A Pursuit Problem in an Infinite System of Second-Order Differential Equations
We study a pursuit differential game problem for an infinite system of second-order differential equations. The control functions of players, i.e., a pursuer and an evader are subject to integral constraints. The pursuit is completed if z(τ) = \( \dot{z} \) (τ) = 0 at some τ > 0, where z(t)...
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Institute of Mathematics, NAS of Ukraine
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| author | Allahabi, F. Ibragimov, G. I. Kuchkarov, A. Аллахабі, Ф. Ібрагімов, Г. І. Кучкаров, А. |
| author_facet | Allahabi, F. Ibragimov, G. I. Kuchkarov, A. Аллахабі, Ф. Ібрагімов, Г. І. Кучкаров, А. |
| author_sort | Allahabi, F. |
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| datestamp_date | 2020-03-18T19:16:44Z |
| description | We study a pursuit differential game problem for an infinite system of second-order differential equations. The control functions of players, i.e., a pursuer and an evader are subject to integral constraints. The pursuit is completed if z(τ) = \( \dot{z} \) (τ) = 0 at some τ > 0, where z(t) is the state of the system. The pursuer tries to complete the pursuit and the evader tries to avoid this. A sufficient condition is obtained for completing the pursuit in the differential game when the control recourse of the pursuer is greater than the control recourse of the evader. To construct the strategy of the pursuer, we assume that the instantaneous control used by the evader is known to the pursuer. |
| first_indexed | 2026-03-24T02:24:29Z |
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UDC 517.9
G. Ibragimov (Univ. Putra Malaysia, Malaysia; Ins. Math., Uzbekistan),
F. Allahabi (Univ. Putra Malaysia, Malaysia),
A. Kuchkarov (Univ. Putra Malaysia, Malaysia; Ins. Math., Uzbekistan)
A PURSUIT PROBLEM IN AN INFINITE SYSTEM OF SECOND-ORDER
DIFFERENTIAL EQUATIONS
ПРОБЛЕМА ПЕРЕСЛIДУВАННЯ В НЕСКIНЧЕННIЙ СИСТЕМI
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ ДРУГОГО ПОРЯДКУ
We study a pursuit differential game problem for an infinite system of second-order differential equations. The control
functions of players, i. e., the pursuer and the evader, are subject to integral constraints.The pursuit is completed if
z(τ) = ż(τ) = 0 at some τ > 0, where z(t) is the state of the system. The pursuer tries to complete the pursuit and
the evader tries to avoid this. A sufficient condition is obtained for completing the pursuit in the differential game when
control recourse of the pursuer greater than that of the evader. To construct the strategy of the pursuer we assume that the
instantaneous control employed by the evader is known to the pursuer.
Вивчається проблема переслiдування в диференцiальнiй грi для нескiнченної системи диференцiальних рiвнянь
другого порядку. Керiвнi функцiї гравцiв, тобто переслiдувача та переслiдуваного, мають деякi обмеження. Пере-
слiдування завершується, коли z(τ) = ż(τ) = 0 для деякого τ > 0, де z(t) — стан системи. Переслiдувач намагається
завершити переслiдування, а переслiдуваний намагається цього уникнути. Встановлено достатню умову завершення
переслiдування в диференцiальнiй грi, коли зворотнe управлiння для переслiдувача бiльше, нiж для переслiдувано-
го. Для побудови стратегiї переслiдувача вважаємо, що миттєве керування, застосоване переслiдуваним, є вiдомим
переслiдувачу.
1. Introduction. The study of two person zero-sum differential games was initiated by Isaacs [1].
Since then many works with various approaches have been done in developing the theory of dif-
ferential games (see, for example, [1 – 4]). Control and differential game problems in systems with
distributed parameters were studied by many researchers (see, for example, [5 – 15]).
Works [12 – 15] concerned with the differential game problems described by the following infinite
system of differential equations:
z̈k(t) + µkzk(t) = −uk(t) + vk(t), (1)
where µk are positive numbers, uk, k = 1, 2, . . . , are control parameters of the pursuer, and vk,
k = 1, 2, . . . , are those of the Evader.
