),;();()),(,( 000000 0
touuuxJDuxJuxJ Uu ντντ
),()(),;( 2002 0
toruxJDC Uu +〉〈+= ττ
taking into account that 0
)(
,0
)( 21 →→
τ
τ UX r
t
tr
with 0→t and 0→τ , are
determined in terms of tC1 and τ2C and, as a result, they change when substitute
t and τ for t− and τ− accordingly.
At the same time there should not be an extreme at the point ),( 00 ux . Ex-
actly this contradiction proves our statement. Consequently, taking into the ac-
count (6), we have
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 4 80
);(Ker 0),;( 0000 00
uxGDhhuxJD xXx ∈∀=〉〈 (7)
and
);(Ker 0),;( 0000 00
uxGDuxJD uUu ∈∀=〉〈 νν . (8)
In other words, );( 000
uxJDx is the element within *X , which is orthogo-
nal to subspace XuxGDx ⊂);(Ker 000
, that is
[ ] *
0000 );(Ker);(
00
XuxGDuxJD xx ∩∈ ⊥ .
Similarly, );( 000
uxJDu is the element within *V , which is orthogonal to
UuxGDx ⊂);(Ker 000
, that is [ ] *
0000 );(Ker);(
00
UuxGDuxJD uu ∩∈ ⊥ . Ac-
cording to the lemma about annihilator [5] we obtain
[ ] [ ]*0000 );(Im);(Ker
00
uxGDuxGD xx =⊥ (9)
and
[ ] [ ]*0000 );(Im);(Ker
00
uxGDuxGD uu =⊥ . (10)
Consequently, if
[ ] *
0000 );(Ker);(
00
XuxGDuxJD xx ∩∈ ⊥ ,
[ ] *
0000 );(Ker);(
00
UuxGDuxJD uu ∩∈ ⊥ ,
then it can be found such functionals **
2
*
1 , Yyy ∈ , such that
[ ]
XxXx xyuxGDxuxJD ,);(),;( *
1
*
0000 00
−= (11)
and
[ ]
UuUu xyuxGDuuxJD ,);(),;( *
2
*
0000 00
−= . (12)
Assuming 121 == λλ and taking into consideration that UXux ×∈),( , we
obtain the expressions (4) and (5), which prove the theorem.
This theorem is an infinitely measurable generalization of Lagrangian coef-
ficients rule, which is known from classical analysis and necessary conditions for
extremal tasks with restrictions.
Let us mention, that the system of equations (3)–(5), which presents neces-
sary conditions for functional optimum (1) with restrictions (2), can be written
(for 121 == λλ ) in the operator form:
yuxG =);( 00 , (13)
[ ] 0);();( *
1
*
0000 00
=+ yuxGDuxJD xx , (14)
[ ] 0);();( *
2
*
0000 00
=+ yuxGDuxJD uu , (15)
where
[ ] ***
00 :);(
0
XYuxGDx → , [ ] ***
00 :);(
0
UYuxGDu → .
Necessary conditions for control of objects with distributed constants
Системні дослідження та інформаційні технології, 2006, № 4 81
Hence, the solution ),,,( *
2
*
100 yyux of system (3)–(5) or (13)–(15) can be
interpreted as a generalized solution.
TASK OF OPTIMAL CONTROL FOR THE OBJECTS WITH DISTRIBUTED
CONSTANTS, WHICH ARE DESCRIBED BY NON-LINEAR DIFFERENTIAL
EQUATIONS — ELLIPTIC TYPE
Let us assume that functions, which determine the state )(ωx of an object and the
control parameter )(ωu are defined in the restricted area NR⊂Ω with the limit
Ω∂ .
We get necessary optimal conditions as solution of functional equation sys-
tem.
Let us consider such optimization task:
∫
Ω
→= inf))(),((),( ωωω duxJuxI , (16)
fuxxxGx
n
N
i i
=
∂
∂
∂
∂
+
∂
∂∑
=
),,...,,,(
11
2
2
ωω
ω
ω
, (17)
0=Ω∂x . (18)
We make the following assumptions:
1. Let us assume that
2 ;,...,1 ),(,, 2
2
≥=Ω∈
∂
∂
∂
∂ pNiLxxx p
ii ωω
; 1),( >Ω∈ rLu r .
2. Let function ×Ω:G ℝ →+2n ℝ belong to the class CAR )CAR( ∈G , that
is if ∈∀ξ ℝ 2+n the function mG );( ξωω∋Ω is measurable, and for almost all
Ω∈ω the function ℝ ),(2 ξωξ Gn ∋+ is continuous.
Let us also assume, that
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
++≤
−+
=
∑ q
r
pN
i
icaG ηξωξω
11
1
)();( ,
where ∈∈=Ω∈ +
+ ηξξξω ,),...,( ),()( 1
21
N
Nq RLa ℝ, 0>c .
