Stages and main tasks of the century-long control theory and system identification development. Part IX. Countercontrol in game problems of dynamics
This paper considers the problem of bringing the trajectory of quasilinear conflict-controlled process to a given cylindrical set. We proceed with representation of a trajectory of dynamic system in the form, in which the block of initial data is separated from the control block. This makes it feasi...
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| author | Chikrii, А. Gubarev, V. Romanenko, V. |
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| citation_txt | Stages and main tasks of the century-long control theory and system identification development. Part IX. Countercontrol in game problems of dynamics / А. Chikrii, V. Gubarev, V. Romanenko // Проблемы управления и информатики. — 2025. — № 4. — С. 24-40. — Бібліогр.: 30 назв. — англ. |
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| description | This paper considers the problem of bringing the trajectory of quasilinear conflict-controlled process to a given cylindrical set. We proceed with representation of a trajectory of dynamic system in the form, in which the block of initial data is separated from the control block. This makes it feasible to consider a wide spectrum of functional-differential systems. The method of resolving functions, based on use of the inverse Minkovski functionals, serves as ideological tool for study. Attention is focused on the case when Pontryagin’s condition does not hold. In this case the upper and lower resolving functions of two types are introduced. With their help sufficient conditions of approach a terminal set in a finite time are deduced. Various method schemes are provided and comparison with Pontryagin’s first direct method is given.
У статті досліджено ігрові задачі приведення траєкторії динамічної системи до заданої циліндричної термінальної множини. Запропоновано представлення траєкторії динамічної системи, де блок початкових даних відділено від блоку керування. Це дає змогу розглядати широкий спектр функціонально-диференціальних систем. Водночас увагу зосереджено на випадку, коли класична умова Понтрягіна не виконується. В цьому разі введено верхні та нижні розв’язувальні функції двох типів, за допомогою яких отримано достатні умови зближення з термінальною множиною за скінченний час для різних класів стратегій. Запропоновано різні схеми методу та надано порівняння з першим прямим методом Понтрягіна.
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| first_indexed | 2026-03-14T14:44:59Z |
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© А. CHIKRII, V. GUBAREV, V. ROMANENKO, 2025
24 ISSN 2786-6491
АДАПТИВНЕ КЕРУВАННЯ ТА МЕТОДИ ІДЕНТИФІКАЦІЇ
UDC 62-50
А. Chikrii, V. Gubarev, V. Romanenko
STAGES AND MAIN TASKS
OF THE CENTURY-LONG CONTROL
THEORY AND SYSTEM IDENTIFICATION
DEVELOPMENT. Part IX. COUNTERCONTROL
IN GAME PROBLEMS OF DYNAMICS
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 NAS of Ukraine and SSA of Ukraine, Kyiv,
https://orcid.org/0000-0001-6284-1866
v.f.gubarev@gmail.com
Viktor Romanenko
National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»,
Institute for Applied Systems Analysis, Kyiv,
romanenko.viktorroman@gmail.com, ipsa@kpi.ua
This paper considers the problem of bringing the trajectory of quasilinear con-
flict-controlled process to a given cylindrical set. We proceed with representa-
tion of a trajectory of dynamic system in the form, in which the block of initial
data is separated from the control block. This makes it feasible to consider a
wide spectrum of functional-differential systems. The method of resolving func-
tions, based on use of the inverse Minkovski functionals, serves as ideological
tool for study. Attention is focused on the case when Pontryagin’s condition
does not hold. In this case the upper and lower resolving functions of two types
are introduced. With their help sufficient conditions of approach a terminal set in
a finite time are deduced. Various method schemes are provided and comparison
with Pontryagin’s first direct method is given.
Keywords: conflict-controlled process, upper and lower resolving functions,
set-valued mapping, Pontryagin’s condition, L B -measurable mapping, Au-
mann’s integral, superposition measurability.
Introduction
There are a number of fundamental methods, which reveal structure of game prob-
lems and provide mathematical constructions for analysis of conflict situations. In the
mailto:g.chikrii@gmail.com
mailto:v.f.gubarev@gmail.com
mailto:romanenko.viktorroman@gmail.com
mailto:ipsa@kpi.ua
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 4 25
theory of conflict-controlled processes, along with method focused on making optimal
decisions, several methods provide a guaranteed result in a conflict confrontation.
Among the former are, first of all, the Isaacs method associated with the basic equation
of the theory of differential games [1], Pontryagin’s method of alternating integral [2]
and the method of Pshenichnyi semigroup T -operators [3]. The guaranteed result is
provided by Pontryagin’s first direct method [2], Krasovskii extreme targeting rule [4]
and the method of resolving functions [5].
The second group of methods, without focusing the research on the problem of op-
timality, justifies well-known from practice methods.
This work is devoted to the further development of the already mentioned method
of resolving functions. If we can say about the rule of extreme targeting that it is based
on the apparatus of support functions, then the method of resolving functions is based
on the inverse Minkowski functionals [5]. In the process of confrontation, the purser
builds his control on the control of the evader based on measurable choice theorems [6],
using the property of superposition measurability.
The attractive feature of the method of resolving functions is that it gives a com-
plete justification of the pursuit along the Euler line of sight, the rule of parallel pursuit,
the proportional navigation method and the rule of pursuit along a ray (motion camou-
flage) [7–10], which are well known to designers of rocket and space technology.