In [15], the numbers µk are assumed to be any positive numbers, and the control functions of
the players are subject to integral constraints. In [12], a differential game described by hyperbolic
equation is reduced to that described by the infinite system of differential equations (1). Here the
numbers µk are generalized eigenvalues of the elliptic operator
Az = −
n∑
i,j=1
∂
∂xi
(
aij(x)
∂z
∂xj
)
and satisfy the conditions
c© G. IBRAGIMOV, F. ALLAHABI, A. KUCHKAROV, 2013
1080 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
A PURSUIT PROBLEM IN AN INFINITE SYSTEM OF SECOND-ORDER DIFFERENTIAL EQUATIONS 1081
0 < µ1 ≤ µ2 ≤ . . .→∞.
Authors studied differential game problems with various constraints on control functions of
players.
The general purpose of the present paper is to investigate differential game problem described by
the following infinite system of differential equations:
ẍk = −αkxk − βkyk − uk1 + vk1, xk(0) = xk0, ẋk(0) = xk1,
ÿk = βkxk − αkyk − uk2 + vk2, yk(0) = yk0, ẏk(0) = yk1,
k = 1, 2, . . . , (2)
where αk, βk are real numbers, xk, yk, xk0, yk0, xk1, yk1, uk1, uk2, vk1, vk2 ∈ R1, u = (u11, u12, u21,
u22, . . .) and v = (v11, v12, v21, v22, . . .) are control parameters of the pursuer and the evader respec-
tively. Note that the system (2) is obtained if we take
zk = xk + iyk, µk = αk − iβk, uk = uk1 + iuk2, vk = vk2 + ivk2
in (1). In other words, we deal with so-called the complex case of the equation (1).
Pursuit is said to be completed in the game described by the infinite system of differential
equations (2) if xk(τ) = 0, yk(τ) = 0, ẋk(τ) = 0, ẏk(τ) = 0, k = 1, 2, . . . at some τ > 0. In the
literature, such differential game problems are called “soft” landing or “soft” capture problems.
In the case of finite dimensional space, a number of works on this subject have been published
see, e. g., [16 – 18]. In the work [16], a “soft landing” game problem is studied, where the dynamics
of the players models the motion of different-type objects in a medium with friction. The goal of the
pursuer is the approach of geometric coordinates and the velocities of the players (soft landing) at a
certain finite instant of time. Sufficient conditions on the parameters of the conflict-control process
were obtained under which the “soft landing” problem is solvable at a finite time.
A game problem of pursuit of a controlled object moving in a horizontal plane, by another
object, moving in a three-dimensional space, is studied in [17]. Sufficient conditions on parameters
of a conflict-controlled object were derived, for which the soft landing may be performed.
In the paper [18], a “soft landing” differential game of many pursuers and one evader described
by the generalized Pontryagin example is studied. By definition the evader is said to be captured if
its state, velocity, and acceleration coincide with those of a pursuer. Under the assumption that the
roots of characteristic equation are real, sufficient capture conditions were obtained in terms of initial
states.
2. Statement of problem. Let λ1, λ2, . . . be a sequence of positive numbers, and r be a fixed
number. We introduce into the consideration the space
l2r =
{
ξ = (ξ1, ξ2, . . .) :
∞∑
i=1
λri ξ
2
i <∞
}
with the inner product and norm
(ξ, η) =
∞∑
i=1
λri ξiηi, ξ, η ∈ l2r , ‖ξ‖l2r =
( ∞∑
i=1
λri ξ
2
i
)1/2
.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1082 G. IBRAGIMOV, F. ALLAHABI, A. KUCHKAROV
From now on λk =
√
α2
k + β2k, k = 1, 2, . . . .
Denote
z(t) =
(
z1(t), z2(t), . . .
)
, zk(t) =
(
xk(t), yk(t)
)
, |zk| =
√
x2k + y2k,
‖z‖2l2r+1
=
∞∑
k=1
λr+1
k
(
x2k + y2k
)
,
z0 = (z10, z20, . . .) = (x10, y10, x20, y20, . . .), zk0 = (xk0, yk0),
z1 = (z11, z21, . . .) = (x11, y11, x21, y21, . . .), zk1 = (xk1, yk1).
We assume that z0 ∈ l2r+1, z1 ∈ l2r .