3. Functional f allows the representation in the form ∑
≤
=
2α
α
ανDf ,
)(Ω∈ qLαν .
Taking into the boundary conditions (18), we obtain
XWWx pp =Ω∩Ω∈ )()( 21 .
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 4 82
Marking
∑
= ∂
∂
=
N
i i
xxL
1
2
2
ω
and ),,...,;;(),(
1
uxxxGuxG
Nωω
ω
∂
∂
∂
∂
= ,
we obtain [1, 4, 7, 8] linear operator )(: Ω→ qLXL , non-linear operator ×)(XG
)()( Ω→Ω× qr LL and non-linear functional RLLI rp →Ω×Ω )()(: .
According to the Lagrange principle, if the pair ),( 00 ux is the solution of
the task (16), (17), then it gives the inferior extreme to the functional
YyuxGyuxIux 〉−〈+=Φ ),(,),(),( * , where ** Yy ∈ , )(Ω= qLY , (19)
which is called Langrangian of the task (16), (17).
We obtain necessary conditions of extreme, when calculating functional
variations (19) and partial variations (partial derivatives according to Gato) and
then separately putting them to zero.
Indeed, for Xh∈∀ (it could be assumed xxxh −+= δ )
YyuhxGyuhxIuhx 〉−+〈++=+Φ=Φ ),(,),(),( * αααα .
Let us find the derivative from ),( uhx α+Φ by the parameter α :
=⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
=
∂
Φ∂
XXYX
hy
x
G
h
x
I
h
x
G
yh
x
I
,,,, *
*
* ααααα
α
X
hy
x
G
x
I ,*
*
⎥⎦
⎤
⎢⎣
⎡
∂
∂
+
∂
∂
= αα .
Hence, when passing on to the limit under 0→α , we obtain the func-
tional Φ variation, that is
X
x hy
x
G
x
I ,*
*
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
+
∂
∂
=Φδ . (20)
Similarly we obtain the variation Φ by u :
)( ,,** Ω=∈∀⎥⎦
⎤
⎢⎣
⎡
∂
∂
+
∂
∂
=Φ r
U
u LUy
u
G
u
I
ννδ . (21)
From the correlations (20) and (21) we get the necessary conditions for the
task (16), (17), which are similar to conditions (4), (5) of extremal task (1), (2). At
the same time there is an element ** Yy ∈ that satisfies equations (4), (5), that is
*
2
*
1
* yyy == .
Let us write down appropriate Langrangian similarly to the task (16)–(18).
=−
∂
∂
∂
∂
+
∂
∂
Ψ+=
Ω=
∑
)(11
2
2
),,...,,,(,),(),(
qLN
N
i i
fuxxxGxyxIyxF
ωω
ω
ω
Necessary conditions for control of objects with distributed constants
Системні дослідження та інформаційні технології, 2006, № 4 83
−
∂
∂
∂
∂
Ψ+∂
∂
∂
Ψ+= ∫∫ ∑
ΩΩ =
ωω
ωω
ωωωω
ω
ω duxxxGxuxI
Ni
N
i i
))(,,...,),(;()()(),(
1
2
2
.)()(∫
Ω
Ψ− ωωω df
Taking into account (16)–(18)
∫∑∫
Ω =Ω
+
∂
Ψ∂
+=
N
i i
dxduxJuxF
1
2
2
)(),(),( ωω
ω
ω
∫∫
ΩΩ
Ψ−
∂
∂
∂
∂
Ψ+ ωωωωω
ωω
ωωω dfduxxxG
N
)()())(,,...,),(;()(
1
.
Let us find partial variations of functional F .
ωδδωωδ
ω
ωδδ ω
ω
dx
x
Gx
x
Gdxdx
x
JF
N
i
N
i i
x i
i
∫ ∑∫∑∫
Ω =Ω =Ω ⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
+
∂
∂
Ψ+
∂
Ψ∂
+
∂
∂
=
11
2
2
)( , (22)
where
Nixx
i
i
,...,3,2,1 , =
∂
∂
=
ωω ,
∫ ∫∫
Ω ΩΩ
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
Ψ+
∂
∂
=
∂
∂
Ψ+
∂
∂
= ωδδωδδ du
u
G
u
Ju
u
Gdu
u
JFu . (23)
Taking into that
iiii
xxxx
ω
δ
ωωω
δ
∂
∂
=
∂
∂
−
∂
∂
=
∂
∂ 0
and
iiii iii
x
Gx
x
Gxxx
x
G
ω
δδ
ω
δ
ωω
δ
ωωω ∂∂
∂
−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂∂
∂
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
⋅
∂
∂
∂
∂ 2
)( ,
and using the integration by pats rule, let us write variation (22) in form:
+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
Ψ+
∂
Ψ∂
+
∂
∂
= ∫ ∑
Ω =
ωδ
ω
δ xd
x
G
x
JF
N
i i
x
1
2
2
∫ ∑
Ω =
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
Ψ+ ω
δω
δδ
ω ωω
d
x
Gx
x
Gx
N
i ii ii1
2
+
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂∂
∂
−
∂
∂
Ψ+
∂
Ψ∂
+
∂
∂
= ∫ ∑
Ω =
ωδ
ωω ω
xd
x
G
x
G
x
J N
i ii i1
2
2
2
∫ ∑
Ω =
=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ ∂
∂
∂
Ψ+
N
i i
d
x
G
i1
ω
δω ω
(24)
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 4 84
+
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂∂
∂
−
∂
∂
Ψ+
∂
∂
∂
Ψ∂
−
∂
Ψ∂
+
∂
∂
= ∫ ∑
Ω =
ωδ
ωωωωω
xdG
x
GG
x
J N
i iiiii1
2
2
2
2
Ω∂
=
∑ ⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
Ψ+
N
i
x
x
G
i1 2
δ
ω
.
Then, putting partial variations (23), (24) to zero, we get necessary optimal
conditions:
0=
∂
∂
Ψ+
∂
∂
u
G
u
J , (25)
0
1
2
2
2
=Ψ
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂∂
∂
−
∂
∂
+
∂
Ψ∂
⋅
∂
∂
−
∂
Ψ∂
+
∂
∂ ∑
=
N
i iii ii
x
G
x
G
x
G
x
J
ωωω ωω
, (26)
0=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
Ψ
Ω∂i
x
G
ω
(or 0=Ψ Ω∂ ). (27)
TASK OF OPTIMAL CONTROL FOR THE OBJECTS WITH DISTRIBUTED
CONSTANTS, WHICH ARE DESCRIBED BY NON-LINEAR DIFFERENTIAL
EQUATIONS - PARABOLIC TYPE
Let us assume, that functions, which determine object’s state ),( ωtx and the con-
trol parameter ),( ωtu , are determined in the restricted domain NR⊂Ω with the
limit Ω∂ at time interval ST =],0[ .
Time-dependent tasks of optimal control for the objects with distributed con-
stants, which are described by non-linear differential equations with particle de-
rivatives of parabolic type, look like:
[ ] inf),(),(),,(
0
→=∫ ∫
Ω
uxIdtdtutxJ
T
ωωω , (28)
),()),(,,...,),,(;,(
11
2
2
ωω
ωω
ωω
ω
tftuxxtxtQx
t
x
N
N
i i
=
∂
∂
∂
∂
+
∂
∂
−
∂
∂ ∑
=
, (29)
0),0( =⋅x , 0),( =Ω∂ωtx . (30)
In this case initial boundary conditions are put to zero. Such assumption does
not affect the general task setting, as non-zero conditions can be put to zero [5, 6].
Considering such tasks we have to deal with functions ),( ωtx , which are
irrespective of time and position, which associate each a pair Ω×∈St ),( ω with
real number or vector ),( ωtx . At the same time, variables t and ω are presented
as independent. There had been used time functions for the convenience of
Necessary conditions for control of objects with distributed constants
Системні дослідження та інформаційні технології, 2006, № 4 85
mathematical description of time dependent processes, associate each a time t
with function ),( ⋅tx of position. Consequently, there had been considered func-
tions, designated on S which have values in some spaces X , that is )( XSx →∈
[4].
Let us introduce
),()( ⋅= txty , ),()( ⋅= tutν ,
t
xty
∂
∂
=′ )( , 0),0()0( =⋅= xy , ),()( ⋅= tftg ,
∑∑
== ∂
∂
=
∂
∂
=
N
i i
N
i i w
wtx
w
tytLy
1
2
2
1
2
2 ),()()( ,
( ) ( )( ) ( ) ( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
∂
∂
= tyytytQtvtyG
N
ν
ωω
,,...,,,,
1
. (31)
Let us present the tasks (28)–(30) in the operator form, taking into account
(31):
( ) inf, →vyI , (32)
( ) ( ) ( ) ( )( ) ( )tgtvtyGtLyty =+−′ , , (33)
( ) 00 =y . (34)
Here the function CAR),...,;( 1 ∈Ntl ξξ , which corresponds to the linear reflec-
tion
∑
= ∂
∂
=
N
i i
tytLy
1
2
2 )()(
ω
,
satisfies the condition
2 ,......),...,;(
1
1
21211 ≥≤+++≤+++= ∑
=
− ptl
N
i
p
iNNN ξξξξξξξξξ .
For the function CAR),...,;( 21 ∈+NtQ ξξ , which corresponds to the non-
linear reflection
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
∂
∂
= )(,,...,),(;))(),((
1
tyytytQttyG
N
ν
ωω
ν ,
we demand the fulfilment of condition
.2 ,0 )),(( ,)(),...,;(
1
1
1
21 ≥>Ω→∈++≤ ∑
+
=
−
+ pcLSactatQ u
N
i
q
rp
iN ηξξξ
On the assumption of boundary conditions and taking into account, that
)()()( 22 Ω⊂Ω∩Ω= ppp LWWX і )(* Ω⊂ qLX , we assume, that
),(),( *XSyXSy →∈′→∈
)()( )),(( )),(( Ω∈Ω→∈Ω→∈
∂
∂
qpp
i
LtgLSLSy
ν
ω
.
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 4 86
Then
→Ω× ))(;();(: rrp LSLXSLI ℝ,
Ω×=→ SQQLXSLL qp ),();(: ,
)();(: QLXSLG qp → . .
We form appropriate Lagrange function for getting necessary conditions of
optimal task (32)–(34), which is equivalent to the task (28)–(30), by introducing
conjugated function ))(( Ω→∈ pLSψ . Then,
=−+−+= ∫
ΩS L
dttgttyGtLy
dt
tdytyIyF
q )(
)())(),(()()(),(),(),( νψνν
+
∂
∂
−+= ∫ ∫ ∑∫ ∫
Ω =Ω
T N
i i
T
dtdtytdtd
dt
tdytttyI
0 1
2
2
0
)()()()())(),(( ω
ω
ψωψν
∫ ∫∫ ∫
ΩΩ
−
∂
∂
∂
∂
+
TT
N
dtdtgtdtdttytytytQt
00 1
)()()(,)(,...,)(),(,()( ωψων
ωω
ψ . (35)
Using the integration by parts rule about first and second integral and chang-
ing integration sequence in first integral, we obtain:
[ ] −
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
−−+= ∫ ∫
Ω
ω
ψ
ψψνν ddt
dt
tdtyTyTyyIyF
T
0
)()()()()0()0(),(),(
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
∂
∂
⋅
∂
∂
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
− ∫ ∫∑∑
Ω =Ω∂=
dtdtyttt
T N
i ii
N
i i0 11
)()()()( ω
ωω
ψ
ω
ψ
ψ
=−
∂
∂
∂
∂
+ ∫ ∫∫ ∫
ΩΩ
dtdtgtdtdttytytytQt
TT
N
ωψων
ωω
ψ
00 1
)()())(,)(,...,)(),(,()(
[ ]∫ ∫
Ω
−
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧ Ψ
−Ψ−Ψ+= ωddt
dt
tdtyTyTyvyI
T
0
)()()()()0()0(),(
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
∂
∂
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
− ∫ ∫ ∑∑∑
Ω =Ω∂=Ω∂=
T N
i i
N
i i
N
i i
dtdttytyttyt
0 1
2
2
11
)()()()()()( ω
ω
ψ
ω
ψ
ω
ψ
∫ ∫∫ ∫
ΩΩ
−
∂
∂
∂
∂
+
TT
N
dtdtgtdtdttytytytQt
00 1
)()())(,)(,...,)(),(,()( ωψων
ωω
ψ . (36)
Taking into account that 0)( =Ω∂ty St∈∀ , it follows
[ ]∫ ∫
Ω
−
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
−−+= ω
ψ
ψψνν ddt
dt
tdtyTyTyyIyF
T
0
)()()()()0()0(),(),(
Necessary conditions for control of objects with distributed constants
Системні дослідження та інформаційні технології, 2006, № 4 87
∫ ∫ ∑ ∫ ∫
Ω = Ω
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
∂
∂
+
∂
∂
−
T N
i
T
Ni
dtdttytytytQtdtdtty
0 1 0 1
2
2
)(,)(,...,)(),(,)()()( ων
ωω
ψω
ω
ψ
∫ ∫
Ω
−
T
dtdtgt
0
)()( ωψ . (37)
Under the condition that 0)0( =y and for conjugated task 0)( =Tψ , we ob-
tain
∫ ∫
Ω
−−=
T
dtd
dt
tdtyyIyF
0
)()(),(),( ω
ψ
νν
∫ ∫∫ ∫ ∑ −
∂
∂
∂
∂
+
∂
∂
−
ΩΩ =
T
N
T N
i i
dtdttytytytQtdtdtty
0 10 1
2
2
))(,)(,...,)(),(,()()()( ων
ωω
ψω
ω
ψ
∫ ∫
Ω
−
T
dtdtgt
0
)()( ωψ . (38)
Then, we find partial variations of the functional ))(),(( ttyF ν
+
∂
∂
−−
∂
∂
= ∫ ∫∑∫ ∫∫ ∫
Ω =ΩΩ
T
ty
N
i i
TT
y dtdtdtdty
dt
tddtdty
ty
JF
0
)(
1
2
2
00
)()()()(
)(
ωδ
ω
ψ
ωδ
ψ
ωδδ
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
+
∂
∂
+ ∫ ∫ ∑
Ω =
dtdty
ty
Qty
ty
Qt
T N
i
i
i
ωδδψ ω
ω0 1
)(
)(
)(
)(
)(
∫ ∫∑∫ ∫∫ ∫ +
∂
∂
−−
∂
∂
=
Ω =ΩΩ
T N
i i
TT
dtdtytdtdty
dt
tddtdty
ty 0 1
2
2
00
)()()()()(
)(
ωδ
ω
ψωδ
ψ
ωδτ
+
∂
∂
+ ∫ ∫
Ω
T
dtdty
ty
Qt
0
)(
)(
)( ωδψ
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂∂
∂
−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
+ ∫ ∫ ∑
Ω =
T N
i ii
dtd
ty
Qtyty
ty
Qt
ii0 1
2
)(
)()(
)(
)( ω
ω
δδ
ω
ψ
ωω
+
∂
∂
−−
∂
∂
= ∫ ∫∑∫ ∫∫ ∫
Ω =ΩΩ
T N
i i
TT
dtdtytdtdty
dt
tddtdty
ty 0 1
2
2
00
)()()()()(
)(
ωδ
ω
ψ
ωδ
ψ
ωδτ
+
∂∂
∂
+
∂
∂
+ ∫ ∫ ∑∫ ∫
Ω =Ω
T N
i i
T
dtd
ty
Qtytdtdty
ty
Qt
i0 1
2
0 )(
)()()(
)(
)( ω
ω
δψωδψ
ω
∫ ∫ ∑
Ω =
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
+
T N
i i
dtdty
ty
Qt
i0 1
)(
)(
)( ωδ
ω
ψ
ω
. (39)
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 4 88
Let us mark the first five integrals through Σ . Using formula of Green, it
follows
=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
+Σ= ∫ ∫ ∑
Ω =
T N
i i
y dtdty
ty
QtF
i0 1
)(
)(
)( ωδ
ω
ψδ
ω
∫ ∫∑∫ ∑
Ω == Ω∂
=
∂
∂
⋅
∂
∂
−
∂∂
∂
+Σ=
T N
i i
T N
i
dtdty
ty
Qtdt
ty
Qtyt
ii 0 10 1
2
)(
)(
)(
)(
)()( ωδ
ω
ψ
ω
δψ
ωω
∫ ∫∑∫ ∫∫ ∫
Ω =ΩΩ
+
∂
∂
−−
∂
∂
=
T N
i i
TT
dtdtytdtdty
dt
tddtdty
ty
J
0 1
2
2
00
)()()()()(
)(
ωδ
ω
ψ
ωδ
ψ
ωδ
+
∂∂
∂
−
∂
∂
+ ∫ ∫ ∑∫ ∫
Ω =Ω
T N
i i
T
dtdty
ty
Qtdtdty
ty
Qt
i0 1
2
0
)(
)(
)()(
)(
)( ωδ
ω
ψωδψ
ω
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
+
Ω∂=
∫ ∑ dtty
ty
Qt
T N
i i0 1
)(
)(
)( δψ
ω
∫ ∫∑
Ω =
=
∂
∂
⋅
∂
∂T N
i i
dtdty
ty
Qt
i0 1
)(
)(
)(
ωδ
ω
ψ
ω
+
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
⎪
⎪
⎪
⎭
⎪⎪
⎪
⎬
⎫
⎪
⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
−
∂∂
∂
+
+
∂
∂
⋅
∂
∂
+
∂
∂
−
∂
∂
= ∫ ∫ ∑
Ω =
T N
i
ii
dtdty
dt
tdt
ty
Q
ty
Q
ty
Qtt
ty
J
i
i
0 1 2
2
2
)(
)()(
)()(
)(
)()(
)(
ωδ
ψ
ψ
ω
ω
ψ
ω
ψ
ω
ω
.)(
)(
)(
0 1
dtty
ty
Qt
T N
i i Ω∂=
∫ ∑
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
+ δψ
ω
(40)
=
∂∂
∂
+
∂
∂
= ∫ ∫ ∑∫
Ω =Ω
T N
i i
u dtdtu
u
Qtdtu
tu
JF
i0 1
2
)()()(
)(
ωδ
ω
ψωδδ
ω
=
∂
∂
+
∂
∂
= ∫ ∫∫ ∫
ΩΩ
dtdtu
tu
Qtdtdtu
tu
J TT
ωδψωδ )(
)(
)()(
)( 00
ωδψ dtdtut
tu
Q
tu
JT
)()(
)()(0
∫ ∫
Ω
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
+
∂
∂
= . (41)
Having put partial variations (40), (41) to zero, we obtain the necessary op-
timal conditions:
0)()()(
1
2
2
2
=
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
−
∂∂
∂
+
∂
∂
∂
∂
+
∂
∂
−
∂
∂
−
∂
∂ ∑
=
N
i iii
t
y
Q
y
Q
y
Qtt
ty
J
ii
ψ
ωω
ψ
ω
ψψ
ωω
, (42)
Necessary conditions for control of objects with distributed constants
Системні дослідження та інформаційні технології, 2006, № 4 89
0)( ,0
1
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
=
Ω∂=
∑
N
i i
y
QtT
ω
ψψ (or 0)( =Ω∂tψ ), (43)
0)( =
∂
∂
+
∂
∂ t
u
Q
u
J ψ . (44)
TASK OF OPTIMAL CONTROL FOR THE OBJECTS WITH DISTRIBUTED
CONSTANTS, WHICH ARE DESCRIBED BY NON-LINEAR DIFFERENTIAL
EQUATIONS - HYPERBOLIC TYPE
Let the state of an object ),( ωtx and control parameter ),( ωtu be determined in
the restricted domain NR⊂Ω with the boundary Ω∂ and time interval
ST =],0[ .
Such task will take place:
inf),()],(),,([
0
→=∫ ∫
Ω
uxIdtdtutxJ
T
ωωω , (45)
),()),(,,...,
1
),,(,,(
1
2
2
2
2
ωω
ωω
ωω
ω
tftu
N
xxtxtQx
t
x N
i i
=
∂
∂
∂
∂
+
∂
∂
−
∂
∂ ∑
=
, (46)
Sttxx
t
tx
ttxx t ∈∀==⋅′=
∂
⋅∂
==⋅′=⋅ Ω∂= ,0|),(,0),0(),(
0|),(,0),0( 0 ω . (47)
Let us introduce the corresponding markings according to the assumption as
for Q and J , as in the task (28)–(30):
0),0()0(,)(,)(),(),(),,()( 2
2
=⋅=
∂
∂
=′′
∂
∂
=′=⋅⋅= xy
t
xty
t
xtytvtutxty ,
∑∑
== ∂
∂
=
∂
∂
=⋅=
∂
⋅∂
=′
N
i i
N
i i
txtytLytfg
t
txy
1
2
2
1
2
2 ),()()(),,()0(,),()0(
ω
ω
ω
,
.)(,,...,
1
),(,))(),((
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
= tv
N
yytytQtvtyG
ωω
(48)
Then, the expressions (45), (46) will look like:
inf),())(),((
0
→=∫ ∫
Ω
vyIdtdtvtyJ
T
ω , (49)
),())(),(()()( tgtvtyGtLyty =+−′′ (50)
.0)0()0( =′= yy (51)
On the assumption of boundary conditions, we assume that
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 4 90
XWWSy pp =Ω∩Ω→∈ ))()(( 21
)).(()),(()),((;, * Ω→∈Ω→∈Ω→∈
∂
∂
∈′′′ pLSgrLSvpLS
i
yXyy
ω
Then
→Ω× ))(;();(: rrp LSLXSLI ℝ,
Ω×=→ SQQLXSLL qp ),();(: ,
)();(: QLXSpLG q→ .
Let us form appropriate Lagrange function, introducing conjugated function:
−
∂
∂
−
∂
∂
+= ∫ ∫ ∑∫ ∫
Ω =Ω
T N
i i
T
dtdtytdtd
t
tytvyIvyF
0 1
2
2
0
2
2 )()()()(),(),( ω
ω
ψωψ
∫ ∫∫ ∫
ΩΩ
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
∂
∂
−
TT
N
dtdtgtdtdtvyytytQt
00 1
)()()(,,...,),(,,)( ωψω
ωω
ωψ , (52)
wherе X∈ψ .
Acquired Langrangian differs from (28)–(30) only for second member
∫ ∫
Ω ∂
∂T
dtd
t
tyt
0
2
2 )()( ωψ .
Assuming that the conditions of Fubini’s theorem are fulfilled [4], and sub-
stituting integration sequence and using the integration by parts rule, we obtain
∫ ∫∫ ∫
ΩΩ
=
∂
∂
=
∂
∂ TT
dtd
t
ytdtd
t
tyt
0
2
2
0
2
2
)()()( ωψωψ
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
∂
∂
−⎜⎜
⎝
⎛
⎟
⎠
⎞
∂
∂
= ∫ ∫ ω
ψ
ψ ddt
t
ty
t
t
t
yt
T
T
0
0
)()()(
=
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
∂
∂
+⎥⎦
⎤
⎢⎣
⎡
∂
∂
−⎥⎦
⎤
⎢⎣
⎡
∂
∂
−
∂
∂
= ∫ ∫
Ω
ωψψ
ψψ ddt
t
tyty
t
t
t
y
t
tyt
T
T
0
2
2
0
)()()()0()0()()(
∫
Ω⎩
⎨
⎧
+
⎭
⎬
⎫
⎩
⎨
⎧
∂
∂
−
∂
∂
−
⎭
⎬
⎫
⎩
⎨
⎧
∂
∂
−
∂
∂
= )0()0()()()0()0()()( y
t
Ty
t
T
t
y
t
tyt ψψ
ψψ
ωψ ddt
t
ty
T
∫ ⎪⎭
⎪
⎬
⎫
∂
∂
+
0
2
2
)( . (53)
Taking into account the initial conditions 0)0()0( =′= yy and assuming for
conjugated task 0)()( =′= tt ψψ , we obtain
Necessary conditions for control of objects with distributed constants
Системні дослідження та інформаційні технології, 2006, № 4 91
∫ ∫∫ ∫
ΩΩ ∂
∂
=
∂
∂ TT
dtd
t
tydtd
t
ty
t
0
2
2
0
2
2
)(
)(
)( ωψωψ . (54)
Then, determining partial variations and putting them to zero, the necessary
optimal conditions are obtained
0)()()(
1
2
2
2
2
2
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
−
∂∂
∂
+
∂
∂
⋅
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂ ∑
=
N
i ii
t
y
Q
y
Q
i
t
i
y
Qt
ty
J
i
ψ
ωψ
ψ
ωω
ψψ
ω
, (55)
,|0)()(,0)()( StttTT ∈∀Ω=′==′= ψψψψ (56)
.0)( =
∂
∂
+
∂
∂ t
u
Q
y
J ψ (57)
CONCLUSION
In this work, we have considered how to find possible optimal solution of task of
control (16)–(18) for the objects with distributed constants, which are described
by non-linear differential equations (elliptic type): we have the system of 3 equa-
tions (17), (25), (26) with boundary conditions (18), (27) for unknown quantities
),,( 000 Ψux . The given system of equations is non-linear as for unknown quanti-
ties ),( ux and linear as for ψ .
First, from (17), (18) we find Uux ∈∀0 . Then 0x , which depends on u, is
substituted in (25)–(27); finally we find appropriate value ),( 00 uψ to 0x .
Then, finding second variation of functional ),( uxF within the interval of
the element ),( 00 ux and checking its sign, we get final answer on optimal ele-
ment UXux ×∈),( 00 .
To find possible optimal solution of task (28)–(30) of control for the objects
with distributed constants, which are described by non-linear differential equa-
tions (parabolic type), we have to solve system of 3 functional equations (29),
(42), (44) with boundary conditions (30), (43) for 3 unknown quantities
0000 ,, ψuyx = .
For the objects with distributed constants, which are described by non-linear
differential equations (hyperbolic type) we have considered the possibility of us-
ing the operator scheme in the absence of restrictions for phase variables and con-
trol functions.
At the same time at first u∀ the non-linear task (45)–(47) is solved, then
(54)–(57). Consequently, a possible optimal solution of task is found by solving a
linear equation system for u,Ψ .
On the other hand, the task of optimal control (45)–(47) with obvious re-
strictions of a kind of integral inequalities generates interest, namely:
let ×Ω×SJi : ℝ×ℝ→ℝ, mi ,...,1= the inequalities
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 4 92
∫ ∫
Ω
=≥=
T
miuxiIdtdtutxtiJ
0
,...,1,0),()),(),,(,,( ωωωω (58)
takes place.
Let us assume, that the functions iJ satisfy the following conditions:
1) the function ×Ω×SJ i : ℝ×ℝ→ℝ is a Borel one for the multitude of
variables;
2) the function ),,,();( uxtJux i ω of class С1 at ℝ 2 Ω×∈∀ St );( ω ;
3) 0, >∈∃ ii cXa , то )|||(|),(||),,,(|| /11 qp
ii uxctauxtJ
i
++≤′ −ωω ,
where );( iuixi JJJ ′′=′ is a gradient of function iJ .
At that conditions the functionals →× )();(: QLXSLI rpi ℝ, mi ,...,1= are
definitely differential and that is why the conditions of the theorem about La-
grange coefficients are satisfied. Then the Lagrange function has the form:
+= ∫ ∫
Ω
T
dtdtvtytJvyF
0
00 )),(),,(,,(),,,( ωωωωλψλ
∑ ∫ ∫
= Ω
++
m
i
T
ii dtdttytJ
1 0
)),(),,(,,( ωωϑωωλ
−
∂
∂
−
∂
∂
+ ∫ ∫ ∑∫ ∫
Ω =Ω
T N
i
T
dtd
t
ytdtd
t
tyt
0 1
2
2
0
2
2
)()()( ωψωψ
∫ ∫∫ ∫
ΩΩ
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
∂
∂
−
TT
N
dtdtgtdtdtyytytQt
00 1
)()()(,,...,),(,,)( ωψωϑ
ωω
ωψ . (59)
It can be found such multitudes ψλ ˆ,ˆ , for which the conditions of Lagrange
function stationary state will be satisfied
0),,ˆ,ˆ(),,ˆ,ˆ( =′+′ uyFuyF uy ψλψλ , (60)
as well as the conditions of sign accordance 0ˆ ≥iλ and the conditions of com-
pletable slackness
miuyiIi ,...,1,0),(ˆ ==λ .
Then, having determined partial variations of function (59), with the help of
(60) an appropriate modification of necessary optimal conditions (55)–(57) can
be obtained.
The obtained results enable to conduct analysis of tasks of optimal control
for the objects with distributed constants, which are described by non-linear dif-
ferential equations of various types, using modern methods of non-linear func-
tional analysis, which imply significant calculations simplification.
Necessary conditions for control of objects with distributed constants
Системні дослідження та інформаційні технології, 2006, № 4 93
REFERENCES
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3. Zgurovsky M.Z., Mel’nik V.S. Nonlinear Analysis and Control of Physical Processes
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Received 28.03.2006
From the editorial Board: The article corresponds completely to submitted manu-
script
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| id | journaliasakpiua-article-154691 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:24:24Z |
| publishDate | 2019 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/53/0ce1e57c871c69e9ecec9505d2ac7f53.pdf |
| spelling | journaliasakpiua-article-1546912019-01-18T15:10:28Z Necessary conditions for control of objects with distributed constants Необходимые условия оптимального управления объектами с распределенными параметрами Необхідні умови оптимального керування об’єктами з розподіленими параметрами Iovane, G. Mizernyy, V. M. Problems of optimal control over objects with distributed constants described by nonlinear differential equations with partial derivatives of elliptic, parabolic and hyperbolic types have been considered. Рассматриваются задачи оптимального управления для объєктов с распределенными параметрами, которые описываются нелинейными дифференциальными уравнениями в частных производных эллиптического, параболического и гиперболического типов. Розглядаються задачі оптимального керування для об’єктів з розподіленими параметрами, які описуються нелінійними диференціальними рівняннями з частинними похідними еліптичного, параболічного та гіперболічного типів. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2019-01-18 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/154691 System research and information technologies; No. 4 (2006); 77-93 Системные исследования и информационные технологии; № 4 (2006); 77-93 Системні дослідження та інформаційні технології; № 4 (2006); 77-93 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/154691/154301 Copyright (c) 2021 System research and information technologies |
| spellingShingle | Iovane, G. Mizernyy, V. M. Необхідні умови оптимального керування об’єктами з розподіленими параметрами |
| title | Необхідні умови оптимального керування об’єктами з розподіленими параметрами |
| title_alt | Necessary conditions for control of objects with distributed constants Необходимые условия оптимального управления объектами с распределенными параметрами |
| title_full | Необхідні умови оптимального керування об’єктами з розподіленими параметрами |
| title_fullStr | Необхідні умови оптимального керування об’єктами з розподіленими параметрами |
| title_full_unstemmed | Необхідні умови оптимального керування об’єктами з розподіленими параметрами |
| title_short | Необхідні умови оптимального керування об’єктами з розподіленими параметрами |
| title_sort | необхідні умови оптимального керування об’єктами з розподіленими параметрами |
| url | https://journal.iasa.kpi.ua/article/view/154691 |
| work_keys_str_mv | AT iovaneg necessaryconditionsforcontrolofobjectswithdistributedconstants AT mizernyyvm necessaryconditionsforcontrolofobjectswithdistributedconstants AT iovaneg neobhodimyeusloviâoptimalʹnogoupravleniâobʺektamisraspredelennymiparametrami AT mizernyyvm neobhodimyeusloviâoptimalʹnogoupravleniâobʺektamisraspredelennymiparametrami AT iovaneg neobhídníumovioptimalʹnogokeruvannâobêktamizrozpodílenimiparametrami AT mizernyyvm neobhídníumovioptimalʹnogokeruvannâobêktamizrozpodílenimiparametrami |