Stages and main tasks of the century-long control theory and system identification development. Part X. Positional conflict-controlled processes
У статті розглядаються конфліктно-керовані процеси, які описуються узагальненими квазілінійними системами диференціальних рівнянь. Досліджується ігрова задача зближення із заданою термінальною множиною циліндричного вигляду за наявності позиційної інформації щодо стану гри. The paper concerns the co...
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| Zitieren: | Stages and main tasks of the century-long control theory and system identification development. Part X. Positional conflict-controlled processes / А. Chikrii, V. Gubarev, V. Romanenko // Проблемы управления и информатики. — 2025. — № 6. — С. 5-28. — Бібліогр.: 23 назв. — англ. |
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| citation_txt | Stages and main tasks of the century-long control theory and system identification development. Part X. Positional conflict-controlled processes / А. Chikrii, V. Gubarev, V. Romanenko // Проблемы управления и информатики. — 2025. — № 6. — С. 5-28. — Бібліогр.: 23 назв. — англ. |
| collection | DSpace DC |
| container_title | Проблеми керування та інформатики |
| description | У статті розглядаються конфліктно-керовані процеси, які описуються узагальненими квазілінійними системами диференціальних рівнянь. Досліджується ігрова задача зближення із заданою термінальною множиною циліндричного вигляду за наявності позиційної інформації щодо стану гри.
The paper concerns the conflict-controlled processes, described by the generalized quasi-linear systems of differential equations. We study the game problem of approaching the terminal set of a cylindrical form, under the positional information on the game state.
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| first_indexed | 2026-03-14T06:31:15Z |
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© А. CHIKRII, V. GUBAREV, V. ROMANENKO, 2025
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 6 5
МЕТОДИ ОПТИМІЗАЦІЇ ТА ОПТИМАЛЬНЕ КЕРУВАННЯ
UDC 62-50
А. Chikrii, V. Gubarev, V. Romanenko
STAGES AND MAIN TASKS
OF THE CENTURY-LONG CONTROL
THEORY AND SYSTEM IDENTIFICATION
DEVELOPMENT. Part X. POSITIONAL
CONFLICT-CONTROLLED PROCESSES
Arkadii Chikrii
V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Kyiv,
https://orcid.org/0000-0001-9665-9085
g.chikrii@gmail.com
Vyacheslav Gubarev
Space Research Institute of the NAS of Ukraine and SSA of Ukraine, Kyiv,
https://orcid.org/0000-0001-6284-1866
v.f.gubarev@gmail.com
Viktor Romanenko
Education and Research Institute for Applied Systems Analysis of National Technical
University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»,
romanenko.viktorroman@gmail.com, ipsa@kpi.ua
The paper concerns the conflict-controlled processes, described by the ge-
neralized quasi-linear systems of differential equations. We study the game
problem of approaching the terminal set of a cylindrical form, under the po-
sitional information on the game state. It is assumed that, in the solution
presentation (an analog of the Cauchy formula), the block of initial data is
separated from the control block. Due to this, it becomes possible to consi -
der the wide class of the game problems in a unique scheme. The research
method is based on the extreme targeting rule. We study both the regular
and the regularized cases. In so doing, the latter is formalized in the frames
of differential inclusions. The general technique is applied to solving the
game problems, in which the process dynamics is described by the integral,
integral-differential equations. The theoretical results are illustrated by the
model examples of game problems.
Keywords: extreme targeting rule, support function, set-valued mapping, measu-
rable choice, integral and integral-differential equations, differential-difference
systems, impulse systems, processes with fractional derivatives.
The strategy of creating solution (control choice) based on information about the cur-
rent position of the dynamic process goes back to the classic works of N.N. Krasovskii.
mailto:g.chikrii@gmail.com
mailto:v.f.gubarev@gmail.com
mailto:romanenko.viktorroman@gmail.com
mailto:ipsa@kpi.ua
6 ISSN 2786-6491
These include his monograph [1] and the subsequent book coauthored with A.I. Sub-
botin [2]. The employment of the extremal targeting rule, developed in [1], allows for
the game termination in the «first absorption» time in the regular and regularized cases.
Similar result was obtained by B.N. Pshenichnyi [3] usіng the convex analysis tech-
nique. On the one hand, this method is the result of extension of the Pontryagin maxi-
mum principle to the game problems. On the other hand, it justifies the Euler law of
goal interception, known as the «line of sight law».
The extremal targeting rule appeared to be very fruitful. Many scientists did re-
search in this direction. B.N. Pshenichnyi, Yu.N. Onopchuk adapted this method
to the case of integral constraints on controls [4]. G.Ts. Chikrii extended the result
of B.N. Pshenichnyi [3] to the non-stationary case [5]. J.S. Rappoport studied the case
of mixed controls. Positional conflict control based on systems of integral and integro-dif-
ferential equations was studied in the papers of V.L. Pasikov, G.Ts. Chikrii and
K. Volyanskij [6]. Complex positional problems of the group pursuit were first stated
and solved in the works of S.I. Tarlinskij [7], A.A. Chikrii [8], A.A. Chikrii and
J.S. Rappoport [9]. A.Ye. Perekatov and A.A. Chikrii [10] examined a positional pursuit
problem involving a group of evaders by controlled moving object (the problem
of «commercial traveler» type).
The investigations of M.S. Gabrielyan and A.V. Kryazhimskii [11] were devoted
studying positional conflict counteraction between groups of controlled objects. It
should be noted that, from mathematical point of view, the extremal targeting rule, as
applied to linear systems, is based on using the apparatus of support functions, the no-
tion of Aumann integral of set-valued mapping and the Lyapunov theorem on vector
measures.
Due to the positional formalization, N.N. Krasovskii [12] proved the theorems
of alternative, dividing the game space into the players’ preference domains. In the case
of failure of the «saddle point condition in small game» the alternative can be formu-
lated in terms of strategy of one player and counterstrategy of another. Under the same
condition, an assertion on alternative can be stated in mixed strategies. Because
of the problem on existence of solution of the differential equations, arising in such
formalization, B.N. Pshenichnyi [13] introduced the notion of -strategy being the de-
velopment of ideas of Pontryagin’s [14] Alternated Integral Method. Thus, there exists
many methods of constructing positional controls in conflict condition.
In this chapter, we suggest a general scheme for investigation of conflict-controlled
processes, illustrated on various types of functional-differential systems.
Note that the application of resolving functions method [15] to similar problems is
based on another assumption on information availability, that leads to control choice
in the class of counter- or quasi-strategies.
1. Problem statement, time of the «first absorption»
Let a state of the conflict-controlled process at arbitrary moment of time ,t
0 0,t t be described by the vector
0
( ) ( ) ( , , ( ), ( )) ,
t
t
z t g t t s u s v s ds= + (1)
where ( ) ,nz t 0( ), :[ , ) ,ng t g t + → is continuous vector-function and
( , , , ),t s u v 0: ( ) ,nt U V → — function, jointly continuous in its variables.
Here 0 0( ) {( , ) : },t t s t s t = + ( ), ( ).n nU K V K
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 6 7
By admissible controls of the first and second players we mean measurable func-
tions, taking their values in U and ,V respectively. Let us introduce the notations:
0 0{ ( ) : ( ) , [ , )}, { ( ) : ( ) , [ , )},U Vu s u s U s t v s v s V s t = + = +
where ( )u s and ( )v s are measurable functions. We denote by ( )u and ( )v elements
of the sets ,U ,V respectively.
Also, a cylindrical terminal set
nM is specified:
0 ,M M M = + (2)
where 0M is a linear subspace in ,n and M is a convex compact subset of the or-
thogonal complement L to 0M in ,n co ( ).M K L We suppose that 0( ) .g t M
The goal of the pursuer u is to bring a trajectory of the object (1) to the terminal
set in the shortest time, whereas the evader v tries to avoid the encounter of the trajec-
tory with the set. The game is terminated at the moment 0T t such that ( )z T M
or ( ) .z T M
Let us take the position of the first pursuer and determine the guaranteed time
of the game termination. We assume that, for 0 ,t t on the half-interval 0[ , )t t
the players employ admissible controls: 0( ) { ( ) : [ , )},tu u s s t t = 0( ) { ( ) : [ , )},tv v s s t t =
( ) , ( ) .U Vu v
Then, for arbitrary finite moments t and ,T such that 0 ,t t T and for fixed
controls ( )tu and ( ),tv the following representation holds:
( ) ( , , ( ), ( )) ( , , ( ), ( )) ,
T
t t
t
z T z T t u v T s u s v s ds= +
where
0
( , , ( ), ( )) ( ) ( , , ( ), ( )) .
t
t t
t
z T t u v g T T s u s v s ds = + (3)
Note that
0 00( , , ( ), ( )) ( ).t tz T t u v g T =
Let us consider the scalar product
( ( ), ) ( ( , , ( ), ( )), ) ( ( , , ( ) ( )), ) , .
T
n
t t
t
z T p z T t u v p T s u s v s p ds p= +
Now we proceed to the analysis of the following expression:
( ) ( )
min max ( ( ), )
V Uv u
z T p
=
( ) ( )
( ( , , ( ), ( )), ) min max ( ( , , ( ) ( )), )
V U
T
t t
v u
t
z T t u v p T s u s v s p ds
= + =
( ( , , ( ), ( )), ) minmax( ( , , , ), ) .
T
t t
v U u U
t
z T t u v p T s u v p ds
= +
8 ISSN 2786-6491
The signs of extremum and integral in the expression above can be interchanged
in view of the properties of function ( , , , )T s u v and the Lyapunov theorem on vector
measures [16]. Denote ( , , ( ), ( ))t tz T t u v = and consider the following function:
( , , , ) ( , ) minmax( ( , , , ), ) ,
T
v V u U
t
W T t p p T s u v p ds
= + 0 ,t t T ,n .np
It is obvious that function ( , , , )W T t p is continuous in the domain of its defini-
tion. Its vector-gradient in ,t has the form
,
minmax ( , ( , , , ))( , , , )
( , , , ) .
( , , , )
t v V u U
t
p T t u vW T t p
W T t p
W T t p p
−
= =
The function ( , , , )W T t p is continuous in , ,T t p due to the joint continuity
of the function ( , , , ).t s u v Consider the support function of the cylindrical set :M
( , ), ,
( , )
, .
C M p p L
C M p
p L
=
+
The subspace L is the barrier cone of the set .M Since M is a compact set lying
in ,L the function ( ; )C M p is continuous on .L
Now we introduce a function
1,
( , , ) min [ ( , , , ) ( , )].
p p L
T t W T t p C M p
=
= + −
(4)
Using the notations
( , ( ), ( )) min{ : ( , , ( , , ( ), ( ))) 0},t t t tT t u v T t T t z T t u v = = (5)
( , , ) { : , 1, ( , , , ) ( , ) ( , , )},T t p p L p W T t p C M p T t = = + − = (6)
let us introduce a set-valued mapping
( )
( , ) ( , , , ( )) .
V
T
v t
W T t M T s U v s ds
= −
(7)
The above expression contains the integral of a set-valued mapping, namely
the Aumann integral [17]. By virtue of the properties of the Aumann integral [18]) the
set-valued mapping ( , )W T t is convex-valued and upper semicontinuous.
The so-called time of the «first absorption» is defined as follows:
0 0 0min{ : , ( ) ( , )}.T T T t g T W T t=
We assume that this minimum is attained.
The support function of the set 0( , )W T t can be written as
0
0( ( , ); ) co ( , ) minmax( , ( , , , )) .
T
v V u U
t
C W T t p C M p p T s u v ds
= + −
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 6 9
It follows that 0( ) ( , )g t W T t if and only if
0
, 1
min [ ( ( , ), ) ( , ( ))] 0.
p L p
C W T t p p g T
=
− +
Since the minimum of a function coincides with the minimum of its convexifica-
tion, we have
0 0
, 1
min [ ( ( , ), ) ( , ( ))] ( , , ( ))
p L p
C W T t p p g T T t g T
=
− + =
.
Thus,
0 00 0 0 0( , ( ), ( )) min{ : ( , , ( )) 0}.t tT T t u v T t T t g T= = =
Let us suppose that the minimum is attained. We assume that the time of the «first
absorption» is finite, i.e., 0 .T + The fact that 0T is the minimal instant at which
0 0 0( ) ( , )g T W T t means that 0T is the minimal time at which the first player, know-
ing in advance the second player’s control over the entire future interval, can steer the
system trajectory to the terminal set .M Let us show that 0 0.T t Using formulas (4)
and (7), we obtain
0 00 0 0 0 0
1,
( , , ( ), ( )) min [ ( , , ( ), ) ( ; )].t t
p p L
t t u v W t t g t p C M p
=
= + −
As 0( ) ,g t M we have 0 0 0( , , ( )) 0;t t g t therefore 0 0.T t
Lemma 1. If ( , ( ), ( )) ,t tT t u v t = then ( ) .z t M
Proof. Consider the half-interval 0[ , ).t t The player use the controls ( ),tu
( )t Uu and ( ), ( ) .t t Vv v If ( , ( ), ( )) ,t tT t u v t = then it follows from defini-
tion (4) that ( , , ( , , ( ), ( ))) 0.t tt t z t t u v From formula (3),
0
( , , ( ), ( )) ( ) ( , , ( ), ( )) .
t
t t
t
z t t u v g t t s u s v s ds = +
Therefore, { ( )} ,z t M and hence ( ) .z t M
2. Sufficient conditions for the game termination
We introduce the notation ( , ( ), ( )).t t tT T t u v= The pair
1
0( , ( , , ( ), ( ))) , ,n
t t tt z T t u v t t+ 0 ,t t
is called the current position of the game. Accordingly, the initial position is given by
0 00 0 0 0 0( , ( , , ( ), ( ))) ( , ( )).t tt z T t u v t g T =
It follows from the definition of the «first absorption» time that the first player is
aware of the second player’s control over the entire future interval. Below we establish
conditions ensuring that the first player can terminate the game no later than at the mo-
ment of the «first absorption», provided that the first player constructs the control based
on information about the current position of the game.
Theorem 1. Let following conditions hold for the game (1), (2):
1) the set-valued mapping ( , )W T t is nonempty, and the time of the «first absorp-
tion» is finite 0( ),T + 0 0;t t T T
10 ISSN 2786-6491
2) for any position ( , ( , , ( ), ( ))),
t t t
t z T t u v
0 0 ,t t T such that 0 ,t
T T
there exists a neighborhood ( , ( , , ( ), ( )))
t t t
t z T t u v
such that for all ( , )t
( , ( , , ( ), ( )))
t t t
t z T t u v
the set ( , , )
t
T t in (6) consists of a unique element
( , , ).
t
p T t
Then the first player can terminate the game from given initial position no later
than at the time 0T under any admissible control of the second player.
Proof. First, we consider the case 0 .t t= For any fixed 0,T t the function
0( , , ( ), ( )), ,t tz T t u v t t T is continuous in .t By assumption 2) of the theorem,
there exists a moment of time 1 0t t such that for all possible 0( , , ( ), ( )),t tz T t u v
0 1[ , ),t t t the set 0 0( , , ( , , ( ), ( )))t tT t z T t u v (see (6)) consists of a unique element
0 0 0( , , ( , , ( ), ( ))).t tp T t z T t u v We now analyze the corresponding set-valued mapping:
0 0 0 0 0 0( , ( , , ), , ( )) { : , max( ( , , ), ( , , , ( )))}e
u U
U T p T t t v t u u U p T t T t u v t
= =
0 0 0 0 1( ( , , ), ( , , , ( ))), [ , ).p T t T t u v t t t t= (8)
Here, we use the notation 0( , , ( ), ( )).t tz T t u v =
The set-valued mapping (8) is compact-valued and upper semicontinuous with
respect to the variables ( , , ),p t v according to the assumptions on the parameters
of the conflict-controlled process (1) (Aubin and Frankowska [18]). Therefore,
by the measurable selection theorem, the set-valued mapping 0 0( , , , )eU T p t v contains
at least one Borel selection 0( , , ).eu p t v
Since the function 0( , , ( ), ( ))t tz T t u v is continuous in ,t the vector-valued function
0 0 0( , , ( , , ( ), ( )))t tp T t z T t u v is also continuous in 0 1[ , ).t t t Because the control ( )v t
is measurable and 0 0 0( , , ( , , ( ), ( )))t tp T t z T t u v is continuous in ,t it follows that
the Borel selection 0( , , )eu p t v is measurable in the sense of superposition. Consequent-
ly, the function: 0 0 0( ) ( ( , , ( , , ( ), ( ))), ( )),e e
t tu t u p T t z T t u v v t= 0 1[ , ),t t t is measurable.
We choose the pursuer’s control on the half-interval 0 1[ , )t t in the form of the selec-
tion ( ).eu t In the case where several measurable selections exist, any one of them
can be used.
According to the properties of the minimum, for each 0,T t the function
( , , ),T t 0 ,t t T ,n is differentiable in t and in any direction. By the se-
cond assumption, the set ( , , )T t consists of a unique element ( , , )p T t for all ( , )t
in some neighborhood ( , )O t of the point ( , ).t Hence, the vector function
( , , )p T t continuously on ( , )t and for all ( , ) ( , ),t O t
,
min max( ( , , ), ( , , , ))
( , , ) .
( , , )
v V u U
t
p T t T t u v
T t
p T t
−
=
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 6 11
Now we show that, under the control ( ),eu t 0 1[ , ),t t t and any arbitrary admissi-
ble control ( ),v t the system (1) reaches the position
1 1 11 1( , ( , , ( ), ( ))),t t tt z T t u v at time 1t
with
10 .tT T To this end, we evaluate the derivative 0 0( , , ( , , ( ), ( ))),e
t t
d
T t z T t u v
dt
0 1[ , ).t t t
To simplify calculations, we denote 0( , , ( ), ( )) .e e
t tz T t u v = By formula (3) and
the fact that, for any fixed 0,T t the function ( , , ( ), ( ))t tz T t u v is differentiable in t
on the half-interval 0 1[ , )t t with
( , , ( ), ( )) ( , , ( ), ( )),t t
d
z T t u v T t u t v t
dt
=
we conclude that
0( , , )e
d
T t
dt
=
0 0 0 0 0 0min max( ( , , ), ( , , ( ), ( ))) ( ( , , ), ( , , ( ), ( )))e e e
v V u U
p T t T t u t v t p T t T t u t v t
= − + =
0 0 0 0 0 0min( ( , , ), ( , , ( ), ( ))) ( ( , , ), ( , , ( ), ( ))) 0.e e e e
v V
p T t T t u t v t p T t T t u t v t
= − +
When taking the derivative of the function 0( , , ),eT t the assumption that
0( , , ( , , ( ), ( )))t tt
T t z T t u v , contains a unique element at each moment plays a crucial
role. Thus,
0 0 0 1( , , ( , , ( ), ( ))) 0, [ , ).e
t t
d
T t z T t u v t t t
dt
Therefore, the function 0 0( , , ( , , ( ), ( )))e
t tT t z T t u v does not decrease on the
half-interval 0 1[ , ).t t Now, we show that, under the control ,etu 0 1[ , ),t t t of
the first player and arbitrary admissible control ( )v t of the second player,
the game reaches the position
1 1 11 1( , ( , , ( ), ( ))),t t tt z T t u v for which
10 .tT T Since
00
0 0 0 0( , , ( , , ( ), ( ))) 0,e
tt
T t z T t u v it follows that
11
0 1 0 1( , , ( , , ( ), ( ))) 0.e
tt
T t z T t u v
The latter inequality implies that, according to (5),
1 0 ,tT T
11
1 0( , ( ), ( )) .e
tt
T t u v T
It is possible that the equality ( , ( ), ( ))t tT t u v t = holds on the half-interval 0 1[ , ),t t
which would imply termination of the game (Lemma 1). Otherwise, by assumptions of
Theorem 1, there exists 2 1t t such that, on the half-interval 1 2[ , ),t t one can repeat the
procedure of construction a selection of the set-valued mapping:
1 10( , ( , , ), , ( ))e
t tU T p T t t v t =
1 1 1 10 0{ : , max( ( , , ), ( , , , ( ))) ( ( , , ), ( , , , ( )))},t t t t
u U
u u U p T t T t u v t p T t T t u v t
= =
where 1 2[ , )t t t and
1
( , , ( ), ( )).t t tz T t u v = It may happen that, within this interval
1 2[ , )t t the equality ( , ( ), ( ))e
t tT t u v t = holds, in which case the game terminates.
12 ISSN 2786-6491
Otherwise, an instant 3 2t t exists such that the procedure of constructing the con-
trol u can be repeated. By continuing this process we obtain a sequence of half-intervals
on each of which the control of the pursuer is chosen in the form of selection ( )eu t
of the set-valued mapping:
0( , ( , , ), , ( ))
k k
e
t tU T p T t t v t =
0 0{ : , max( ( , , ), ( , , , ( ))) ( ( , , ), ( , , , ( )))},
k k k kt t t t
u U
u u U p T t T t u v t p T t T t u v t
= = (9)
where ( , , ( ), ( )),
kt t tz T t u v = 1[ , ),k kt t t + 1 .
kk tt T+
Since the set-valued mapping 0( , , , )eU T p t v is compact-valued and upper
semi-continuous with respect to 0( , , )p t v for any ,T it contains a Borel selection
0( , , ),eu p t v which is a measurable function in the sense of superposition. Taking into
account that 0( , , )
kt
p T t is continuous in ( , )t and that ( )t = is continuous in ,t
we conclude that 0( , , ( ))
kt
p T t t is continuous in .t Therefore, ( )eu t appears as
a measurable function on each of half-intervals 1[ , ).k kt t +
According to the results of Natanson [19], the function ( )eu t is measurable
on each of half-intervals 1[ , ), 0,1, 2, ;k kt t k+ = therefore, it is measurable on their
countable union, that is, for all 0.t t Let us now show that the following relationship
holds:
11
1( , ( ), ( )) ( , ( ), ( )).
k kk k
e e
k t k tt t
T t u v T t u v
++
+ (10)
To this end, we evaluate 1( , , ( , , ( ), ( ))), [ , ).
k k kk
e
t t t k kt
d
T t z T t u v t t t
dt
+ Let us denote
( , , ( ), ( )) .
k kk
e e
t tt
z T t u v = The procedure analogous to that performed on the half-inter-
val 0 1[ , )t t yields:
0 0( , , ) min max( ( , , ), ( , , , )) ( ( , , ), ( , , , ))
k k k k k
e e e
t t t t t
v V u U
d
T t p T t T t u v p T t T t u v
dt
= − + =
0 0min ( ( , , ), ( , , ( ), )) ( ( , , ), ( , , ( ), )).
k k k k
e e e e
t t t t
v V
p T t T t u t v p T t T t u t v
= − +
Thus,
1( , , ( , , , ( ))) 0, [ , );
k k
e
t t t t k k
d
T t z T t u v t t t
dt
+
the function ( , , ( , , , ( )))
k k
e
t t t tT t z T t u v do not decrease on the half-interval 1[ , ).k kt t +
By definition,
( , , ( , , ( ), ( ))) 0.
k k kk
e
t k t tt
T t z T t u v
Hence,
11
1 1( , , ( , , ( ), ( ))) 0.
k k kk
e
t k t k tt
T t z T t u v
++
+ +
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 6 13
From this, inequality (10) follows, which means that 0( , ( ), ( )) ,e
t tT t u v T 0 0.t t T
On the other hand, according to (5), we have ( , ( ), ( )) ,e
t tT t u v t 0 0.t t T Thus, we
obtain
0 0 0( , ( ), ( )) , .e
t tt T t u v T t t T (11)
Therefore, the equality
( , ( ), ( ))e
t tT t u v t = (12)
which indicates the termination of the game (Lemma 1), is satisfied no later than at the
time 0.t T=
Suppose that this procedure continues indefinitely long. Then we obtain a sequence
of half-intervals 0 1 1 2 2 3[ , ), [ , ), [ , ), .t t t t t t It is evident that 0,kt T 0,1, 2, ;k =
otherwise, the equality (12) would be satisfied on one of these half-intervals, and
the game would be terminated. The sequence { },kt 0,1, 2, ,k = is monotonically in-
creasing and bounded above, which implies that there exists a moment of time t such
that lim .
k
t
→+
= It is clear that 0( , ( ), ( )) .e
t tT t u v T The procedure of constructing
the control u can therefore be extended on some half-interval [ , ), ,t t t t beginning
from the position ( , ( , , ( ), ( ))).e
t t tt z T t u v
Note that various controls ( ),e
tu 1[ , ),k kt t t + chosen as measurable selections
of the set-valued mapping (9), generate different system trajectories. Evidently,
the double inequality (11) holds for all motions, induced by the controls ( ),e
tu
1[ , ),k kt t t + 0,1, 2, .k = In view of this inequality, condition (12), which charac-
terizes the termination of the game, is satisfied no later than at the time 0 .T The theo-
rem is proved.
Remark 1. Let an analog of the «small game» saddle-point condition (Krasovskii
and Subbotin [2]) be satisfied in the game under consideration, namely:
0 0min max ( , ( , , , )) maxmin ( , ( , , , )), 0 .
v V v Vu U u U
p T t u v p T t u v t T
=
Then in the proof of Theorem 1, instead of the set-valued mapping (9), one can
analyze the set-valued mapping of the form
0 0 0( , , ) { , :min max( , ( , , , )) min( , ( , , , ))}.e
v V v Vu U
U T p t u v U V p T t u v p T t u v
= =
In this case, the control ( )etu is independent of ( ).v t
3. Integro-differential approach
Let us consider the problem of approach for the conflict-controlled process whose
evolution is described by the system of integro-differential equations:
0 0
1 2( ) ( ) ( ) ( , ) ( ) ( ) ( ( ), ( )) ( , ) ( ( ), ( )) ( ),
t t
t t
z t A t z t K t s z s ds B t u t v t C t s u s v s ds f t= + + + + (13)
where 0 0 0( ) , 0, .nz t z t t z= The matrix functions ( )A t and ( )B t have dimensions
1n n and 2,n n respectively, and are continuous on the semi-axis
0 0{ : }.t t t t=
14 ISSN 2786-6491
The vector-function ( )f t is continuous on
0
.t The matrix kernels ( , ),K t s and
( , ),C t s of dimensions n n and 2,n n are continuous in the closed triang
le 0{( , ): }.t s t s t = + The vector-functions 1
1( , ) :
n
u v U V → and
2
2 ( , ) :
n
u v U V → are jointly continuous in their variables. The control sets ,U V
are compact subsets of 1n and 2 ,
n respectively. The players’ controls ( )u and ( )v
are measurable functions taking their values in U and ,V respectively; that is,
( ) ,Uu ( ) .Vv The terminal (target) set of cylindrical form (2) is given
by 0 ,M M M = + where 0M is a linear subspace of ,n co ( ),M K L and
0L M ⊥= is the orthogonal complement of 0M in .n
Our task is to apply the technique described in Subsection 1 to the systems of the
form (13). Note that expression (13) additionally involves the integral control term.
Lemma 2. Let the function ( )g t be continuous on the half-interval 0[ , ).t T Then
the equation
0
( ) ( ) ( ) ( , ) ( ) ( )
t
t
z t A t z t K t s z s ds g t= + + (14)
has a unique solution, is continuous on 0[ , ),t T which can be written in the form
0
( ) ( ) ( , ) ( ) ,
t
t
z t g t R t s g s ds= +
where
0
0 0( ) ( , ) ( , ) ( ) ,
t
t
g t H t t z H t g d= + (15)
and ( , )H t s is the fundamental matrix of the homogeneous system
0 0 0( ) ( ) ( ), ( ) , [ , ).z t A t z t z t z t t T= = (16)
The function ( , )R t s is the resolvent of the matrix ( , ),K t s which is a matrix function
continuous on the set 0{( , ): }T t s t s t T = and determined by a series uniformly
convergent on :T
1
( , ) ( , ),n
n
R t s K t s
=
= (17)
where
1 1( , ) ( , ), ( , ) ( , ) ( , ) , 2, 3, ,
( , ) ( , ) ( , ) .
t
n n
s
t
s
K t s K t s K t s K t K s d n
K t s H t K s d
−= = =
=
(18)
Proof. A unique continuous solution of equation (14) exists under the assumptions
on the parameters of system (13) [20]. Let us introduce the function
( )
0
( , ) ( ) ( ).
t
t
F t K t s z s ds g t= +
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 6 15
It follows from the assumptions of Lemma 2 that the function ( )F t is continuous
on the half-interval 0[ , ).t T Let us consider the Cauchy problem for the system
0 0( ) ( ) ( ) ( ), ( ) , [ , ).z t A t z t F t z t z t t T= + =
This problem has a unique solution, which can be expressed by the Cauchy formu-
la Gaishun [21]
0
0 0( ) ( , ) ( , ) ( ) .
t
t
z t H t t z H t F d= +
Here ( , )H t s is the fundamental matrix of the homogeneous system (16). Substi-
tuting the expression for ( )F t into the above formula we obtain
0 0 0
0 0( ) ( , ) ( , ) ( , ) ( ) ( , ) ( ) .
t t
t t t
z t H t t z H t K s z s ds d H t g d
= + +
Applying the Dirichlet formula [22], we deduce
0 0 0
0 0( ) ( , ) ( , ) ( , ) ( ) ( ) ( , ) ( ) .
t t
t t t
z t H t t z H t K s z s d z s ds H t g d
= + +
By (15), (18), this expression can be rewritten in the form of a second-order
Volterra equation:
0
0( ) ( ) ( , ) ( ) , [ , ).
t
t
z t g t K t s z s ds t t T= + (19)
Here ( , )K t s and ( )g t are functions continuous in their respective domains. Equa-
tion (19) has a unique solution [22]
0
0( ) ( ) ( , ) ( ) , [ , ),
t
t
z t g t R t s g s ds t t T= +
where ( , )R t s is given by formulas (17) and (18). The lemma is proved.
Corollary 1. The equation (13) has a unique (continuous) solution on an arbitrary
interval 0[ , ],t T which can here presented in the form
0 0
0 1 2( ) ( ) ( , ) ( ( ), ( )) ( , ) ( ( ), ( )) ,
t t
t t
z t g t M t s u s v s ds N t s u s v s ds= + + (20)
where
0
0 0 0( ) ( ) ( , ) ( ) ;
t
t
g t f t R t s f s ds= + (21)
0
0 0 0 0( ) ( , ) ( , ) ( ) , [ , );
t
t
f t H t t z H t s f s ds t t T= + (22)
16 ISSN 2786-6491
( , ) ( , ) ( , ) ( , ) ;
t
s
M t s B t s R t B s d= + (23)
( , ) ( , ) ( , ) ( , ) ;
t
s
N t s C t s R t C s d= + (24)
( , ) ( , ) ( );B t s H t s B s= (25)
( , ) ( , ) ( , ) .
t
s
C t s H t C s d= (26)
Proof. Lemma 2 ensures for the existence and uniqueness of the solution. Let us define
0
1 2( ) ( ) ( ( ), ( )) ( , ) ( ( ), ( )) ( ).
t
t
g t B t u t v t C t s u s v s ds f t= + +
Thus, following representation holds:
0
0( ) ( ) ( , ) ( ) , [ , ).
t
t
z t f t R t s f s ds t t T= +
Here,
0 0
0 0 1( ) ( , ) ( , ) ( ) ( , ) ( ) ( ( ), ( ))
t t
t t
f t H t t z H t s f s ds H t s B s u s v s ds= + + +
0
2( , ) ( , ) ( ( ), ( )) .
t t
t s
H t C s d u s v s ds
+
Now we rewrite the previous formula taking into account notations (22), (25), (26):
0 0
0 1 2( ) ( ) ( , ) ( ( ), ( )) ( , ) ( ( ), ( )) .
t t
t t
f t f t B t s u s v s ds C t s u s v s ds= + +
Then,
0 0 0 0
0 1( , ) ( ) ( , ) ( ) ( , ) ( , ) ( ( ), ( ))
t t t s
t t t t
R t s f s ds R t s f s ds R t s B s u v d ds= + +
0 0
2( , ) ( , ) ( ( ), ( )) .
t s
t t
R t s C s u v d ds+
Let us apply the Dirichlet formula to the last two terms of the above sum. After changing
the order of integration and renaming the variables , s as ,s , respectively, we obtain:
0 0 0
1 1( , ) ( , ) ( ( ), ( )) ( , ) ( , ) ( ( ), ( )) ,
t s t t
t t t s
R t s B s u v d ds R t B s d u s v s ds
=
0 0 0
2 2( , ) ( , ) ( ( ), ( )) ( , ) ( , ) ( ( ), ( )) .
t s t t
t t t s
R t s C s u v d ds R t C s d u s v s ds
=
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 6 17
Considering (23) and (24) we derive formula (20). Hence, the solution of equa-
tion (13) can be represented in the form (1). Herewith, 0( ),g t defined by (21), is con-
tinuous for
0
.tt Consequently,
1 2 0( , , , ) ( , ) ( , ) ( , ) ( , ), ( , ) ( ), , .t s u v M t s u v N t s u v t s t u U v V = + (27)
The function ( , , , )t s u v satisfies all the requirements stated in Section 2, since
( , )M t s and ( , )N t s are continuous on the closed infinite triangle , and, by assump-
tion, the functions 1( , )u v and 2( , )u v are jointly continuous in their variables.
Therefore, Theorem 1 holds for the game whose dynamics are described by equa-
tion (13). It should be noted that, in the analyzed game, the position of the system is
represented by the pair ( , ( , , ( ), ( ))).t t tt z T t u v
The function ( , , ( ), ( ))t t tz T t u v is given by
0
0 0( , , ( ), ( )) ( ) ( , , ( ), ( )) , .
T
t t t
t
z T t u v g T T s u s v s ds t t T = +
Here, the function ( , , , )T s u v is defined by formulas (27), (23) and (24), and the
initial position corresponds to the pair
0
0 0
0
0 0 0 0 0 0 0 0 0 0 0( , ( , , ( ), ( ))) ( , ( )) , ( ) ( , ) ( ) .
T
t t
t
t z T t u v t g T t f T R T s f s ds
= = +
In the above expressions, the function 0( )f t is defined by formula (22) and
( , )R t s — by formulas (17) and (18). For the case under consideration, the func-
tions ( , , )T t and ( , , , ),W T t p as well as the time of «first absorption» 0 ,T and
the set-valued mapping ( , ),W T t take the following forms:
1,
( , , ) min [ ( , , , ) ( ; )];
p p L
T t W T t p C M p
=
= + −
( ) ( ) ( )( ) 0, , , , minmax , , , , , , , ;
T
n n
v V u U
t
W T t p p T s u v p ds t t T p
= +
0 0 0min{ : , ( ) ( , )};T T T t g T W T t=
0
0
( )
( , ) ( , , , ( )) .
V
t
v t
W T t M T s U v s ds
= −
Thus, with the representation (20) for solution of the system of integro-differential
equations (13), one can easy formulate an analog of the Theorem 1.
4. Example of the integro-differential game approach
Let us consider the game approach problem for the following system:
0 0
0
( ) ( ) ( ) ( ) ( ( ) ( )) , 0, (0) , .
t
nz t z t u t v t u s v s ds t z z z= + − + − = = (28)
18 ISSN 2786-6491
The terminal set 0M is a linear subspace. In the previous notations 1 2 1 2 ,r r n n n= = = =
{ : , , 1},nU u u u a a= { : , 1}.nV v v v= The system parameters
are ( ) ,A t E= ( ) ,B t E= ( , ) ,C t s E= where E is the n n identity matrix. The func-
tions 1 2( , ) ( , ) ,u v u v u v = = − and ( )f t is the zero n-dimensional column vector.
In view of formula (13), the matrix ( , )K t s is zero, and according to (18), the same
holds for ( , ).K t s The fundamental matrix of the homogeneous system is
( )( , ) .t sH t s e E −= We will analyze two cases.
Case A. 0.
From formulas (21) and (22) it follows that 0 0 0( ) ( ) .tf t g t e z= = According
to (25) and (26), we obtain
( )( ) ( ) ( )1
( , ) , ( , ) 1 .
t
t s t t s
s
B t s e E C t s e d E e E − − −
= = = −
Using expressions (23) and (24), we find ( )( , ) ,t sM t s e E −= ( )( )1
( , ) 1 .t sN t s e E −= −
From (20) we have
( )
0
1 1
( ) 1 ( ( ) ( )) , 0.
t
t T s
s
z t e z e u s v s ds t −
= + + − −
Taking into account formula (27) and the expression for ( , , ( ), ( ))t tz T t u v we
deduce:
( ) 1 1
( , , , ) 1 ( );t st s u v e u v −
= + − −
( )
0
1 1
( , , ( ), ( )) 1 ( ( ) ( )) , 0.
t
t T s
t t
s
z T t u v e z e u s v s ds t −
= = + + − −
Then the function ( , , , )W T t p takes the form
( ) 1 1
( , , , ) ( , ) minmax , 1 ( ) ,
T
T s
v S u aS
t
W T t p p p e u v ds −
= + + − −
{ : 1}.S v v=
Since
( ) 1 1
1 0, [0, ],T se s T −
+ −
for any 0 (it suffices to examine
separately cases 0 and 0), we have
( ) 1 1
( , , , ) ( , ) ( 1) 1
T
T s
t
W T t p p a p e ds −
= + − + − =
( )( )1 1
( , ) ( 1) 1 1 ( ) .T tp a p e T t −
= + − + − − −
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 6 19
According to the construction scheme, we obtain
( , , ( , , ( ), ( )))t tT t z T t u v =
( )( )
, 1
1 1
min ( ( , , ( ), ( )), ) ( 1) 1 1 ( )T t
t t
p L p
z T t u v p a e T t −
=
= + − + − − − =
( )( )1 1
( , , ( ), ( )) ( 1) 1 1 ( ) .T t
t tz T t u v a e T t −
= − + − + − − −
By definition, ( , ( ), ( ))t t tT t u v is the least root of the following equation in :T
( )( )1 1
( , , ( ), ( )) ( 1) 1 1 ( ) , .T tz T t u v a e T t T t −
= − + − − −
(29)
Setting 0t = in (29), we obtain the equation for the time of the «first absorption»:
( )0
1 1
( 1) 1 1 .T Te z a e T
= − + − −
(30)
Note that 0 0.z Let us rewrite equation (30) in the form
0
1 1 1 1 1
( 1) 1 ( 1) ( 1) 1 .Te a z a T a
− + − = − + − +
(31)
Now we find the solution to the following inequality in :
0
1 1
( 1) 1 0.a z
− + −
This inequality is equivalent to
2
0 ( 1) ( 1) 0, 0,z a a − − − −
whose solution is given by
2
0
0 0
1 1
, 0 0, , ( 1) 4 ( 1) .
2 2
a d a d
d a z a
z z
− − − +
= − + −
A solution of inequality (31) can be written as
0( ) ,Te b z kT b − = + (32)
where
1 1 1
( 1) 1 , ( 1) , 0.b a k a T
= − + = −
Using the notations 1 0( ) ( ),Tf T e b z= − 2( ) ,f T kT b= + equation (32) can be
written in the form
1 2( ) ( ), 0.f T f T T= (33)
Several possible cases are considered below.
1.
0
1
0, .
2
a d
z
− +
Then, in equation (32) 0 0b z− and 0.k
Consequently, the functions 1( )f T and 2( )f T are monotonically increasing:
20 ISSN 2786-6491
1( )f T is exponential, while 2( )f T is linear. Moreover, 1 2(0) (0),f f since
1 0(0)f b z= − and 2(0) .f b= Therefore, equation (33) has a unique root.
2.
0
1
.
2
a d
z
− +
=
Then, in equation (32) 0 0b z− = and 0,k while
1( ) 0,f T 0,T and 2(0) ,f b= 2( )f T is monotonically increasing function for
0.T In this case, equation (32) has no roots.
3.
0
1
.
2
a d
z
− +
Then, in equation (32) 0 0b z− and 0.k We have
1 0(0)f b z= − and 2(0) .f b= The function 1( )f T is monotonically decreasing,
whereas 2( )f T is monotonically increasing. In is evident that, in this case, equa-
tion (33) also has no roots.
4.
0
1
, 0 .
2
a d
z
− −
In this case, from equation (32) we have 0 0b z− and
0.k Then 1 0(0)f b z b= − and 1 2(0) (0) .f f b = The exponential function
1( )f T is monotonically decreasing and converges to zero as .T → + The linear
function 2( )f T is monotonically decreasing (since 0)k and diverges to − as
.T → + Hence, equation (33) has a unique root.
5.
0
1
2
a d
z
− −
=
. Then 0 0b z− = and 0.k In this case 1( ) 0,f T while
the linear function 2( )f T is monotonically increasing since 2(0) 0.f b= Therefore,
equation (33) possesses a unique root.
6.
0
1
, .
2
a d
z
− +
−
Then 0 0b z− and 0.k In this case, the function
1( )f T is monotonically decreasing and converges to zero as ,T → + while 2( )f T is
monotonically increasing and diverges to infinity. Since 1 0 2(0) (0) .f b z f b= − =
Therefore, equation (33) has a unique root.
The analysis of cases 1–6 shows that equation (30) has a solution; that is, the time
of the «first absorption» is finite, for
0
1
( ,0) 0, .
2
a d
z
− +
−
Moreover, if ( , , ( ), ( )) 0,t tz T t u v then the minimum in the expression
for ( , , ( , , ( ), ( )))t tT t z T t u v is achieved at the unique element
0 0 1 0 1
0 1
1
( , , ( , , ( ), ( ))) ( , , ( ), ( )).
( , , ( ), ( ))
t t t t
t t
p T t z T t u v z T t u v
z T t u v
= −
Otherwise, if ( , , ( ), ( )) 0,t tz T t u v = the minimum is achieved for any .p L
Let t be an arbitrary moment of time at which the game has not yet been terminated
(that is, ( ) ).z t M Then, by Lemma 1 and definition (5), tT t and the following
inequality holds:
( )( )1 1
( 1) 1 1 ( ) 0.tT t
ta e T t
−
− + − − −
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 6 21
Since tT is the root of equation (29), it follows that ( , , ( ), ( )) 0.t t tz T t u v
Because the function ( , , ( ), ( ))t tz T t u v is continuous with respect to t, there exists
a half-interval [ , ),t t + where 0, such that ( , , ( ), ( )) 0,tz T u v
[ , ).t t + Therefore, for all [ , )t t + the set ( , , ( , , ( ), ( )))t tT z T u v consists
of a unique element:
0
1
( , , ( , , ( ), ( ))) ( , , ( ), ( )).
( , , ( ), ( ))
t t t
t
p T z T u v z T u v
z T u v
= −
Thus, it has been shown that condition 2) of the Theorem 1 is satisfied. On the
half-interval [ , )t t + the set-valued mapping of the form (8) consists of a unique ele-
ment 0 0( , , ) ( , , ( , , ( ), ( ))),e
t t tu T p a p T z T u v = [ , ).t t +
Case B. 0. =
It is evident that, in this case, the fundamental matrix is the unit matrix, that is,
( , ) .H t s E= From formulas (21) and (22) we have 0 0( ) ,f t z= 0 0( ) .g t z= From for-
mulas (25) and (26) we infer ( , ) ( ) ,C t s t s E= − ( , ) ,B t s E= since ( )B t E= and
( , ) .C t s E= Moreover, from (22) and (23) we obtain ( , ) ,M t s E= ( , ) ( ) .N t s t s E= −
Therefore, the trajectory of the system (20) in the case (28) takes the form
0
0
( ) ( 1)( ( ) ( )) , 0.
t
z t z t s u s v s ds t− = + − + −
By formula (27), the function ( , , , ) ( 1)( )t s u v t s u v = − + − and therefore
0
0
( , , ( ), ( )) ( 1)( ( ) ( )) .
t
t tz T t u v z T s u s v s ds = + − + −
Then the following expression for ( , , , )W t s p holds:
( , , , ) ( , ) minmax( ,( 1))( ) .
T
v S u aS
t
W T t p p p T s u v ds
= + − + −
Since 1 0T s− + for 0 ,t s T we have
( , , , ) ( , ) ( 1) ( 1) ( , ) ( 1) ( ) 1 .
2
T
t
T t
W T t p p a p T s ds p a p T t
−
= + − − + = + − − +
Then, from formula (29), taking into account that {0},M = we obtain:
( )( ) ( )
1,
( , , ( , , ( ), ( ))) min , , ( ), ( ) , ( 1)( ) 1
2t t t t
p p L
T tT t z T t u v z T t u v p a T t
=
− = + − − + =
( ) ( ), , ( ), ( ) , ( 1) ( ) 1 .
2t t
T tz T t u v p a T t −= − + − − +
According to (29) ( , ( ), ( ))t tT t u v is defined as the least root of the following equa-
tion with respect to :T
( ) ( )( 1) ( ) 1 , , ( ), ( ) , , .
2 t t
T ta T t z T t u v p T t−− − + =
22 ISSN 2786-6491
Hence, the time «first absorption» 0T appears as the least positive root of the equation
( ) 0( 1) 1 , 0.
2
Ta T z T− + =
It is easy to see that 0
0
2
1 1 .
1
z
T
a
= − + +
−
Evidently, the time of the «first absorption» is finite, regardless of the initial state
of the game 0 0( , ).z z Analogously to Case A ( 0), it can be shown that
the condition 2) of Theorem 1 is fulfilled, and therefore the required positional control
is determined. Combining Cases A and B, we come to the following conclusion: if
( ) ( )
2
0
0
1
, , 1 4 1 ,
2
a d
d a z a
z
− +
− = − + −
then approaching the terminal set can be performed within the class of positional con-
trols in a time less than or equal the «first absorption» time.
5. Quasilinear positional integral games approach
Let us consider the conflict-controlled process, governed by a system of linear in-
tegral Volterra equations of the second kind:
0 0
( ) ( ) ( , ) ( ) ( , ) ( ( ), ( )) , 0.
t t
z t f t K t s z s ds Q t s u s v s ds t= + + (34)
Here ;nz ( )f t is a vector-function, continuous on the positive semi-axis
{ : 0};t t+= ( , )K t s and ( , )Q t s are n n and n r matrices, respectively, continu-
ous on the triangle {( , ) : 0 }.t s s t = +
Control domains are compact sets 1,
r
U 2 .
r
V The vector function
( , ): ,ru v U V → is jointly continuous in its variables.
The terminal set has a cylindrical form: 0 ,M M M= + where 0M is a linear sub-
space in ,n co ( ),M K L and 0 .L M⊥=
Admissible controls of the players are assumed to be measurable functions.
The goals of the players are the same as in the previous section. We assume that
(0) (0) .z f M =
Lemma 3. Let the elements of the matrix ( , )K t s be continuous on the closed tri-
angle and let the function ( )g t be continuous on the closed interval [0, ].T Then the
equation
0
( ) ( ) ( , ) ( ) , [0, ],
t
z t g t K t s z s ds t T= +
has a unique solution of the form
0
( ) ( ) ( , ) ( ) , [0, ].
t
z t g t R t s g s ds t T= +
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 6 23
where ( , )R t s is the resolvent of the matrix ( , ).K t s The function ( , )R t s is continuous
on T and is defined by the following series, which converges on :
1 1
1 0
( , ) ( , ), ( , ) ( , ), ( , ) ( , ) ( , ) , 2,3, ... .
t
n n n
n
R t s K t s K t s K t s K t s K t K s d n
−
=
= = = = (35)
Corollary 2. Equation (34) has a unique continuous solution on an arbitrary in-
terval [0, ],T 0 :T
1
0 0
( ) ( ) ( , ) ( ) ( , ) ( ( ), ( )) ,
t t
z t f t R t s f s ds N t s u s v s ds= + + (36)
where
1( , ) ( , ) ( , ) ( , )
t
s
N t s Q t s R t Q s ds= +
is a continuous matrix function.
Proof. The first statement immediately follows from Lemma 2. Then, by the same
lemma, since
0
( ) ( ) ( , ) ( ( ), ( )) ,
t
g t f t Q t s u s v s ds= +
we have
0 0 0
( ) ( ) ( , ) ( ( ), ( )) ( , ) ( ) ( , ) ( ( ), ( ))
t t s
z t f t Q t s u s v s ds R t s f t Q s u v d ds
= + + + =
0 0 00
( ) ( , ) ( ) ( , ) ( ( ), ( )) ( , ) ( , ) ( ( ), ( )) .
t t t s
f t R t s f s ds Q t s u s v s ds R t s Q s u v d ds= + + +
Let us apply the Dirichlet formula to the last term. Then, successively replacing
the notations for s and s for , we obtain:
0 0
( ) ( ) ( , ) ( ) ( , ) ( ( ), ( ))
t t
z t f t R t s f s ds Q t s u s v s ds= + + +
0 0
( , ) ( , ) ( ( ), ( )) ( ) ( , ) ( )
t t t
R t s Q s u v dsd f t R t s f s ds
+ = + +
0 0
( , ) ( ( ), ( )) ( , ) ( , ) ( ( ), ( ))
t t s
s
Q t s u s v s ds R t Q s d u s v s ds
+ + =
0 0
( ) ( , ) ( ) ( , ) ( , ) ( , ) ( ( ), ( ))
t t t
s
f t R t s f s ds Q t s R t Q s d u s v s ds
= + + + =
1
0 0
( ) ( , ) ( ) ( , ) ( ( ), ( )) .
t t
f t R t s f s ds N t s u s v s ds= + +
Thus, the solution to the system of integral equations (34) can be represented
in the form (1), where 0 0:t =
1
0
( ) ( ) ( , ) ( ) , ( , , , ) ( , ) ( , ).
t
g t f t R t s f s ds t s u v N t s u v= + =
24 ISSN 2786-6491
Now we provide examples of resolvent for several types of continuous kernels
on 1, obtained from representation (35). Let A be a square matrix of order ,n and Ae
be a matrix exponent. Then, the following assertions hold:
1) if ( , ) ,K t s A= then ( )( , ) ;t s AR t s Ae −=
2) if ( , ) ( ) ,K t s t s A= − then
2 1
1
( )
( , ) ;
(2 1)!
n
n
n
t s
R t s A
n
−
=
−
=
−
— for example, in the case ,A E= we have ( , ) sh( ) ,R t s t s E= −
2 1
1
sh ;
2 (2 1)!
t t n
n
e e t
t
n
− −
=
−
= =
−
3) if ( )( , ) ,t s AK t s e −= then ( ) ( )( , ) ;t s t s AR t s e Ae− −=
4) if
( ) ( )( , ) ,t s t s AK t s e e − −= where , , are real numbers, then
( )( ) ( )( , ) .t s t s AR t s e + − + −=
Since the solution to (36) can be presented in the form (1), we can apply the con-
struction outlined in the Section 2. Evidently, theorem 1 is also valid in this case. Con-
sidering the notations of Section 3, we obtain
0
0
( , , ( ), ( )) ( , ) ( ( ), ( )) ,
t
t t Tz T t u v z N T s u s v s ds = +
where 0
0
( , ) ( ) .
T
Tz R T s f s ds=
It is easy to see that for the process evolving according to (34),
1( , , ( ), ( )), ( , , , ) ( , ) minmax( , ( , ) ( , )) ,
T
t t
v V u U
t
z T t u v W T t p p p N t s u v ds
= = +
1,
( , , ) min [ ( , , , ) ( , )],
p p L
T t W T t p C M p
=
= + −
1
( ) 0
( ) ( , ) ( , ( )) ,
V
t
v
W t M N t s U v s ds
= −
0 min{ 0: ( ) ( )} min{ 0: ( ,0, ( ))}.T t g T W t t t g t= =
We see that the integral game (34) is a specific case of the game (1). Then the ana-
log of Theorem 1 is true under the conditions for the non-emptiness of ( )W t for any
0t and the uniqueness of extremal element in the expression defining ( , , ).T t
Remark 2. If ( , ) ,u v u v = − + then
1 1
0 0
( ) ( , ) ( , ) .
t t
W t M N t s U ds N t s Vds
= + −
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 6 25
6. Case of the pursuit problem
We examine the dynamic system with separated motions of the players. Let
the motions of the players at [0, )t + be described, respectively, by the second-kind
linear integral Volterra equations:
1 1 1
0 0
( ) ( ) ( , ) ( ) ( , ) ( ) ,
t t
x t f t A t s x s ds B t s u s ds= + + (37)
2 2 2
0 0
( ) ( ) ( , ) ( ) ( , ) ( ) .
t t
y t f t A t s y s ds B t s v s ds= + + (38)
Here 1,
n
x 2 ,
n
y 1( )f t and 2( ),f t [0, ),t + are 1n - and 2n -dimensional
continuous functions; 1( , ),A t s 1( , ),B t s 2( , ),A t s 2( , )B t s are matrices of orders
1 1,n n 1 1,n r 2 2n n and 2 2 ,n r respectively, which are continuous on the closed
infinite triangle {( , ): 0 }.t s s t = + The controls of the players ( )u t and ( )v t
are, respectively, 1r- and 2r -measurable vector functions, chosen in , .U V
By Corollary 2 to Lemma 2 and under the assumptions concerning equations (37)
and (38), right-hand sides ensure the existence of the unique continuous solutions
( ( ), ( ))x t y t to these equations on an arbitrary interval [0, ], 0.t t It is assumed that
at the initial moment 0,t = 1 2{ (0)} { (0)} { (0)} { (0)} ,m m m mx y f f− = − where
{ }mx is a vector consisting of the first m coordinates of vector ,x 1 2min( , )m n n
and 0 is a given number.
The task of the first player is to choose an admissible control in such a way that
the following inequality is satisfied in the shortest time: { ( )} { ( )} ,m mx T y T− under
any arbitrary control of the second player. Now we introduce the following notations:
1 11
2 2 2
( , )( )( )
( ) , ( ) , ( , ) ,
( ) ( ) ( , )
A t s Of tx t
z t f t A t s
y t f t O A t s
= = =
1 21 1
2 2
( , )
( , ) , ( , ) : .
( , )
r rB t s O
Q t s u v U V
O B t s
+
= →
Here 1,O 2O are zero matrices of the dimensions 1 2 2 1 1 2, ,n n n n n r , and 2 1,n r
respectively.
Thus, we come to the approach problem examined in the previous subsection, with
the terminal set 0 ,M M M = + where
1 21
0 1 2
2
, , , ,
n m n mm
a
a
M a a a
a
a
− −
=
1 21
1 2
2
, , , , .
2
n m n mm
p
o
M p p o o
p
o
− −
=
−
(39)
Here 1,o 2o are zero vectors.
26 ISSN 2786-6491
Therefore, to solve the problem under study, Theorem1 can be applied. Let us de-
note, for 0 ,t T
1 1 1
0 0
( , , ( )) ( ) ( , ) ( , ) ( ) ,
T t
tx T t u f T R T s ds N T s u s ds = + +
2 2 2
0 0
( , , ) ( ) ( , ) ( , ) ( ) .
T t
ty T t v f T R T s ds N T s v s ds= + +
Here
1 1 1 1 2 2 2 2( , ) ( , ) ( , ) ( , ) , ( , ) ( , ) ( , ) ( , ) ,
t t
s s
N t s B t s R t B s ds N t s B t s R t B s ds= + = +
where 1( , )R t s and 2( , )R t s are the resolvents of the matrices 1( , )A t s and 2( , ),A t s re-
spectively. In the new notations,
1 1 1
0 0
2 2 2
0 0
( ) ( , ) ( , ) ( ) ,
( , , ( ))
( , , ( ), ( )) .
( , , ( ))
( ) ( , ) ( , ) ( ) .
T t
t
t t
T t
t
f T R T s ds N T s u s ds
x T t u
z T t u v
y T t v
f T R T s ds N T s v s ds
+ +
= =
+ +
(40)
The function ( , , ( , , ( ), ( ))),t tT t z T t u v taking into account notations (39) and (40),
has the following form:
( , , ( , , ( ), ( )))t tT t z T t u v =
( ) 1
, 1
1
min ,{ ( , , ( ))} { ( , , ( ))} max( ,{ ( , ) } )
2 m
T
t m t m m
u Up p
t
p x T t u y T t v p N T s u ds
=
= − + −
2max( ,{ ( , ) } ) .
T
m
v V
t
p N T s v ds
− +
Let us denote
( , , ( , , ( )), ( , , ( )))t tT t x T t u y T t v =
( ) 1
, 1
min ,{ ( , , ( ))} { ( , , ( ))} max( ,{ ( , ) } )
m
T
t m t m m
u Up p
t
p x T t u y T t v p N T s u ds
=
= − + −
2max( ,{ ( , ) } ) .
T
m
v V
t
p N T s v ds
− +
( , ( ), ( )) min{ : ( , , ( , , ( )), ( , , ( ))) 0},t t t tT t u v T t T t x T t u y T t v = =
0 min{ 0: ( ,0, ( ,0, , ), ( ,0, , )) 0},T T T x T y T= =
( , , ( , , ( )), ( , , ( ))) { : , 1, ( , , , , ) ( , , , )},m
t tT t x T t u y T t v p p p W T t x y p T t x y = = + =
( ) ( ) ( )1 2( , , , , ) ,{ } { } max ,{ ( , ) } max ,{ ( , ) } ,
T T
m m m m
u U v V
t t
W T t x y p p x y p N T s u ds p N T s v ds
= − + −
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 6 27
( , ( ), ( )),t t tT T t u v= where ( , , ( )),tx x T t u= ( , , ( )).ty y T t v= In what following triple
will be referred to as the game position: ( , ( , , ( )), ( , , ( ))).t tt x T t u y T t v
Theorem 2. Let, in the game of approach for the dynamic system with distributed
motions of the players (37), (38):
1) the «first absorption» time 0T is finite;
2) for any position 1 2 1( , ( , , ( )), ( , , ( ))) ,
n n
t t t t
t x T t u y T t v
+ + 00 ,t T with
0 ,t
T T there exists a neighborhood ( , ( , , ( )), ( , , ( ))),
t t t t
t x T t u y T t v
such that for
all ( , , ) ( , ( , , ( )), ( , , ( ))),
t t t t
t t x T t u y T t v
1,
n
2 ,
n
the set
( , , , )
t
T t consists of a unique element 0 ( , , , ).
t
p T t
Then the first player can terminate the game no later than at the time 0T for any
admissible control of the second player.
The proof is analogous to that of the Theorem 1. Here we only note that control of
the first player on the half-interval 1 0[ , ), 0, 1, 2, ... , 0,k kt t k t+ = = is to be constructed
in the form of measurable selection ( )eu t of the set-valued mapping:
0 0 1( , ( , , , ), ) { : ,max( ( , , , ),{ ( , ) } )}
k k k k
e
t t t t m
u U
U T p T t t u u U p T t N T t u
= =
0 1 1( ( , , , ),{ ( , ) } ), [ , ),
k kt t m k kp T t N T t u t t t += ( , , ( )), ( , , ( ))).
k k
e e
t t t tx T t u x T t v = =
So, the theorem is proven.
А.О. Чикрій, В.Ф. Губарев, В.Д. Романенко
ЕТАПИ ТА ОСНОВНІ ЗАДАЧІ СТОЛІТНЬОГО
РОЗВИТКУ ТЕОРІЇ СИСТЕМ КЕРУВАННЯ
ТА ІДЕНТИФІКАЦІЇ. Частина 10. ПОЗИЦІЙНІ
КОНФЛІКТНО-КЕРОВАНІ ПРОЦЕСИ
Чикрій Аркадій Олексійович
Інститут кібернетики імені В.М. Глушкова НАН України, м. Київ,
g.chikrii@gmail.com
Губарев Вячеслав Федорович
Інститут космічних досліджень НАН України та ДКА України, м. Київ,
v.f.gubarev@gmail.com
Романенко Віктор Демидович
Навчально-науковий інститут прикладного системного аналізу Національного тех-
нічного університету України «Київський політехнічний інститут імені Ігоря Сі-
корського»,
romanenko.viktorroman@gmail.com, ipsa@kpi.ua
У статті розглядаються конфліктно-керовані процеси, які описуються уза-
гальненими квазілінійними системами диференціальних рівнянь. Досліджу-
ється ігрова задача зближення із заданою термінальною множиною циліндрич-
ного вигляду за наявності позиційної інформації щодо стану гри. Вважа-
ється, що у запропонованому розв’язку системи, який є аналогом формули
Коші, блок початкових даних відділений від блока керування. Внаслідок
цього можна розглядати широкий клас ігрових задач в єдиній схемі. Метод
дослідження ґрунтується на правилі екстремального прицілювання. Роз-
mailto:g.chikrii@gmail.com
mailto:v.f.gubarev@gmail.com
mailto:romanenko.viktorroman@gmail.com
mailto:ipsa@kpi.ua
28 ISSN 2786-6491
глянуто регулярний та регуляризований випадки. Останній формалізовано
в межах диференціальних включень. Загальна методика застосовується
до розв’язання ігрових задач, у яких динаміка процесу описується інтег-
ральними, інтегро-диференціальними рівняннями. Теоретичні результати
ілюструються модельними прикладами ігрових задач.
Ключові слова: правило екстремального прицілювання, опорна функція,
багатозначне відображення, вимірний вибір, інтегральне та інтегро-дифе-
ренціальне рівняння, диференціально-різницеві ігри, імпульсні системи,
процеси з дробовими похідними.
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Submitted 26.05.2025
https://www.mathnet.ru/links/7c1e37f6847466cf7bd01bfa1101744c/dan40630.pdf
https://doi.org/10.1016/0021-8928(79)90096-0
https://www.mathnet.ru/links/be6af4ca6a06f14
https://doi.org/%0b10.1007/978-94-017-1135-7
https://doi.org/%0b10.1007/978-94-017-1135-7
https://doi.org/10.1016/s0168-2024%2809%29x7002-5
https://doi.org/10.1016/0022-247X(65)90049-1
https://archive.org/details/smirnov-a-course-of-higher-mathematics-vol-4-integral-and-partial-dif%1fferential-equations/page/811/mode/2up
https://archive.org/details/smirnov-a-course-of-higher-mathematics-vol-4-integral-and-partial-dif%1fferential-equations/page/811/mode/2up
|
| id | nasplib_isofts_kiev_ua-123456789-211468 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0572-2691 |
| language | English |
| last_indexed | 2026-03-14T06:31:15Z |
| publishDate | 2025 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Chikrii, А. Gubarev, V. Romanenko, V. 2026-01-03T09:43:48Z 2025 Stages and main tasks of the century-long control theory and system identification development. Part X. Positional conflict-controlled processes / А. Chikrii, V. Gubarev, V. Romanenko // Проблемы управления и информатики. — 2025. — № 6. — С. 5-28. — Бібліогр.: 23 назв. — англ. 0572-2691 https://nasplib.isofts.kiev.ua/handle/123456789/211468 62-50 10.34229/1028-0979-2025-6-1 У статті розглядаються конфліктно-керовані процеси, які описуються узагальненими квазілінійними системами диференціальних рівнянь. Досліджується ігрова задача зближення із заданою термінальною множиною циліндричного вигляду за наявності позиційної інформації щодо стану гри. The paper concerns the conflict-controlled processes, described by the generalized quasi-linear systems of differential equations. We study the game problem of approaching the terminal set of a cylindrical form, under the positional information on the game state. en Інститут кібернетики ім. В.М. Глушкова НАН України Проблеми керування та інформатики Методи оптимізації та оптимальне керування Stages and main tasks of the century-long control theory and system identification development. Part X. Positional conflict-controlled processes Етапи та основні задачі столітнього розвитку теорії систем керування та ідентифікації. Частина 10. Позиційні конфліктно-керовані процеси Article published earlier |
| spellingShingle | Stages and main tasks of the century-long control theory and system identification development. Part X. Positional conflict-controlled processes Chikrii, А. Gubarev, V. Romanenko, V. Методи оптимізації та оптимальне керування |
| title | Stages and main tasks of the century-long control theory and system identification development. Part X. Positional conflict-controlled processes |
| title_alt | Етапи та основні задачі столітнього розвитку теорії систем керування та ідентифікації. Частина 10. Позиційні конфліктно-керовані процеси |
| title_full | Stages and main tasks of the century-long control theory and system identification development. Part X. Positional conflict-controlled processes |
| title_fullStr | Stages and main tasks of the century-long control theory and system identification development. Part X. Positional conflict-controlled processes |
| title_full_unstemmed | Stages and main tasks of the century-long control theory and system identification development. Part X. Positional conflict-controlled processes |
| title_short | Stages and main tasks of the century-long control theory and system identification development. Part X. Positional conflict-controlled processes |
| title_sort | stages and main tasks of the century-long control theory and system identification development. part x. positional conflict-controlled processes |
| topic | Методи оптимізації та оптимальне керування |
| topic_facet | Методи оптимізації та оптимальне керування |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211468 |
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