Let L2(0, T ; l
2
r) be the space of functions f(t) = (f1(t), f2(t), . . .), f : [0, T ]→ l2r , with measu-
rable coordinates fk(t) =
(
fk1(t), fk2(t)
)
, 0 ≤ t ≤ T, such that
∥∥f(·)∥∥
L2(0,T ;l2r)
=
∞∑
k=1
λrk
T∫
0
(
f2k1(t) + f2k2(t)
)
dt <∞,
where T is any given positive number.
Let ρ0, ρ, and σ be given positive numbers.
Definition 1. A function w(·) ∈ L2(0, T ; l
2
r) subjected to the the condition
∞∑
k=1
λrk
T∫
0
(
w2
1k(s) + w2
2k(s)
)
ds ≤ ρ20
is referred to as the admissible control. We denote the set of all admissible controls by S(ρ0).
Definition 2. A function u(·) ∈ S(ρ) (respectively v(·) ∈ S(σ)) is referred to as the admissible
control of the pursuer (the evader).
Definition 3. A function
u(t, v) = (u1(t, v), u2(t, v), . . .), u : [0, T ]× l2r → l2r , uk(t, v) =
(
uk1(t, v), uk2(t, v)
)
,
of the form
uk(t, v) = vk(t) + ωk(t), ω(·) =
(
ω1(·), ω2(·), . . .
)
∈ S(ρ− σ), ωk(·) =
(
ωk1(·), ωk2(·)
)
,
where v(·) ∈ S(σ), is called a strategy of the pursuer.
Definition 4. If there exists a strategy u(·) of the pursuer such that z(τ) = 0, ż(τ) = 0 at
some τ, 0 ≤ τ ≤ ϑ, for any control of the evader, then we say that differential game (2) can be
completed for the time ϑ.
The pursuer tries to complete the game as soon as possible while the aim of the evader is
opposite.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
A PURSUIT PROBLEM IN AN INFINITE SYSTEM OF SECOND-ORDER DIFFERENTIAL EQUATIONS 1083
Definition 5. Let w(·) = (w1(·), w2(·), . . .) ∈ L2(0, T, l
2
r), wk(·) =
(
wk1(·), wk2(·)
)
. The
function z(t) =
(
z1(t), z2(t), . . .
)
, 0 ≤ t ≤ T, where each coordinate zk(t)
1) is continuously differentiable on (0, T ) and satisfies the initial conditions zk(0) = zk0,
żk(0) = zk1,
2) has the second derivative z̈k(t) almost everywhere on (0, T ) satisfying the equation
z̈k(t) = Dkzk(t) + wk(t), Dk =
[
−αk −βk
βk −αk
]
,
almost everywhere on [0, T ] is called the solution of the system
z̈k(t) = Dkzk(t) + wk(t), zk(0) = zk0, żk(0) = zk1, k = 1, 2, . . . . (3)
Let
Ak1(t) = er1kt
[
cos(r2kt) − sin(r2kt)
sin(r2kt) cos(r2kt)
]
, Ak2(t) = Ak1(−t), Rk =
[
r1k −r2k
r2k r1k
]
,
Ak(t) =
1
2
(
Ak1(t) +Ak2(t)
)
, Bk(t) =
1
2
R−1k
(
Ak1(t)−Ak2(t)
)
,
r1k =
√√√√−αk +√α2
k + β2k
2
, r2k =
√√√√αk +
√
α2
k + β2k
2
, k = 1, 2, . . . .
Clearly, rk =
√
r21k + r22k =
4
√
α2
k + β2k =
√
λk.
It can be shown that the matrices Ak1(t), Ak2(t) have the following properties:
Ak1(t+ h) = Ak1(t)Ak1(h) = Ak1(h)Ak1(t),
∣∣Ak1(t)zk∣∣ = ∣∣A∗k1(t)zk∣∣ = er1kt|zk|,
Ak2(t+ h) = Ak2(t)Ak2(h) = Ak2(h)Ak2(t),
∣∣Ak2(t)zk∣∣ = ∣∣A∗k2(t)zk∣∣ = e−r1kt|zk|,
where A∗ denotes the transpose of the matrix A, and E2 does the identity (2× 2)-matrix.
It is easy to verify that
Ȧk1(t) = RkAk1(t), Ȧk2(t) = −RkAk2(t), Ȧk(t) = R2
kBk(t), Ḃk(t) = Ak(t).
By using the properties
Ak1(t+ h) = Ak1(t)Ak1(h), Ak2(t+ h) = Ak2(t)Ak2(h),
Ak1(t)Ak2(h) = Ak1(t− h), Ak1(t) = Ak2(−t)
it can be easily proved that
A2
k(t)−R2
kB
2
k(t) = E2, (4)
Ak(t)Bk(t− s)−Bk(t)Ak(t− s) = −Bk(s), (5)
Ak(t)Ak(t− s)−R2
kBk(t)Bk(t− s) = Ak(s). (6)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1084 G. IBRAGIMOV, F. ALLAHABI, A. KUCHKAROV
3. Control problem. Consider the infinite system of differential equations (3). Let C(0, T ; l2r) be
the space of continuous functions z(t), 0 ≤ t ≤ T, with the values in the space l2r . It can be shown
similarly to [19] that the following assertion is true.
Assertion. If {r1k}k∈N is a bounded above sequence, then the infinite system of differential
equations (3) has a unique solution z(·) ∈ C(0, T ; l2r+1) defined by
zk(t)
.
= Ak(t)zk0 +Bk(t)zk1 +
t∫
0
Bk(t− s)wk(s)ds, k = 1, 2, . . . . (7)
It can be verified that
żk(t) = R2
kBk(t)zk0 +Ak(t)zk1 +
t∫
0
Ak(t− s)wk(s)ds.
We transform the system
zk(t) = Ak(t)zk0 +Bk(t)zk1 +
t∫
0
Bk(t− s)wk(s)ds,
żk(t) = R2
kBk(t)zk0 +Ak(t)zk1 +
t∫
0
Ak(t− s)wk(s)ds,
k = 1, 2, . . . , (8)
by setting ηk(t)
ξk(t)
=
Ak(t) −RkBk(t)
−RkBk(t) Ak(t)
Rkzk(t)
żk(t)
,
ηk0
ξk0
=
Rkzk0
zk1
. (9)
Then using (4), (5) and (6), we obtain
ηk(t) = RkAk(t)zk(t)−RkBk(t)żk(t) =
= Rk
(
A2
k(t)−R2
kB
2
k(t)
)
zk0 +Rk
(
Ak(t)Bk(t)−Bk(t)Ak(t)
)
zk1+
+
t∫
0
Rk (Ak(t)Bk(t− s)−Bk(t)Ak(t− s))wk(s)ds =
= Rkzk0 −
t∫
0
RkBk(s)wk(s)ds = ηk0 −
t∫
0
RkBk(s)wk(s)ds,
ξk(t) = −R2
kBk(t)zk(t) +Ak(t)żk(t) =
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
A PURSUIT PROBLEM IN AN INFINITE SYSTEM OF SECOND-ORDER DIFFERENTIAL EQUATIONS 1085
=
(
−R2
kBk(t)Ak(t) +Ak(t)R
2
kBk(t)
)
zk0 +
(
−R2
kB
2
k(t) +A2
k(t)
)
zk1+
+
t∫
0
(
−R2
kBk(t)Bk(t− s) +Ak(t)Ak(t− s)
)
wk(s)ds =
= zk1 +
t∫
0
Ak(s)wk(s)ds = ξk0 +
t∫
0
Ak(s)wk(s)ds.
Our goal is to realize ηk(t) = 0, ξk(t) = 0, for all k = 1, 2, . . . , at some time t. They are
equivalent to
ηk0 =
t∫
0
RkBk(s)wk(s)ds,
− ξk0 =
t∫
0
Ak(s)wk(s)ds,
k = 1, 2, . . . . (10)
We shall find a condition on ηk0, ξk0, k = 1, 2, . . . , to be found a control
(
w1(·), w2(·), . . .
)
∈
∈ S(σ) guaranteeing (10). To this end we study some properties of the set
Xk(ϑ, σk) =
(η, ξ) | η =
ϑ∫
0
RkBk(s)wk(s)ds, ξ =
ϑ∫
0
Ak(s)wk(s)ds, wk(.) ∈ Sϑ(σk)
,
where
∞∑
k=1
σ2k = σ2, σk ≥ 0 and Sϑ(σk) =
wk(·)
∣∣∣∣∣
ϑ∫
0
∣∣wk(s)∣∣2ds ≤ σ2k
.
Let ψk ∈ Xk(ϑ, σk) be any point and e ∈ R4 be a unit vector. We find a control wk(·) for that
ψk = (ηk, ξk) belongs to the boundary of Xk(ϑ, σk). By using the Cauchy – Schwartz inequality we
get
〈ψk, e〉 =
ϑ∫
0
〈(
RkBk(s)
Ak(s)
)
wk(s), e
〉
ds =
=
ϑ∫
0
〈Ck(s)wk(s), e〉 ds =
ϑ∫
0
〈wk(s), C∗k(s)e〉 ds ≤
≤ σk
ϑ∫
0
|C∗k(s)e|2ds
1/2 = σkF
1/2
k (ϑ, e),
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1086 G. IBRAGIMOV, F. ALLAHABI, A. KUCHKAROV
where
Ck(s) =
(
RkBk(s)
Ak(s)
)
, Fk(ϑ, e) =
ϑ∫
0
∣∣C∗k(s)e∣∣2ds.
Note that the equality occurs if
wk(s) =
σk√
Fk(ϑ, e)
C∗k(s)e, (11)
almost everywhere on [0, ϑ]. It can be shown that the point ψk is on the boundary ∂Xk(ϑ, σk) of the
set Xk(ϑ, σk) whenever the control wk(·) has the form (11).
We obtain
Fk(ϑ, e) =
ϑ∫
0
∣∣C∗k(s)e∣∣2ds =
〈
e,
ϑ∫
0
Ck(s)C
∗
k(s)e ds
〉
= 〈e, Pk(ϑ)e〉, (12)
where
Pk(ϑ) =
ϑ∫
0
Ck(s)C
∗
k(s)ds =
c1k 0 c2k c3k
0 c1k c3k c2k
c2k c3k c4k 0
−c3k c2k 0 c4k
,
c1k =
1
4r1k
sinh(2r1kϑ)−
1
4r2k
sin(2r2kϑ), c2k =
1
2r1k
sinh2(r1kϑ),
c3k =
1
2r2k
sin2(r2kϑ), c4k =
1
4r1k
sinh(2r1kϑ) +
1
4r2k
sin(2r2kϑ).
We study now some properties of eigenvalues and eigenvectors of the matrix Pk(ϑ). It is not difficult
to verify that the eigenvalues of the matrix Pk(ϑ) are
m1(ϑ) = m2(ϑ) =
1
4r1k
sinh(2r1kϑ)−
√
1
4r21k
sinh4(r1kϑ) +
1
4r22k
sin4(r2kϑ),
m3(ϑ) = m4(ϑ) =
1
4r1k
sinh(2r1kϑ) +
√
1
4r21k
sinh4(r1kϑ) +
1
4r22k
sin4(r2kϑ).
Eigenvectors associated with these eigenvalues are
e1(ϑ) =
c2km1(ϑ)− c2kc4k
c22k + c23k
c3km1(ϑ)− c3kc4k
c22k + c23k
1
0
, e2(ϑ) =
−c3km1(ϑ) + c3kc4k
c22k + c23k
c2km1(ϑ)− c2kc4k
c22k + c23k
0
1
,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
A PURSUIT PROBLEM IN AN INFINITE SYSTEM OF SECOND-ORDER DIFFERENTIAL EQUATIONS 1087
e3(ϑ) =
c2km3(ϑ)− c2kc4k
c22k + c23k
c3km3(ϑ)− c3kc4k
c22k + c23k
1
0
, e4(ϑ) =
−c3km3(ϑ) + c3kc4k
c22k + c23k
c2km3(ϑ)− c2kc4k
c22k + c23k
0
1
.
Hence, Pk(ϑ)ei(ϑ) = mi(ϑ)ei(ϑ), i = 1, 2, 3, 4. Note that ei(ϑ), i = 1, 2, 3, 4, is an orthonormal
system in R4.
Property 1. The eigenvalues of the matrix Pk(ϑ) are positive for all ϑ > 0.
Proof. It is sufficient to show that m1(ϑ) > 0 for all ϑ > 0.
We have
m1(ϑ)m3(ϑ) = g(ϑ)h(ϑ),
where
g(ϑ) =
1
2r1
sinh(r1ϑ)−
1
2r2
sin2(r2ϑ), h(ϑ) =
1
2r1
sinh(r1ϑ) +
1
2r2
sin2(r2ϑ).
It is obvious that
g′(ϑ) =
1
2
(cosh(r1ϑ)− sin(2r2ϑ)) > 0, ϑ > 0,
since cosh(t) > 1 ≥ sin(t) for all t > 0. As g(0) = 0, g′(ϑ) > 0, ϑ > 0, then g(ϑ) > 0, ϑ > 0.
Since g(ϑ) > 0, m3(ϑ) > 0, and h(ϑ) > 0 for all ϑ > 0, we obtain m1(ϑ) > 0.
Property 2. The set ∂Xk(ϑ, σk), k ∈ {1, 2, . . .}, is an ellipsoid in R4.
Proof. Let ψk ∈ ∂Xk(ϑ, σk) and e(ϑ) =
∑4
i=1
diei(ϑ), where the numbers di satisfy the
condition
∑4
i=1
d2i = 1. By (11) we obtain
ψk =
ϑ∫
0
Ck(s)wk(s)ds =
σk√
Fk(ϑ, e(ϑ))
ϑ∫
0
Ck(s)C
∗
k(s)ds
e(ϑ) =
σk√
Fk(ϑ, e(ϑ))
Pk(ϑ)e(ϑ).
It follows from (12) that
Fk(ϑ, e(ϑ)) = 〈 e(ϑ), Pk(ϑ)e(ϑ)〉 =
〈
4∑
i=1
diei(ϑ),
4∑
i=1
mi(ϑ)diei(ϑ)
〉
=
4∑
i=1
mi(ϑ)d
2
i . (13)
Hence
ψk =
σk√
Fk(ϑ, e(ϑ))
(
4∑
i=1
mi(ϑ)diei(ϑ)
)
.
Let ψik(ϑ) = 〈ψk, ei(ϑ)〉, i = 1, 2, 3, 4. Then
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1088 G. IBRAGIMOV, F. ALLAHABI, A. KUCHKAROV
ψik(ϑ) =
σkmi(ϑ)di√
Fk(ϑ, e(ϑ))
.
Combining this formula with (13) we conclude that
4∑
i=1
(
ψik(ϑ)
σk
√
mi(ϑ)
)2
=
∑4
i=1
mi(ϑ)d
2
i
Fk(ϑ, e(ϑ))
= 1, (14)
and hence
∂Xk(ϑ, σk) =
{
ψk
∣∣∣∣∣
4∑
i=1
ψ2
ik(ϑ)
σ2kmi(ϑ)
= 1, ψk =
4∑
i=1
ψik(ϑ)ei(ϑ)
}
so it is an ellipsoid.
Property 3. The eigenvalue m1(ϑ) is bounded above.
Proof. It follows from the boundedness of the limit
lim
ϑ→∞
m1(ϑ) = lim
ϑ→∞
(
1
4r1
sinh(2r1ϑ)−
1
2
√
1
r21
sinh4(r1ϑ) +
1
r22
sin4(r2ϑ)
)
=
1
4r1
and the fact that m1(ϑ) is a continuous function of ϑ.
Proposition 1. If 0 ≤ ϑ1 < ϑ2, then
X(ϑ1, σ) ⊂ X(ϑ2, σ), X(ϑ, σ) =
⋃
(σ1,σ2,...)
∞∏
k=1
Xk(ϑ, σk),
where union is taken over all the sequences
(σ1, σ2, . . .), σi ≥ 0, i = 1, 2, . . . ,
∞∑
k=1
σ2k = σ2.
Proof. Assume that (η, ξ) = (η1, ξ1, η2, ξ2, . . .) ∈ X(ϑ1, σ). Then there exists an admissible
control w(·) = (w1(·), w2(·), . . .), wk(·) ∈ Sϑ1(σk) such that
(ηk, ξk) =
ϑ1∫
0
RkBk(s)wk(s)ds,
ϑ1∫
0
Ak(s)wk(s)ds
∈ Xk(ϑ1, σk), k = 1, 2, . . . .
Now define a new control w̃(·) =
(
w̃1(·), w̃2(·), . . .
)
as follows:
w̃k(t) =
wk(t), 0 ≤ t ≤ ϑ1,
0, ϑ1 < t ≤ ϑ2.
It is obvious that w̃k(·) ