Про моделювання систем змішаного типу в задачах аналізу та керування
The paper presents the research of the mathematical models of mixed systems and considers the principal tasks of analysis, controlling and evaluation of objects’ states parameters, described by nonlinear integral and differential equations with partial derivatives.
Gespeichert in:
| Datum: | 2019 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2019
|
| Online Zugang: | https://journal.iasa.kpi.ua/article/view/165001 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | System research and information technologies |
| Завантажити файл: |  |
Institution
System research and information technologies| _version_ | 1867334350232616960 |
|---|---|
| author | Iovane, G. Mizerny, V. M. |
| author_facet | Iovane, G. Mizerny, V. M. |
| author_institution_txt_mv | [
{
"author": "G. Iovane",
"institution": null
},
{
"author": "V. M. Mizerny",
"institution": null
}
] |
| author_sort | Iovane, G. |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2019-04-23T15:44:17Z |
| description | The paper presents the research of the mathematical models of mixed systems and considers the principal tasks of analysis, controlling and evaluation of objects’ states parameters, described by nonlinear integral and differential equations with partial derivatives. |
| first_indexed | 2025-07-17T10:24:34Z |
| format | Article |
| fulltext |
© G. Iovane, V.M. Mizernyy, 2006
Системні дослідження та інформаційні технології, 2006, № 3 63
TIДC
ТЕОРЕТИЧНІ ТА ПРИКЛАДНІ ПРОБЛЕМИ
ІНТЕЛЕКТУАЛЬНИХ СИСТЕМ ПІДТРИМКИ
ПРИЙНЯТТЯ РІШЕНЬ
UDK 517.9
TO THE QUESTION OF MIXED TYPE SYSTEM SIMULATION IN
THE TASKS OF ANALYSIS AND CONTROL
G. IOVANE, V.M. MIZERNYY
The paper presents the research of the mathematical models of mixed systems and
considers the principal tasks of analysis, controlling and evaluation of objects’ states
parameters, described by nonlinear integral and differential equations with partial
derivatives.
INTRODUCTION
In the context of making tasks on simulation and controlling for complex systems,
which arise in different branches of physics, chemistry, economics and so on, dif-
ferent approaches were developed based upon the conception of mixed systems
[1–4]. The number of control processes in mathematical models, which contain
systems of equations of different types, as for instance differential and integral
equations, contain blocks with distributed and concentrated parameters, multi-
variable and discrete-continued systems [5–6], etc. The progress of research of
mathematical modelling for objects with distributed parameters is due to link to
the great development of nonlinear analysis method, which is applicable in differ-
ent spheres of mathematics [7–9]. Thus, it is quite natural to reduce the study of
these models to nonlinear operator, differential-operator equations, variable ine-
qualities and systems, which contain the above objects. With this approach, the
results for specific objects will be the consequence of operator methods.
TASK SETTING
For the description of some nonstationary processes, which take place in the di-
mensional sphere NR⊂Ω during the time S , we operate with functions of time
and coordinates, that is, with the function z , which brings the actual number of
vector ),( ωtz to conformity with each pair Ω×⊂ St ),( ω . The variables t and ω
are independent.
Another convenient approach to the mathematical description of non-
stationary processes allows to work with functions, which bring the coordinate
function ),( ⋅tz determined on S to conformity with each moment of time t ,
with the determination in some space Z , that is )( ZSz →∈ .
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 3 64
Let us consider some nonstationary tasks, whose description is made with
the help of nonlinear functional equations system.
Task 1. Let Ω be the restricted sphere of NR with regular bound Ω∂ and
the S time interval 0],,0[ >= TTS ; we consider
),())),(),,((,(),(),( wtgdwwtxwtzwhwKtx =+ ∫
Ω
ωω , (1)
),()),(),,(,(),(),(
1,
ωωωωω
ωω
ω tftztxQtzat
t
z
j
ij
N
ji i
=+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂ ∑
=
(2)
,0),(),(),0(,),( StxtzzSt ∈∀==Ω×∈∀ ∑ωγωω (3)
where S×Ω∂=∑ , the factors ija are constant values.
Through Sttz ∈⋅),,( we label a function, designated on Ω×S , which has
the fixed variable t .
The classical solution for the system (1)–(3) is function of ),(),,( ωω tztx ,
designated on Ω×S ; and the function ),( ωtz has to be continuously differenti-
able with respect to t and double differentiable with respect to ω , and should
fulfil the conditions (3). The function ),( ωtz is differentiable with respect to ω .
Typically, proving the theorems of existence for classical solution of tasks
(2)–(3) requires the application of complicated mathematic technique. That is
why the proceeding from the classical task (1)–(3) to the corresponding task in
functional and analytical setting is quite logical. Consequently, let us introduce
some functional spaces with )( ZS → [5] and the following symbols:
),()();,()( ⋅=⋅= tzttxty ψ ,
dwwtxwKtyDtyDtBy ∫
Ω
=⋅= ),(),(),)(();,)(()( ωω ,
( ) ( ) )()(;),(),,(,)(),( tgtbtxtzhtytF =⋅⋅⋅=ψ ,
)),(),,(,())(),(( ⋅⋅⋅= tztxQttyG ψ ,
∑
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
−=⋅=
N
ji j
ij
i
tzatEztEztL
1,
),(),)(();,()( ω
ωω
ωψ , (4)
),()();()0();,()( ⋅=⋅=⋅
∂
∂
=′ tftt
t
zt ϕγψψ .
Considering the above relations, the system (1)–(3) with the initial and
boundary conditions can be represented in the terms of operator equations:
)())(),(()( tbtytFBty =+ ψ , (5)
)())(),(()()( tttyGtLt ϕψψψ =++′ , (6)
)()0( ⋅= γψ . (7)
To the question of mixed type system simulation in the tasks of analysis and control
Системні дослідження та інформаційні технології, 2006, № 3 65
Let us assume, that the abstract functions belong to classes
))(()( Ω→∈ pLSty ,
[ ] ⎟
⎠
⎞⎜
⎝
⎛ Ω→∈Ω→∈
*
)()());(()( m
p
m
p WStWSt ψψ ,
and the operators, which belong to (5)–(6), act according to the rules:
)()()(: Ω→Ω×Ω qp
m
p LLWF ,
)()(: Ω→Ω qp LLB ,
[ ])()(: Ω→Ω m
p
m
p WWL ,
[ ])()()(: Ω→Ω×Ω m
p
m
pp WWLG ,
where G and F are nonlinear reflections, BL, are linear, )(ΩpL is the space of
р-integrable functions, )(Ωm
pW is Sobolev space [5], [ ]*)(Ωm
pW is the integrated
space.
We note that the operators GLBF ,,, do not depend explicitly on vari-
able t .
Task 2. Instead of the system (1)–(3), we consider the following system:
),())),(),,((,(),(),( wtgdwwtxwtzwhwKtx =+ ∫
Ω
ωω , (8)
),()),(),,(,(),(),(
1,
ωωωωω
ωω
ω tftztxQtzat
t
z N
ji j
ij
i
=+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
−
∂
∂ ∑
=
(9)
,0),(;)(),0(;),( StxtzzSt ∈∀==Ω×∈∀ ∑ωγωω (10)
where the factors ija are constant values.
We introduce the following functions and expressions:
),()();,()();,()( ⋅
∂
∂
=′⋅=⋅= t
t
zttzttxty ψψ ,
dwwtxwKtYtDtYtDtytB ∫
Ω
⋅=⋅⋅= ),(),,(),(),();,(),()()( ωωω ,
),()());,(),,(,,())(),()(( ⋅=⋅⋅⋅= tgtbtxtzthtyttF ψ ,
)),(),,(,,())(),(()( ⋅⋅⋅= tztxtQttytG ψ ,
∑
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
−=⋅=
N
ji j
ij
i
tzatzEtzEtL
1,
),(),()(),,()()( ω
ωω
ωψ ,
),()();()0( ⋅=⋅= tftϕγψ
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 3 66
and from (8)–(10) we proceed to the system of operator equations:
)())(),(()()()( tbtyttFtBty =+ ψ , (11)
)())(),(()()()( tttytGtLt ϕψψψ =++′ , (12)
γψ =)0( , (13)
where the operators, which act on abstract functions, depend on variable t explic-
itly. In detail, )(),(),( tGtFtB act in the spaces, determined by the task (5)–(7).
In the given example )( *ZSZ →∈′ shall be understood as follows: the de-
rivative from Z with the respect to t in the sense of space devision ),( ** ZSD can
be represented with the help of the function )( *ZS → [5].
Task (1)–(3) or (8)–(10) with the initial and boundary conditions and the
corresponding tasks (5)–(7) and (11)–(13) are equivalent in some sense.
This can be proved with the application of such lemma:
Lemma 1 [5]. The following relation
Sttzt ∈∀⋅= ),()(ψ (14)
determines the mutual correspondence in a unique fashion ψ→Z between
the function )( Ω×∈ SCz and the function ))(;( Ω∈ CSCψ . The function
)( Ω×∈ SCz has partial derivative )( Ω×∈
∂
∂ SC
t
z and )(),( tt
t
z ψ ′=⋅
∂
∂ for each a
St∈ .
Let us consider the conditions for a possible realization of the models of
mixed objects, which are described by system of nonlinear integer equations and
evolution equations with integro-differential operators.
Let Ω be a restricted sphere in Euclidean space NR and ZX , be Banach
spaces of functions on Ω with the norm X⋅ , Z⋅ and [ ]TS ,0= , 0>T .
Let us consider the equation system
∫
Ω
=+ ),()),(),,(,,(),(),( ωωω tgdwwtxwtzwthwKtx , (15)
=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
∂
∂
∂
∂
−
∂
∂
∫
Ω
dw
w
z
w
z
w
z
w
zwtzwtxtQwL
t
tz
NN
2
2
2
1
2
1
,...,,,...,),,(),,(,,),(),( ωωω
),( ωtp= (16)
Ω∂×=Σ=∈=⋅Ω×∈∀ Σ T)(0, ,0),( ,Z)(0, ,),( 0 ωω tzZzSt , (17)
which we will write [5] in the kind of operator
)())(),(()()( ttyttBFty ϕψ =+ , (18)
S )())(),()(()(' ∈∀=+ ttfttytGt ψψ (19)
with the initial condition γψ =)0( .
To the question of mixed type system simulation in the tasks of analysis and control
Системні дослідження та інформаційні технології, 2006, № 3 67
We stress that in the first equation the variable t acts as a parameter.
Here { })(tFF = , { })(tGG = , St ∈ are in the families of nonlinear opera-
tors, acting in spaces *:)( XXZtF →× , S :)( ∈∀→× tZZXtG , and the op-
erator XXB →*: is a linear one. The functions )(tyt → and )(tt ψ→ are de-
termined for St ∈ and belong to spaces );( XSC and );( ZSC accordingly.
Let us assume that the family of operators { })t(FF = , { })t(GG = satisfies
the following conditions:
а) for each a Xy ∈ and );( ZSC∈ψ , the function ∈∋ )),()(( yttFtS ψ
*X∈ is of class );( *XSC ;
b) for each a Zz ∈ and );( XSCh∈ , the function ∈∋ )),()(( zthtGtS
Z∈ is of class );( ZSC ;
c) the operators )(),()( *XXtF →∈⋅ψ and )(),()( ZZytG →∈⋅ are equally
continuous, that is, there are the constants 1r and 2r , which are independent
from t , and follow the condition
XyyyyrytFytF
X
∈∀∈∀−≤− 2121121 ,Z, ),()(),)((
X*
ψψψ
and
ZXyrytGytG
ZZ
∈∀∈∀−≤− 2121221 ,, ),()(),()( ψψψψψψ .
Let us assume that { }))(),(()(),( ttytGtyG ψ= .
Lemma 2. Let the family of operators { })(tGG = satisfy the conditions b),
c) );( XSCy∈∀ and );( ZSC∈ψ .
Then );(),( ZSCyG ∈ψ .
Proof. Let { } Stn ⊂ be any sequence, with 0ttn → when ∞→n . By virtue
of execution of condition c)
−− ))(),()(())(),(()( 000 ttytGttytG nnn ψψ
≤+−
Z
ttytGttytG nnnn ))(),()(())(),(()( 00 ψψ
.))(),()(())(),()(()()( 000002 ZZ
ttytGttytGttr nnn ψψψψ −+−≤
When 0ttn → the augend in the right summand tends to zero due to the con-
tinuous function ψ, and the addend tend to zero according to the condition b).
This proves for lemma.
The analogical lemma can be formed as for the family of operators
{ })(tFF = .
Lemma 3 [5]. The norm
{ } 0 ,)(sup),( ≥= −
∈
kte
Z
kt
St
kC ψψ , (20)
is equivalent to the norm
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 3 68
Z
t
St
ZSC )(sup);( ψψ
∈
= .
Theorem 1. Let the conditions а)÷c), related to the family of the operators
F , G , come true and the operator В be linear and continuous, whose norm satis-
fies the inequality
1
1
r
B < , where const1 =r is constant.
Then the task
)())(),(()()( ttyttBFty ϕψ =+ ,
)())(),(()()(' tfttytGt =+ ψψ
with the initial condition γψ =)0( has solution for any );( ZSC∈ϕ , );( ZSf ∈
and Z∈γ .
Proof. We obtain the previous result by using the principle of fixed point.
Integrating the second system equation on the interval ];0[ t , we obtain
[ ]∫ −−=
t
dssfssysGt
0
)())(),(()()( ψγψ . (21)
The integral here is considered in the sense of Bohner. We designate
[ ]∫ −−=
t
dssfsssGtU
0
)())(),(()())(( ψγψ , (22)
))(),()(()()()(0 tyttBFttytU ψϕ −= . (23)
By virtue of lemma 2 and differentiality of an indeterminate integral of
Bohner [5], the operator U acts from );();( ZSCXSC × into );(1 ZSC .
We show that with each );( XSCy∈ the reflection of U with some 0≥k
is compressed into );( kC — to the norm of the space );( ZSC .
According to c), from (22) it follows that for any );(, 21 ZSC∈ψψ
−≤− ∫
t
ssysGtUtU
Z
0
121 ))(),(()())(())(( ψψψ
≤−≤⋅− ∫ −−
t
ksksksks dseerdseessysG
ZZ
0
2122 ))(),()(( ψψψ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
−≤
k
er
kt
kC
1
),(212 ψψ .
Then
≤−−≤− −−
),(21
2
21 )1())(())((
kCZ
ktkt e
k
r
etUtU ψψψψ
),(21
2 )1(
kC
kte
k
r ψψ −−≤ − .
To the question of mixed type system simulation in the tasks of analysis and control
Системні дослідження та інформаційні технології, 2006, № 3 69
Taking in the left part the upper bound on the St ∈ , we obtain
.)1(
),(21
2
21 kCZ
kte
k
r
UU ψψψψ −−≤− −
If we choose 2rk ≥ , the reflection of U will be compressed. So for each
);( XSCy ∈ there is the element );(0 ZSC∈ψ , which is fixed point of reflection
U , that is 00 ψψ U= .
By the virtue of (22) we have
[ ]∫ ∈∀−−=
t
StdssfssysGt
0
0 )())(),()(()( ψγψ . (24)
Considering that the right part of this expression has the continuous deriva-
tive on t, then );(1
0 ZSC∈ψ and
SttfttytGt ∈∀=+ )())(),(()()( 00 ψψ ,
and )()0(0 ωγψ = .
Then we consider the operator 0U with the fixed ψ . According to c) from
(23) it follows that for );(, 21 XSCyy ∈∀
=−=−
XX
tyttBFtyttBFtytUtytU ))(),(()())(),(()()()()()( 212010 ψψ
[ ] −⋅≤−= ))(),()(())(),(()())(),()(( 121 tyttFBtyttFtyttFB
X
ψψψ
XX
tytyrBtyttF )()())(),(()( 2112 *
−⋅⋅≤− ψ .
If
11 <⋅ rB , (25)
then the operator 0U will be compressed in space );( XSC with each
);( ZSC∈ψ .
In the system of equations,
( ))(),()()()( tyttBFtty ψϕ −= , (26)
[ ]∫ −−=
t
o
dssfssysGt )())(),(()()( ψγψ , (27)
which is received from task (18)–(19) with the initial condition γψ =)0( , assum-
ing that 1yy = , 1ψψ = , where );( 11 ψy — is any pair from );();( ZSCXSC × ,
we have
))(),(()()()( 112 tyttBFtty ψϕ −= .
Then we substitute the pair );( 22 ψy into the equation (13). Consequently,
we get
[ ]∫ −−=
t
dssfssysGt
0
122 )())(),(()()( ψγψ .
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 3 70
We substitute the pair );( 22 ψy into (26). Having determined 3y , we substi-
tute the pair );( 23 ψy into (27). Then we determine 3ψ . By repeating the previous
procedure, we get an iteration process
))(),(()()()(1 tyttBFtty nnn ψϕ −=+ (28)
and
[ ]∫ −−=+
t
nnn dssfssysGt
0
1 )())(),()(()( ψγψ . (29)
We prove that the succession { } { }nn ,y ψ converge to the fixed point
);( 00 ψy , which is the solution of the system (26)–(27) and, as a consequence, of
the system (18)–(19).
With any n we have
=−=− −+ XX
tytUtytUtyty nnnn )()()()()()( 1001
=−= −− X
tyttBFtyttBF nnnn )))(),(()()(),(()( 11ψψ
[ ] ≤−= −− X
tyttFtyttFB nnnn ))(),(()())(),()(( 11ψψ
≤−⋅≤ −− *
))(),(()())(),(()( 11
X
tyttFtyttFB nnnn ψψ
XX
tytytytyrB nnnn )()()()( 111 −− −=−≤ α ,
where 11 <= rBα .
Generally, the following chain of inequalities holds:
XX
tytytytytyty n
nnnn )()(...)()()()( 12
1
11 −≤≤−≤− −
−+ αα .
We show that the sequence { }ny is fundamental for );( XSC . Using the ine-
quality of triangle and previous inequalities with m>n, we have
≤−+−+−+−=− +−−−− );(12211);( ... XSCnnmmmmmXSCnm yyyyyyyyy
≤−++−+−≤ +−−− );(1);(21);(1 ... XSCnnXSCmmXSCmm yyyyyy
=−++−+−≤ −−−
);(12
1
);(12
3
);(12
2 ... XSC
n
XSC
m
XSC
m yyyyyy ααα
.)...( );(12
132
XSC
nmm yy −+++= −−− ααα
We designate the expression between brackets through
231
1 ... −−− ++++=Σ mmnn αααα .
Analogically through
...... 231
2 ++++++=Σ −−− mmmnn ααααα .
we designate the infinite sum.
To the question of mixed type system simulation in the tasks of analysis and control
Системні дослідження та інформаційні технології, 2006, № 3 71
Then
...1
121 +++Σ=Σ<Σ − mm αα ,
where 2∑ is the sum of members of infinite decreasing geometric progression,
whose first member is 1−nα with the denomination 1<α .
As well known,
α
α
−
=Σ
−
1
1
2
n
,
then
≤−Σ≤−Σ≤− );(122);(121);( XSCXSCXSCnm yyyyyy
.
1 );(12
1
XSC
n
yy −
−
≤
−
α
α
Considering that with ∞→n the value in the right part of inequality tends
to zero with any nm > , then 0);( →− XSCnm yy thus, the consequence { }ny is
fundamental. In Banach space );( XSC the fundamental consequence { }ny has
the limit 0y . In analogical way, we obtain the sequence { }nψ , fundamental in
);( XSC with the limit 0ψ .
Proceeding to the limit ∞→n in the equations (28)–(29) and considering
the condition а) (it needs in the condition of the given theorem), and using the fact
that the operator B in the first equation and integral operator in the second equa-
tion are linear and continuous, we obtain:
))(),(()()( 000 tyttBFty ψϕ −= ,
[ ]∫ −−=
t
dssfssysG
0
000 .)())(),(()( ψγψ
Thus, the theorem is proved.
Let us consider the system of integro-differential equations of the following
type
∫
Ω
=+ ),()),(),,(,,(),,(),( ωωω tgdwwtxwtzwthwtKtx , (30)
),(
,...,...,
,...),,(),,(,,
),(),(
2
2
2
1
2
1
ωωω tpdw
w
z
w
z
w
z
w
zwtzwtxwt
QwLt
t
z
NN
=
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
∂
∂
∂
∂
∂
∂
∂
∂
−
∂
∂
∫
Ω
(31)
0)(t, Z,)(0, ,),( =∈=⋅Ω×∈∀ Σωγω zzSt , (32)
or in the operator form
)())(),(()()()( ttyttFtBty ϕψ =+ , (33)
S )())(),(()()(' ∈∀=+ ttfttytGt ψψ (34)
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 3 72
with the initial condition γψ =)0( .
We stress that in the first equation the variable t acts as a parameter.
The family of nonlinear operators
{ } { } SttGGtFF ∈== ,)( ,)(
are the same as in (4)–(5).
XXtB →*:)( explicitly depend on St ∈ .
Let F and G satisfy the conditions а)÷c), and the family of operators )(tB
is such that St
r
tB ∈∀< 1)(
1
, where the constant 1r comes from condition c).
For the system (33)–(34) with the initial condition γψ =)0( such statement
comes true.
Theorem 2. Let us assume that the conditions a)÷c) come true as for the
family of operators )(tF , )(tG and the norm of the family of linear operators
)(tB satisfy the inequality St
r
1)t(B
1
∈∀< , where const1 =r from the condi-
tion c). Then the differential-operator system
)())(),(()()()( ttyttFtBty ϕψ =+ ,
)())(),(()()(' tfttytGt =+ ψψ
with initial condition γψ =)0( has a solution in );();( ZSCXSC × for any
);( ),;( ZSCfXSC ∈∈ϕ and Z∈γ .
The proof of theorem 2 is obtained by the analogy with theorem 1.
Let us consider the conditions of possible realization of mixed objects mod-
els, which contains evolution equations of the second order.
In the restricted domain nR⊂Ω with the bound Ω∂ on the interval
),0( TS = , the system of equations is searched
( )∫
Ω
=+ ),(),(),,(,,)(),( ωωω tgdwwtxwtzwthKtx , (35)
),,(,...,,...,
,...),,(),,(,,),(),(
2
2
2
1
2
1
2
2
ω
ωωω
tpdw
w
z
w
z
w
z
w
zwtzwtxtQwL
t
tz
NN
=⎟
⎟
⎠
⎞
∂
∂
∂
∂
∂
∂
⎜⎜
⎝
⎛
∂
∂
−
∂
∂
∫
Ω (36)
where Ω×∈St ),( ω , with initial and boundary conditions
,),0(),(),0( 10 ωγωωγω =
∂
∂
=
t
zz (37)
Ω∂×=Σ=Σ Stz ,0),( ω . (38)
To the question of mixed type system simulation in the tasks of analysis and control
Системні дослідження та інформаційні технології, 2006, № 3 73
In the first equation the variable t acts as a parameter. Through ZX , we
designate the true functional spaces and consider the families { }SttFF ∈= ),(
{ }SttGG ∈= ),( of nonlinear operators, which act in spaces *:)( XXZtF →× ,
ZZXtG →×:)( . As in the previous cases we rewrite the task (35)–(38) in op-
erator form
( ) )()(),()()()( ttyttFtBty ϕψ =+ , (39)
( ) )()(),()()('' tfttytGt =+ ψψ St∈∀ (40)
with the initial conditions
1)0(',)0( γψγψ == , (41)
where XXtB →*:)( is the family of linear reflections.
Task (39)–(40) with initial conditions (41) can be brought to the task, which
was considered in the theorem 2, by introducing designation )()(' tqt =ψ . Here
instead of the families of the operators )(tF , )(tG , )(tB , which act as
*:)( XXtF → , XXtB →*:)( , ZZtG →:)( , we consider the following opera-
tors )()(: ZSZSG →→→ з ( ));();( ZSLZSL qp → , 111, =+> qpqp .
Thus, we get more common task setting, considering that each family { })(tG
of operators with )( ZZ → can be given the correspondent one trajectory operator
( ))()( ZSZSG →→→∈ according to rule SttztGtzG ∈∀= )()()()( .
Not each operator )()(: ZSZSG →→→ are interesting for our purpose.
An example is given by the operators, which contain the so-called Volterra’s op-
erators a ( ))()( ZSZSG →→→∈ (which play an important role in different
practical appendixes). They are characterized by the fact that the value ))(( tGz
can depend upon the values of function z in the interval ];0[ t , that is on “previous
history”.
Considering the above question, let us give more modified task setting. Let
Z be a reflective Banach space, continually and snugly put into the Hilbert space
H and the operator of Volterra G works as
111,1),;();(: * =+>→ qppZSLZSLG qp , );( *ZSLf q∈ .
Then the following task setting will be right: HfG ∈==+ γψψψ )0(,' .
Indeed, considering that *ZHZ ⊂⊂ (see [5], paragraph 6, ch. 1), from
);( ZSLp∈ψ it follows that );( ** ZSD∈ψ . That is why the equation
fG =+ ψψ ' can be understood as the equation in );( ** ZSD . If );( ZSLp∈ψ
satisfies the given equation, then );(' *zSLq∈ψ (see [5], theorem 1.17, ch. 4)
);( HSC∈ψ , that is the initial condition H∈= γψ )0( makes sense. The results
of theorems 1, 2 can be generalized for the specific case, when instead of opera-
tors from family { } SttG ∈,)( there are the operators of Volterra’s type. The
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 3 74
meaning of this operator can be specified more wider than in the previous consid-
erations.
Denotation 1 [5]. Let 21, ZZ be linear spaces and [ ] 0,;0 >= TTS . The re-
flection ( ) )()(,)()( 12 ZSGDZSGDG →⊂→→∈ is called operator of Volter-
rif from the equation )()( ss ϕψ = almost for all [ ] Stts ∈∈ ,;0 , it follows that
))(()()( sGsG ϕψ = .
Firstly, we consider operators of Volterra, which reflect the space
);( ZSC into themselves. For this case in denotation 1 it needs to put
ZZZ == 21 and );()( ZSCGD = , and the expression “almost for all” change for
“for all”.
In this case the condition of Lipshitz for the operator of Volterra G looks
like:
);(, 21 ZSC∈∀ ψψ
( ) const, 2;212);(21 =−≤− rrGG ZSCZSC ψψψψ . (42)
As for the system of operator equations, which are under consideration, the
generalization of the conditions (a) and (b) shall be:
(aa) for each );( XSCy∈ and );( XSC∈ψ the functions ×→ )(tFt
( ))(),()( tytt ψ× and ( ))(),()( ttytGt ψ→ are determined St∈∀ and belong to
);( *XSC , );( ZSC accordingly.
(bb) for );( XSCy∈ , );( ZSC∈ψ the operators ( →∈⋅ );(),( ZSCyG
));( ZSC→ , ( ));();(),( *XSCXSCF →∈⋅ψ and equally of Lipshitz, that is there
are the constant 1r and 2r , that the conditions come true:
);(211);(21 *),(),( XSCXSC yyryFyF −≤− ψψ
);(,),;( 21 ZSCyyZSC ∈∀∈∀ψ ,
);(212);(21 ),(),( ZSCZSC ryGyG ψψψψ −≤−
);(,),;( 21 ZSCXSCy ∈∀∈∀ ψψ .
We give for example the known statement, which is used with the applica-
tion of principle of fixed point to the tasks under consideration.
Lemma 4 [5]. If the operators G satisfy the conditions (42), then for any
);(, 21 ZSC∈ψψ and St∈∀
[ ] [ ] );;0(212);;0(21 ZtCZtC rGG ψψψψ −≤− , const2 =r .
Let us give the example for Volterra’s operators, which satisfy (42).
1. Let tthSCh ≤≤∈ )(0),( for St∈∀ and { }SttQ ∈),( be the family of
operators from XX → , which satisfies the conditions:
• for each Xx∈ the function xtQt )(→ is determined for St∈ and be-
longs to );( XSC ;
To the question of mixed type system simulation in the tasks of analysis and control
Системні дослідження та інформаційні технології, 2006, № 3 75
• the operators { } )()( XXtQ →∈ (equally-relatively St∈ ) is Lipshitz’s
continuous, i.e. there is such a constant r (independent on t), so for any Xyx ∈,
Lipshitz condition comes true.
XX yxrytQxtQ −≤− )()( . (43)
If we put
( ))()()()( thutQtuQ = , );( XSCu∈ ,
then Q will be Lipshitz’s-continuous operator of Volterra from ( →);( XSC
));( XSC→ .
2. Let the operator of Volterra V satisfy the conditions (42). Then for
);( XSCu∈ we have
StSSCkdssuVstktuQ
t
∈×∈= ∫ ),(,)()(),())(( 1
0
;
the operator Q will also be an operator of Volterra, which satisfies (8).
3. Let Q , V be operators of Volterra, which satisfy (8). Then their linear
combinations and composition VQ have the same properties.
The following theorem is true.
Theorem 3. Let GF , be operators of Volterra, which satisfy the conditions
(aa), (bb), and the family of linear operators )(tB satisfy the inequality <)(tB
St
r
∈∀<
1
1 , 1r — Lipshitz’s constant from (bb).
Then the task
( ) StttyFtBty ∈∀=+ )()(),()()( ϕψ , (44)
( ) ),()(),()(' tftyGt =+ ψψ (45)
);(,)0( 1 ZSC∈= ψγψ (46)
has the solution );();(),( 1 ZSCXSCy ×∈ψ for any );( XS∈ϕ , );( ZSCf ∈ and
Z∈γ .
Proof. Due to the fact that in this case the principle of fixed point is used
(but in other Banach spaces), the proof of the theorem is conducted analogically
to the proof of the theorem 1. Integrating equations (45) in ];0[ t and considering
the initial conditions we obtain:
( )[ ]∫ −−=
t
dssfsyGt
0
)()(),()()( ψωγψ ,
);(),;(, ZSCXSCySt ∈∈∈ ψ .
Here the integral is considered in the sense of Bohner.
We introduce the operator U , which follows the rule
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 3 76
( )[ ]∫ −−=
t
dsSfsyGtU
0
,)()(),()()( ψγψ (47)
where the function );( ZSCf ∈ and the operator ( ));();( ZSCZSCG →∈ . Pro-
ceeding from theorem on differentiality of nondesignated integral of Bohner (see
[5], theorem 1.9, ch. 4), we obtain ( ));();( 1 ZSCZSCU →∈ .
We mark in equation (44)
( ) SttyFtBttyU ∈−= ),(),()()()()( 0 ψϕ . (48)
We show that U , as the reflection of space );( ZSC in itself, with some
0≥k , is compressed in ),( kC -norm, which is determined in lemma 2.
By the virtue of condition (bb) from (47), it follows that for any
);(, 21 ZSC∈ψψ
=−≤− ∫
t
ZSCZSC dsyGyGUU
0
);(21);(21 ),(),( ψψψψ
{ } ≤−= ∫ −
≤≤
t
ksks
Z
s
dseer
0
21
0
2 )()(sup τψτψ
τ
[ ]{ } =−≤ ∫ −
≤≤
t
ksks
ZsC
s
dseer
0
);;0(21
0
2 sup ψψ
τ
)1(1
),(212
0
),(212 −−≤−= ∫ kt
kC
t
ks
kC e
k
rdser ψψψψ .
Having multiplied both parts of this inequality by kte− , we obtain
kC
kT
kC
ktkt
ZSC e
k
r
e
k
r
eUU ,21
2
),(21
2
);(21 )1()1( ψψψψψψ −−≤−−≤− −−− .
Considering in the left part the upper margine as for St∈ , we obtain
),(21
2
),(21 )1( kC
kT
kC e
k
rUU ψψψψ −−≤− − .
If we choose 2rk ≥ , then the reflection U in ),( kC -norm will be com-
pressed, i.e. there is an element );(0 ZSC∈ψ , which is a fixed point for this re-
flection:
00 ψU=Ψ .
Considering (47), we obtain
( )[ ] StdssfsyGt
t
∈∀−−= ∫
0
00 )()(),()( ψγψ . (49)
To the question of mixed type system simulation in the tasks of analysis and control
Системні дослідження та інформаційні технології, 2006, № 3 77
Due to the fact that the right part of this equation has the continuous deriva-
tive as for t , then );(1
0 ZSC∈ψ and ( ) )()(),()(' 00 tftyGt =+ ψψ St∈∀ ,
γψ =)0(0 .
We consider the conditions under which the operator 0U will be com-
pressed.
According to (bb) from (48), it follows that );(, 21 XSCyy ∈∀
( ) ( ) =−=− XX tyFtBtyFtBtyUtyU )(),()()(),()()()( 212010 ψψ
[ ] ( ) ( ) ≤−⋅≤−= *)(),()(),()()(),()(),()( 2121 XX tyFtyFtBtyFtyFtB ψψψψ
StyyrtB XSC ∈∀−≤ );(211)( .
Following the condition of the theorem
1
1)(
r
tB < , the operator 0U is com-
pressed.
In the system of the equation
( ) )(),()()( tyFtBty ψϕ −= , (50)
( )[ ]∫ −−=
t
dssfsyGt
0
)()(),()( ψγψ , (51)
giving in the right part of (50) the values 11, ψψ == yy , where ),( 11 ψy is
some pair, we get ( ) )(),()()()( 112 tyFtBtty ψϕ −= , from which −= γψ )(2 t
( )[ ]∫ −−
t
dssfsyG
0
12 )()(),( ψ .
Analogically, substituting the obtained pair ),( 22 ψy , we find 3y . Then,
substituting the pair ),( 23 ψy in (51), we find 3ψ . Repeating such procedure, we
find
( ) )(),()()(1 tyFtBty nnn ψϕ −=+ (52)
and
( )[ ] .)()(),()(
0
11 ∫ −−= ++
t
nnn dssfsyGt ψγψ (53)
Then, like in theorem 1, we have proved that the sequences { } { }nny ψ, are
fundamental and, as a result, proceed to the fixed point { }00 ,ψy , which is the
solution for the system of equation under consideration.
CONCLUSIONS
In this paper we have shown some results in the context of the mathematical
models of mixed systems. They are the necessary base when setting and solving
G. Iovane, V.M. Mizernyy
ISSN 1681–6048 System Research & Information Technologies, 2006, № 3 78
tasks for the optima control and the evaluation of parameters of objects states,
which are described trough nonlinear initial-boundary tasks for integral and dif-
ferential equations and their systems with partial derivatives, developing methods
and algorithms of regularization of optimization tasks, composing termi-
nal-measurable approximations and averaging-out schemes, synthesis of applied
control systems for different processes etc.
REFERENCES
1. Akbarov D., Mizernyy V.M. Optimization problems for the objects described by the
system of operating equations of the Hammerstein type // Modelling and optimi-
zation of distributed parameter systems with applications to engineering. — War-
saw, Poland. — 1995. — P. 6–7.
2. Mizernyy V.M., Yasinsky V.V. Tasks and methods of control of mixed systems. Pro-
ceedings of 5th International scientific and technical conference «Control in
Complex Systems». — Vinnytsia. — 1999. — Part 1. — P. 89–93.
3. Akbarov D.E., Mel’nik V.S., Yasinsky V.V. Methids for controlling over mixed sys-
tems. Operator approach. — K.: Vyriy, 1998. — 224 p.
4. Mizernyy V.M. Analysis of the tasks for optimum control over singular mixed sys-
tems // Naukovi visti NTUU «KPI». — 2005. — № 2. — P. 41–47.
5. Gaevsky K., Greger K., Zakharias K. Nonlinear operator equations and operator dif-
ferential equations. — М.: Mir, 1978. — P. 336.
6. Ivanenko V.I., Mel’nik V.S. Variation methods in the control tasks for systems with
distributed constants. — K.: Nauk. dumka, 1988. — 288 p.
7. Zgurovsky M.Z., Mel’nik V.S. Nonlinear analyses and controlling over infinite meas-
urable systems. — K.: Nauk. dumka, 1999. — 630 p.
8. Zgurovsky M.Z., Mel’nik V.S., Novikov A.N. Applied methods for analyses and con-
trolling over nonlinear processes and fields. — K.: Nauk. dumka, 2004. — 590 p.
9. Zgurovsky M.Z., Mel’nik V.S. Nonlinear Analysis and Control of Physical Processes
and Fields. — Berlin; Heidenberg; New York: Springer-Verlag, 2004. — 508 p.
Received 28.03.2006
From the Editorial Board: The article corresponds completely to submitted manu-
script.
|
| id | journaliasakpiua-article-165001 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:24:34Z |
| publishDate | 2019 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/0c/fccb437c862a483d840f99535e14300c.pdf |
| spelling | journaliasakpiua-article-1650012019-04-23T15:44:17Z To the question of mixed type system simulation in the tasks of analysis and control О моделировании систем смешанного типа в задачах анализа и управления Про моделювання систем змішаного типу в задачах аналізу та керування Iovane, G. Mizerny, V. M. The paper presents the research of the mathematical models of mixed systems and considers the principal tasks of analysis, controlling and evaluation of objects’ states parameters, described by nonlinear integral and differential equations with partial derivatives. Приведены результаты исследования математических моделей смешанных систем. Рассмотрены основные задачи анализа, управления и оценивания параметров состояний объектов, которые описываются нелинейными интегральными и дифференциальными уравнениями в частных производных. Наведено результати дослідження математичних моделей змішаних систем. Розглянуто основні задачі аналізу, керування та оцінювання параметрів станів об’єктів, які описуються нелінійними інтегральними та диференціальними рівняннями з частинними похідними. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2019-04-23 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/165001 System research and information technologies; No. 3 (2006); 63-78 Системные исследования и информационные технологии; № 3 (2006); 63-78 Системні дослідження та інформаційні технології; № 3 (2006); 63-78 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/165001/164035 Copyright (c) 2021 System research and information technologies |
| spellingShingle | Iovane, G. Mizerny, V. M. Про моделювання систем змішаного типу в задачах аналізу та керування |
| title | Про моделювання систем змішаного типу в задачах аналізу та керування |
| title_alt | To the question of mixed type system simulation in the tasks of analysis and control О моделировании систем смешанного типа в задачах анализа и управления |
| title_full | Про моделювання систем змішаного типу в задачах аналізу та керування |
| title_fullStr | Про моделювання систем змішаного типу в задачах аналізу та керування |
| title_full_unstemmed | Про моделювання систем змішаного типу в задачах аналізу та керування |
| title_short | Про моделювання систем змішаного типу в задачах аналізу та керування |
| title_sort | про моделювання систем змішаного типу в задачах аналізу та керування |
| url | https://journal.iasa.kpi.ua/article/view/165001 |
| work_keys_str_mv | AT iovaneg tothequestionofmixedtypesystemsimulationinthetasksofanalysisandcontrol AT mizernyvm tothequestionofmixedtypesystemsimulationinthetasksofanalysisandcontrol AT iovaneg omodelirovaniisistemsmešannogotipavzadačahanalizaiupravleniâ AT mizernyvm omodelirovaniisistemsmešannogotipavzadačahanalizaiupravleniâ AT iovaneg promodelûvannâsistemzmíšanogotipuvzadačahanalízutakeruvannâ AT mizernyvm promodelûvannâsistemzmíšanogotipuvzadačahanalízutakeruvannâ |