It also allows effective use of the modern apparatus of set-valued mappings and
their selections [11] in the justification of game structures and obtaining meaningful re-
sults based on them.
This paper is devoted to the problem of pursuit in the case when condition does not
hold. The upper and the lower resolving functions of two types are introduced that
makes it possible to broaden the class of problems solvable on the basis of the above
mentioned ideology.
1. Problem statement and players goals
Let us consider in finite-dimensional Euclidian space ,nE 2,n a conflict-cont-
rolled process whose evolution is given by the equation
0
( ) ( ) ( , ) ( ( ), ( )) ,
t
z t g t t u v d 0,t (1)
where ( ) ,nz t E the function of initial data ( ),g t : ,ng E E { : 0},E t t is the
Lebesgue measurable and bounded for 0,t the matrix function ( , ),t 0,t is
bounded almost everywhere, measurable in t and summable in for every .t E
Control unit includes function ( , ), : ,nu v U V E which is assumed to be jointly
continuous in its variables on direct product of nonempty compacts U and ,V
( ),mU K E ( ), , ,lV K E n m l are natural integers.
Admissible controls of the players, ( ),u : ,u E U ( ),v : ,v E V are
measurable functions of time.
Let the terminal set
*,M having the cylindrical form be given
*
0 ,M M M (2)
where 0M is linear subspace from ,nE аnd ( ),M K L 0L M is orthogonal com-
plement to 0M in .nE
26 ISSN 2786-6491
Goals of the players are opposite, moreover, the performance criterion is the time.
The first one ( )u tries in the shortest time to bring a trajectory (1) to the terminal
set (2), and the second one ( )v tries to delay maximally the instant of trajectory hitting
the set *M or avoid the hitting. General representation of the dynamical system solu-
tion in the form (1) makes it possible to consider in the frames of unified scheme a wide
range of quasilinear functional-differential systems, functioning under the condition of
conflict, in particular, the systems of ordinary differential, integral-differential and dif-
ferential-difference equations as well as the systems of equations with classical frac-
tional derivatives of Riemann–Liouville, regularized fractional derivatives of Dzhar-
bashian–Nersesian–Caputo, sequential fractional derivatives of Miller-Ross, the
generalized Hilfer derivatives [12] and the impulse systems [13]. Similar representation
in the discrete case makes it feasible to study multistep processes, in particular, the dis-
crete systems of fractional order by Grunwald–Letnikov [14] and the hybrid systems.
Specific forms of the function ( )g t and the matrix function ( , )t define the type
of a conflict–controlled process.
Let us take the side of the first player. If the game (1), (2) is evolving on the inter-
val [0, ],T then we select control at time instant t in the form of measurable function
( ) ( ( ), ( )),tu t u g T v ( ) ,u t U [0, ],t T (3)
where ( ) { ( ) : [0, ]},tv v s s t is a prehistory of admissible control of the second player
until the time instant t or, in the form of a counter-control,
( ) ( ( ), ( )),u t u g T v t ( ) ,u t U [0, ].t T (4)
In the case (3) we speak about quasi-strategy of the first player, in the case (4) —
about Hajek stroboscopic strategy [15], that prescribes counter-control by Krasovskii [4].
The goal of the paper is to establish sufficient conditions for termination of ap-
proach game (1), (2) in certain guaranteed time, using the second player control prehis-
tory, as well as in the class of stroboscopic strategies by comparing guaranteed time of
game termination with guaranteed time of Pontryagin’s first direct method [2, 16].
2. First direct method
Let us denote by the operator of orthogonal projecting from nE into .L We set
( , ) { ( , ) : }U v u v u U
and introduce set — valued mapping
( , , ) ( , ) ( , )W t v t U v
on the set ,V where
{( , ) : 0 }t t
is a plane cone.
Condition 1. Mapping ( , , )W t v is closed–valued on the direct product of cone
and compact .V
Let us consider
( , ) ( , , ),
v V
W t W t v
0 ,t
and denote [17]
dom {( , ) : ( , ) }.W t W t
Pontryagin’s condition. dom .W
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 4 27
If Condition 1 and Pontryagin’s condition hold, then mapping ( , )W t is closed-
valued and measurable in , [0, ].t Therefore [6], there exists at least one measura-
ble in selection, namely, Pontryagin’s selection.
Let us consider Pontryagin’s function, under the above mentioned conditions,
( ( )) inf { 0 : ( ( ))},p g t t P g
where
0
( ( )) 0 : ( ) ( , )
t
P g t g t M W t d
. (5)
Here by the integral of set-valued mapping is meant Aumann’s integral [18]. If in (5)
the inclusion in braces does not hold for all ,t 0,t then we set
( ( )) ,P g аnd ( ( )) .p g
Theorem 1. Let for conflict-controlled process (1), (2) Condition 1 and Pont-
ryagin’s condition hold, and ( ( )) .P P g
Then a trajectory of the process (1) can be brought into the terminal set (2) at in-
stant P by means of the Hajek stroboscopic strategy, which prescribes the correspon-
ding Krasovskii’ counter-control.
Proof. From the assumptions of Theorem 1 and the inclusion in the relation (5)
there follows that
0
( ) ( , ) .
P
g P M W P d
This means that there exist a point m, ,m M and, by definition of Aumann’s integral,
Pontryagin’s measurable selection ( , ),P ( , ) ( , ),P W P [0, ],P such that
0
( ) ( , ) .
P
g P m P d (6)
Let us consider the set-valued mapping
0( , ) { : ( , ) ( , ) ( , ) 0},U v u U P u v P [0, ],P .v V
The mapping 0 ( , )U v is closed-valued and L B -measurable [19, 11]. Therefore,
by the theorem on measurable choice [6], there exists at least one L B -measurable se-
lection 0 ( , ),u v 0 0( , ) ( , ),u v U v which appears as a superposition measurable func-
tion [19, 6]. Let us select control of the first player in the form of measurable function
0( ) ( , ( )),u u v [0, ],P where ( )v is an admissible control of the second player.
From the representation (1), in view of the relation (6) and equality in the expres-
sion for 0 ( , ),U v it immediately follows it that ( ) .g P m M
In next section we do not assume that Pontryagin condition holds and focus our in-
vestigations on development of the ideology of resolving functions [5, 19] in this case.
3. Schemes of the method of resolving functions
Let ( , ),t : ,L be some function, which is almost everywhere bounded,
measurable in t and summable in , [0, ]t for each .t E In the sequel, it will be
called the shift function.
28 ISSN 2786-6491
Let us denote
0
( ) ( , ( ), ( , )) ( ) ( , )
t
t t g t t g t t d
and consider the set-valued mapping
( , , ) { 0 :[ ( , ) ( , ) ( , )] [ ( )] },t v t u v t M t R (7)
: 2 .
E
V R
Condition 2. Set-valued mapping ( , , )t vR has nonempty images on the set .V
Under this condition, we consider scalar upper and lower resolving functions of the
first type
*( , , ) sup{ : ( , , )},t v t v R
*
( , , ) inf { : ( , , )}.t v t v R
If, in addition, Pontryagin’s condition holds, then, to emphasize the role of the
Minkowski functional [20, 17] and its inverse in the method scheme, we represent
the upper resolving function *( , , )t v in another way. To this end, we introduce
the inverse Minkowski functional [5] for closed set ,X ,nX R 0 : ( )XX p
sup{ 0 : },p X .np E
Then [19] *
( , , ) ( , )( , , ) sup ( ( )).W t v t
m M
t v m t
Since image of the mapping ( , , )t vR present itself numerical sets on the semi-
axis ,E then the upper resolving function is the support function of this mapping in
the direction +1. Taking into account properties of the conflict-controlled process (1), (2),
Conditions 1 and 2, and the theorem on characterization and inverse image [6], we can
show [19, 11] that closed — valued mapping ( , , )t vR is jointly L B -measurable in
( , ),v [0, ],t ,v V and the upper and the lower resolving functions are jointly
L B -measurable in ( , )v by virtue of the theorem on support function [6].
Let ( )V be a set of measurable functions ( ),v [0, ), taking their values in .V
Since at fixed t the function *( , , )t v is jointly L B -measurable in ( , ),v then it is
superposition measurable [19], i.e. *( , , ( ))t v is measurable in , [0, ],t for
each measurable function ( ) ( ).v V The lower resolving function possesses the
same property.
The upper resolving function *( , , ),t v in essence, reflects maximal gain of the
first player in the game at the time instant on interval [0, ]t under instantaneous
counter-action .v We associate with this function the set
*
( ) ( )
0
( ( ), ( , )) 0 : inf ( , , ( )) 1
t
v V
T g t t v d
(8)
and its least element is
( ( ), ( , )) inf { : ( ( ), ( , ))}.t g t t T g
If for some ,t 0,t *( , , ) ,t v [0, ],t ,v V then in this case it is natu-
ral to set the value of integral in the relation (8) to be equal to . Then the corre-
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 4 29
sponding inequality is readily satisfied and ( ( ), ( , )).t T g In case the inequality
in (8) does not hold for all 0,t we set ( ( ), ( , ))T g and correspondingly
( ( ), ( , )) .t g
Let us denote
( , ) ( , , ),
v V
t t v
R R ( , ) .t
Condition 3. The set-valued mapping ( , )t R has nonempty image on the cone .
Under this condition we introduce upper and lower resolving functions of the sec-
ond type
*( , ) sup{ : ( , )},t R t
*
( , ) inf { : ( , )}.t R t
Lemma. Let conditions 1 and 3 hold for the conflict-controlled process (1), (2), the
mapping ( , , )t vR be compact-valued, and the upper resolving function *( , , )t v be
bounded on the set .V Then the following inequality is true:
* *inf ( , , ) ( , ),
v V
t v t
( , ) .t (9)
If, in addition, the mapping ( , , ),R t v ( , ) ,t ,v V is convex-valued, then the
inequality (9) turns into equality.
Proof. By construction, the studied functions have the forms
*inf ( , , ) infsup{ : ( , , )},
v V v V
t v t v
R
*( , ) sup : ( , , ) ,
v V
t R t
0 .t
Let us denote * *( , ).t Since the mapping ( , , )t vR is compact-valued, then
( , )t R is compact-valued in *, * ( , , ),t v R for each .v V This implies
* sup{ : ( , , )} ,t v v V R
and
* *infsup{ : ( , , )} inf ( , , ).
v V v V
R t v t v
From the assumption that mapping ( , , )t vR is convex-valued and compact-valued
there follows that on the set V
*
*
( , , ) [ ( , , ), ( , , )].t v t v t v R
Then the non-emptity of the images of mapping ( , ),t R ( , ) ,t means that
*
*
( , ) [ ( , ), ( , )],t t t R moreover
* *
* *
( , ) sup ( , , ) inf ( , , ) ( , ),
v Vv V
t t v t v t
( , ) .t
Thereby equality in the relation (9) is proved.
Let us introduce into consideration the numerical functions
30 ISSN 2786-6491
* *
0
( ) ( , ) ,
t
t t d
* *
0
( ) ( , ) ,
t
t t d
( ( ), ( , )) inf { : ( ( ), ( , ))},g t t g
where
*( ( ), ( , )) { 0 : ( ) 1}.g t t
Theorem 2. Let for the conflict-controlled process (1), (2) Conditions 1, 3 hold, in
addition co ,M M moreover, for the given function ( )g and the shift ( , )
( ( ), ( , )) .T T g
Then, if
*
( ) 1T then a trajectory of the process (1) can be brought to the termi-
nal set (2) at the time instant ,T by means of control of the form (3), if, otherwise,
*( ) 1,T then by means of counter-control, for arbitrary admissible counteractions of
the second player.
Proof. Let ( ),v :[0, ]v T V be a measurable function. Let us assume that
*( , , ) ,T v [0, ],T .v V
We consider the testing function
*
*
0
( ) 1 ( , , ( )) ( , ) ,
t T
t
h t T v d T d [0, ].T
As it was mentioned above, the function *( , , )t v is L B -measurable in
( , ),v [0, ],T ,v V therefore, it is superposition measurable and, consequently,
*( , , ( ))T v is a function, measurable in , [0, ].T Function
*
( , )T is also
measurable in . Hence, ( )h t is absolutely continuous on the interval [0, ].T Since
* *
0
(0) 1 ( , ) 1 ( ) 0,
T
h T d T
and, by definition of moment of ,T
*
0
( ) 1 ( , , ( )) 0,
T
h T T v d
then, by the well-known theorem of mathematical analysis, there exists a time instant
*
,t
*
( ( )),t t v
*
[0, ],t T such that
*
( ) 0.h t Note that the instant of switching
*
t de-
pends on control prehistory of the second player
*
( ).tv
We shall call the time intervals
*
[0, )t and
*
[ , ]t T «active» and «passive», respec-
tively. Let us describe how the first player chooses its control on each of them. To this
end, let us consider the compact-valued mappings
1( , ) { : ( , ) ( , ) ( , )U v u U T u v T
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 4 31
*
*
( , , )[ ( )]}, [0, ),T v M T t
(10)
2( , ) { : ( , ) ( , ) ( , )U v u U T u v T
* *
( , )[ ( )]}, [ , ].T M T t T
From construction of the mappings ( , , )T vR and ( , )T R there follows that
( , ),iU v 1, 2,i have nonempty images.
By virtue of the theorem on inverse image [6] the set-valued mappings 1( , ),U v
2 ( , )U v are L B -measurable [6], therefore, by the theorem on measurable choice [6],
in both there exist at least one L B -measurable selection 1( , ),u v 2 ( , ),u v which are
superposition measurable functions.
Let us denote 1 1( ) ( , ( )),u u v 2 2( ) ( , ( )).u u v
We set control of the first player on the «active» interval to be equal to 1( ),u and
on the «passive» one — to 2 ( ).u Thus, despite the fact that the first player on each of
the interval does not use directly prehistory of control of the second player but only
its instantaneous control, in order to determine the switching instant
*
t the prehisto-
ry is required.
Under chosen controls, we have from representation (1)
*
1
0 0
( ) ( ) ( , ) ( ( ), ( )) ( ) ( , ) ( ( ), ( ))
tT
z T g T T u v d g T T u v d
*
2
0
( , ) ( ( ), ( )) ( , ) .
T T
t
T u v d T d
Using the relations (10), from above formula we deduce
*
*
*
*
0 0
( ) ( ) ( , , ( ))[ ( )] ( , )[ ( )] ( , )
t T T
t
z T g T T v M T d T M T d T d
* *
* *
* *
* *
0 0
( )(1 ( , , ( )) ( , ) ) ( , , ( )) ( , )
t tT T
t t
T T v d T d T v Md T Md
*
*
*
*
0
( , , ( )) ( , ) ) .
t T
t
T v d T d M M
Here the equality
*
( ) 0h t is twice taken into account, and transition of set-valued
mappings with set M can be substituted by usage of the support functions tech-
nique [17].
As follows from the expression (7), the case *( , , )T v for some [0, ],T
,v V is possible only under the conditions
0 ( ),M T 0 ( , ) ( , ) ( , ).T U v T
32 ISSN 2786-6491
It is evident that in this case
( , , ) [0, ),T v R [0, ],T ,v V
and
*
( , ) 0,T [0, ].T
This makes it possible to select as resolving functions at the points [0, ],T
where
*( , , ( )) ,T v an arbitrary finite, superposition measurable function,
which takes its values on the semi-infinite interval [0, ) with the only condition that
the final resolving function on the interval [0, ]T provides the relation
*
( ) 0h t for cer-
tain switching instant
*
,t
*
[0, ].t T Thereby the construction of control on «active»
and «passive» intervals is reduced to the previous case.
The case when
*( , , )T v for all [0, ],T ,v V corresponds to the first
Pontryagin method [2, 16]. Indeed, the inclusion
0 ( , ) ( , ) ( , ) [0, ], ,T U v T T v V
provides the non-emptiness of the set of images of Pontryagin’s mapping ( , )W T on
the game interval [0, ],T and the shift function ( , )T appears as measurable selection
of the mapping ( , ),W T i.e., the Pontryagin selection.
The inclusion 0 ( )M T implies the relation
0
( ) ( , ) ,
T
g T M W T d
from which, by virtue of Theorem 1, there follows that the game (1), (2) can be termi-
nated in the class of stroboscopic strategies.
Let us consider separately the case *( ) 1,T
*
( ) 1.T We introduce the testing
function
*
1 *
0
( ) 1 ( , ) ( , ) .
t T
t
h t T d T d
It is natural to examine only the case
*( , ) ,T [0, ].T Then
1 *
(0) 1 ( ) 0,h T *
1( ) 1 ( ) 0,h T T
and, by virtue of continuity of the function 1( ),h t there exists a moment
1
* ,t 1
* [0, ],t T
such that 1
1 *( ) 0.h t It should be noted that the time instant 1
*t does not depend on ( )v
in this case.
We consider set-valued mappings (10) on «active» and «passive» intervals 1
*
[0, )t
and
1
*
[ , ],t T with the function *( , )T replaced for *( , , )T v in the expression for
1( , ).U v Using the property for L B -measurability of compact-valued mappings
1( , ),U v 2 ( , ),U v we select in them L B -measurable selections, which determine
admissible controls on both intervals. Final considerations are similar to the conclusions
in previous case.
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 4 33
Let us introduce into consideration the functions
*
0
( ) inf ( , , ) ,
t
v V
t t v d
*1
( , ) inf ( , , ),
( ) v V
t t v
t
0 .t
Condition 4. For the chosen shift function ( , ),t ( , ) ,t the function
*inf ( , , )
v V
t v
is measurable in , [0, ],t 0,t and
* *
( ) ( )
0 0
inf ( , , ) inf ( , , ) ,
t t
v V v V
t v d t v d
0.t
Assumptions on the function *( , , ),t v ensuring fulfillment of the above equality
are analyzed, for example, in [11].
Theorem 3. Let for the game problem (1), (2) with some shift function ( , ),t
( , ) ,t ( ( ), ( , )) ,T T g Conditions 1, 3, 4 hold and the mapping ( , , )t vR be
convex-valued on the set ,V co .M M Let us suppose that the following inequality
is true:
*
( , ) sup ( , , ),
v V
T T v
[0, ].T (11)
Then a trajectory of the process (1) can be brought to the set (2) at the time-instant T
with help of certain counter-control.
Proof. It suffices to consider the case *( , , ) ,T v [0, ],T .v V Since,
by virtue of the inequality in (8), ( ) 1,T then
* *1
( , ) inf ( , , ) inf ( , , ),
( ) v V v V
T T v T v
T
[0, ],T
and, thereby,
*( , ) ( , , ) ,T T v v V [0, ].T
Taking into account the inequality (11), one can draw conclusion that ( , )T
( , , )R T v for ,v V [0, ],T and, consequently, ( , ) ( , ),T T R [0, ].T
Let us consider the set-valued mapping
( , ) { : ( , ) ( , ) ( , ) ( , )[ ( )]},U v u U T u v T T M T [0, ],T .v V (12)
Analogously to the previous case, the mapping ( , )U t is compact-valued and
L B -measurable, therefore, by the theorem on measurable choice [6], there exists a
measurable selection ( , ),u t ( , ) ( , ),u t U t which appears as superposition measura-
ble function. Then, if ( ),v ( ) ( ),v V is an arbitrary admissible control of the second
player, then we set control of the first player to be equal to
( ) ( , ( )),u u v [0, ].T
34 ISSN 2786-6491
From representation (1), with account of the inclusion in (12), we obtain
0 0
( ) ( ) 1 ( , ) ( , ) .
T T
z T T T d T Md
Since M is a convex compact, and ( , ),t [0, ],T is a nonnegative function,
moreover,
0
( , ) 1,
T
T d then
0
( , )
T
T Md M and, therefore, ( ) .z T M
4. Scheme with fixed points of the terminal set solid part
One of the assumptions of the above mentioned theorems, concerning the method
of resolving functions, is convexity of the set M (solid part of the terminal set). Now
we provide one scheme of the method of resolving functions without this assumption, in
which the targeting point of the set remains the same as time goes on.
For simplicity sake, we assume that Pontryagin’s condition holds. It immediately
follows that the lower resolving function equals zero and the upper one coincides with
the ordinary resolving function [19]. As previously, in this case ( , )t is a selection of
the set-valued mapping ( , ).W t We fix some point , ,m m M and set ( , ) ( ) ,t m t m
0.t Let us introduce a set-valued mapping
( , , , ) { 0: ( , ) ( , ) ( , ) ( , )}t v m t m t U v t R
and its support function in the direction +1
( , , , ) sup{ : ( , , , )}, 0, , .t v m R t v m t v V m M
Note that if ( , ) 0t m then ( , , , ) [0, )R t v m for [0, ], ,t v V and, con-
sequently, ( , , , ) .t v m
Let us consider the set
( )
0
( ( ), , ( , )) 0 : inf ( , , ( ), ) 1
t
v
T g m t t v m d
and its least element
( ( ), , ( , )) inf { 0 : ( ( ), , ( , ))},t g m t t T g m
under assumption that this element exists. If the function ( ( ), , ( , ))t g m is lower
semi-continuous in , ,m m M then it generates the marginal set
( ( ), ( , )) { : ( ( ), , ( , ) ( ( ), ( , ))},M g m M t g m t g
where
( ( ), ( , )) min ( ( ), , ( , )).
m M
t g t g m
For the sake of comparison, note that [19]
( )
0
( ( ), ( , )) inf 0 : inf sup ( , , ( ), ) 1 ,
t
v m M
t g t t v m d
( )
0
( ( ), , ( , )) inf 0 : sup inf ( , , ( ), ) 1 ,
t
vm M
t g m t t v m d
where sup ( , , , ) ( , , ).
m M
t v m t v
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 4 35
Theorem 4. Let for the conflict-controlled process (1), (2) Pontryagin’s condition
hold, and for given function ( ),g t some selection ( , ), ( , ) ( , )t t W t and point
, ,m m M
( ( ), , ( , )) .mT T g m
Then the projection of a trajectory (1) on the subspace L can be brought into the point
m at the moment mT with the help of certain control, prescribed by the quasi-strategy.
If, in addition, ( ( ), ( , )),m M g then it will occur at the moment ( ( ), ( , )).t g
Without going into details of the proof we only describe the procedure of control
construction. It should be mentioned that the line of reasoning is analogous, for exam-
ple, to that of the proof of Theorem 2.
If ( , ) 0,mT m then we divide the interval [0, ]mT into the active and passive
parts [0, )mt and [ , ],m mt T where the moment of switching mt is a root of the testing
function
0
( ) 1 ( , , ( ), ) .
t
m mh t T v m d
Then control of the first player on the active part [0, )mt is chosen in the form of
superposition measurable selection of the set-valued mapping
( , , ) { : ( , ) ( , ) ( , ) ( , , , ) ( , )},m m m mv m u U T u v T T v m T m U
and on the passive part [ , ]m mt T — analogously, but with ( , , , ) 0.mT v m
If, otherwise, ( , ) 0,mT m then on all interval [0, ]mT control of the first player
is chosen in the same way as on the passive part.
5. Connection of the first direct method with the method of resolving functions
Let us deduce relations between guaranteed times of the above mentioned methods.
Proposition 1. Let the game problem (1), (2) be given. Then for fulfillment of
Pontryagin’s condition: ( , ) ,W t ( , ) ,t it is necessary and sufficient that there
exists a shift function ( , )t such that
0 ( , , ) ( , ) ,T v t R .v V
Proof. Let the condition ( , ) ,W t ( , ) ,t hold. Then, by virtue of closure and
measurability of the mapping ( , ),W t from the theorem on measurable choice [6] there fol-
lows that there exists a measurable selection ( , ),t ( , ) ( , ).t W t This implies that
0 ( , ) ( , ) ( , )W t t t
or
0 ( , , ) ( , ) ( , ) ,W T v t t .v V
Thereby, zero value of in expression (7) provides non-emptiness of the sets in-
tersection and, therefore,
0 ( , , ) ( , ) ,T v t R .v V (13)
Consideration in inverse order results in the required conclusion.
36 ISSN 2786-6491
Thus, within the framework of Proposition 1 the shift function ( , )t appears as
the Pontryagin selection. We denote the set of such selections by . In addition, we
have that
0 ( , ) ( , )t t R
and the corresponding lower resolving functions are
* *
( , , ) ( , ) 0 ( , ) ,t v t t .v V (14)
Proposition 2. Let for some 0.t ( , ) ,W t [0, ].t In this case the inclusion
0
( ) ( , )
t
g t M W t d (15)
holds if and only if there exists a measurable in Pontryagin’s selection, such that
( , ( ), ( , )) .t g t t M
Proof. Let the inclusion (15) hold. Then, by definition of the Aumann integral there
exists a Pontryagin selection, such that
0
( ) ( , ) ( ) .
t
g t t d t M (16)
Conversely, if for some Pontryagin’s selection the relation (16) is true, then by
transferring the integral of selection into the right-hand side of (16) we, all the more, get
the inclusion (15).
Thus, if for some time instant t and the chosen Pontryagin’s selection the inclu-
sion (16) holds, then
( , , ) [0, ) ( , ) ,t v t R .v V
Thereby
( , ) [0, ),t R ( , ) .t
Hence, in this case the corresponding upper resolving functions of both types coincide:
* *( , , ) ( , ) ,t v t ( , ) ,t ,v V
and the corresponding lower resolving functions is equal to zero.
Before comparing guaranteed times of the considered methods let us deduce the
so-called functional form [19] of the first direct method, expressed through special re-
solving functions.
Let for conflict-controlled process (1), (2) Pontryagin’s condition hold. We analyze
the set-valued mapping
( , ) { 0 :[ ( , ) ( , )] [ ( )] },t W t t M t
where ( , )t is the Pontryagin’s selection which is measurable in , [0, ].t Its sup-
port function in the direction +1 is
*( , ) sup{ : ( , )},t t 0.t
Since 0 ( , )t for ( , ) ,t by virtue of the Pontryagin condition, then
*
( , ) inf { : ( , )} 0,t t 0.t
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 4 37
If ( ) ,t M then, by the theorem on characterization and inverse image [6], the
mapping ( , )t is closed-valued and measurable in , [0, ].t Correspondingly, on
the basis of the theorem on support function [6], the function
*( , )t is also measura-
ble in , [0, ].t In the case ( )t M for some 0t
( , ) [0, ),t [0, ],t
and *( , )t for all [0, ].t
The resolving function
*( , )t generates the guaranteed time
*
1
0
( ( ), ( , )) inf 0 : ( , ) 1
t
p g t t d
. (17)
Theorem 5. Let for conflict-controlled process (1), (2) the Pontryagin condition
hold, coM M and let for the function ( )g and the Pontryagin selection ( , )
1 1( ( ), ( , )) ,p p g
with infimum in the relation (17) attained.
Then a trajectory of the process (1) can be brought into the terminal set (2) at the
time instant 1p by some counter-control, therewith
1
( , )
inf ( ( ), ( , )) ( ( )).p g p g
. (18)
Proof of the first part of statement follows from the proof of Theorem 2, and the
equality (18) is obtained in the case of differential games in [21, Theorem 6]. In the case
under study the proof is similar.
From the above provided constructions, under assumptions of Theorem 1, 2, it
readily follows the inequality
( , )
inf ( ( ), ( , )) ( ( )).t g p g
(19)
The case of equality (19) is highlighted by the following statement.
Theorem 6. Let for the conflict-controlled process (1), (2) Pontryagin’s condition
hold and let us suppose that for some shift function the mapping ( , , ),t vR ( , ) ,t
,v V is convex-valued, besides set M is convex.
Then, if for some ,T ( ( ), ( , ))T T g the condition
[ ( , , ) ( , ) ] ( , ) ( , ) ,
v V
W T v T M W T T M
0 ,T (20)
holds, then
( , ) ( , )
inf ( ( ), ( , )) inf ( ( ), ( , )) ( ( )).t g g p g
Proof. It will suffice to prove the inequality
( , )
( ( )) inf ( ( ), ( , )).p g t g
(21)
38 ISSN 2786-6491
Since ( , ) ( , , ),T T v R ( , ) ,t ,v V then, by virtue of the property for
convexity of the mapping ( , , ),T vR we have from the relation (7)
[ ( , , ) ( , )] ( , )[ ( )] ,W T v T T M T [0, ],T .v V
The latter is equivalent to the inclusion
0 ( , , ) ( , ) ( , ) ( ) ( , ),W T v T M T T T [0, ],T ,v V
or
0 [ ( , , ) ( , ) ] ( , ) ( ) ( , ),
v V
W T v T M T T T
[0, ].T
Taking into account the condition (20) we get
0 ( , ) ( , ) ( , ) ( ) ( , ),W T T M T T T [0, ].T
Upon integration from 0 to T both parts of this inclusion and taking into account
the condition co ,M M we have
0
( ) ( , ) .
T
g T M W T d
This implies the inequality (21) and, therefore, the theorem is proved.
Applications of approach methods for conflict-controlled processes to the design of
control systems are presented, for example, in [22], for differential-difference system
in [23, 24], for the game problems with terminal payoff, integral constraints, for objects
with different inertia [25, 26]. The upper and lower resolving function are introduced
in [27]. The methodology is developed in [28]. The alternative approach deals with time
dilation and is presented in [29, 30].
Conclusion
Various schemes of the method of resolving functions for investigation of the qua-
si-linear game dynamic problems are presented in the paper. In so doing, the case, when
the classic Pontryagin’s condition does not hold, is analyzed. To tackle this difficulty,
we take into consideration the upper and the lower resolving functions of two types.
This makes it feasible to derive sufficient conditions for approach a cylindrical terminal
set by a trajectory of conflict-controlled process in the class of quasi- and stroboscopic
strategies. That a solution of dynamic system is presented in a rather general form al-
lows to encompass in unified form a wide spectrum of functional-differential systems.
The comarison of guaranteed times of the schemes of the first direct method and the
method of resolving functions is made.
А.О. Чикрій, В.Ф. Губарев, В.Д. Романенко
ЕТАПИ ТА ОСНОВНІ ЗАДАЧІ СТОЛІТНЬОГО
РОЗВИТКУ ТЕОРІЇ СИСТЕМ КЕРУВАННЯ
ТА ІДЕНТИФІКАЦІЇ. Частина 9. КОНТРКЕРУВАННЯ
В ІГРОВИХ ЗАДАЧАХ ДИНАМІКИ
Чикрій Аркадій Олексійович
Інститут кібернетики імені В.М. Глушкова НАН України, м. Київ,
g.chikrii@gmail.com
mailto:g.chikrii@gmail.com
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 4 39
Губарев Вячеслав Федорович
Інститут космічних досліджень НАН України та ДКА України, м. Київ,
v.f.gubarev@gmail.com
Романенко Віктор Демидович
Національний технічний університет України «Київський політехнічний інститут
імені Ігоря Сікорського», Навчально-науковий інститут прикладного системного
аналізу, м. Київ,
romanenko.viktorroman@gmail.com, ipsa@kpi.ua
У статті досліджено ігрові задачі приведення траєкторії динамічної
системи до заданої циліндричної термінальної множини. Запропонова-
но представлення траєкторії динамічної системи, де блок початкових
даних відділено від блоку керування. Це дає змогу розглядати широ-
кий спектр функціонально-диференціальних систем. Водночас увагу
зосереджено на випадку, коли класична умова Понтрягіна не виконує-
ться. В цьому разі введено верхні та нижні розв’язувальні функції
двох типів, за допомогою яких отримано достатні умови зближення з
термінальною множиною за скінченний час для різних класів страте-
гій. Запропоновано різні схеми методу та надано порівняння з першим
прямим методом Понтрягіна.
Ключові слова: конфліктно-керований процес, верхні та нижні розв’я-
зувальні функції, опорна функція, багатозначне відображення, умова Понт-
рягіна, L B -вимірне відображення, інтеграл Ауманна, суперпозиційна
вимірність.
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https://www.taylorfrancis.com/search?contributorName=Vyacheslav%20F.%20Gubarev&contributorRole=editor&redirectFromPDP=true&context=ubx
https://doi.org/%0b10.1201/9781003339229
https://doi.org/%0b10.1201/9781003339229
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| id | nasplib_isofts_kiev_ua-123456789-211408 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0572-2691 |
| language | English |
| last_indexed | 2026-03-14T14:44:59Z |
| publishDate | 2025 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Chikrii, А. Gubarev, V. Romanenko, V. 2026-01-01T22:35:40Z 2025 Stages and main tasks of the century-long control theory and system identification development. Part IX. Countercontrol in game problems of dynamics / А. Chikrii, V. Gubarev, V. Romanenko // Проблемы управления и информатики. — 2025. — № 4. — С. 24-40. — Бібліогр.: 30 назв. — англ. 0572-2691 https://nasplib.isofts.kiev.ua/handle/123456789/211408 62-50 10.34229/1028-0979-2025-4-2 This paper considers the problem of bringing the trajectory of quasilinear conflict-controlled process to a given cylindrical set. We proceed with representation of a trajectory of dynamic system in the form, in which the block of initial data is separated from the control block. This makes it feasible to consider a wide spectrum of functional-differential systems. The method of resolving functions, based on use of the inverse Minkovski functionals, serves as ideological tool for study. Attention is focused on the case when Pontryagin’s condition does not hold. In this case the upper and lower resolving functions of two types are introduced. With their help sufficient conditions of approach a terminal set in a finite time are deduced. Various method schemes are provided and comparison with Pontryagin’s first direct method is given. У статті досліджено ігрові задачі приведення траєкторії динамічної системи до заданої циліндричної термінальної множини. Запропоновано представлення траєкторії динамічної системи, де блок початкових даних відділено від блоку керування. Це дає змогу розглядати широкий спектр функціонально-диференціальних систем. Водночас увагу зосереджено на випадку, коли класична умова Понтрягіна не виконується. В цьому разі введено верхні та нижні розв’язувальні функції двох типів, за допомогою яких отримано достатні умови зближення з термінальною множиною за скінченний час для різних класів стратегій. Запропоновано різні схеми методу та надано порівняння з першим прямим методом Понтрягіна. en Інститут кібернетики ім. В.М. Глушкова НАН України Проблеми керування та інформатики Адаптивне керування та методи ідентифікації Stages and main tasks of the century-long control theory and system identification development. Part IX. Countercontrol in game problems of dynamics Етапи та основні задачі столітнього розвитку теорії систем керування та ідентифікації. Частина 9. Контркерування в ігрових задачах динаміки Article published earlier |
| spellingShingle | Stages and main tasks of the century-long control theory and system identification development. Part IX. Countercontrol in game problems of dynamics Chikrii, А. Gubarev, V. Romanenko, V. Адаптивне керування та методи ідентифікації |
| title | Stages and main tasks of the century-long control theory and system identification development. Part IX. Countercontrol in game problems of dynamics |
| title_alt | Етапи та основні задачі столітнього розвитку теорії систем керування та ідентифікації. Частина 9. Контркерування в ігрових задачах динаміки |
| title_full | Stages and main tasks of the century-long control theory and system identification development. Part IX. Countercontrol in game problems of dynamics |
| title_fullStr | Stages and main tasks of the century-long control theory and system identification development. Part IX. Countercontrol in game problems of dynamics |
| title_full_unstemmed | Stages and main tasks of the century-long control theory and system identification development. Part IX. Countercontrol in game problems of dynamics |
| title_short | Stages and main tasks of the century-long control theory and system identification development. Part IX. Countercontrol in game problems of dynamics |
| title_sort | stages and main tasks of the century-long control theory and system identification development. part ix. countercontrol in game problems of dynamics |
| topic | Адаптивне керування та методи ідентифікації |
| topic_facet | Адаптивне керування та методи ідентифікації |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211408 |
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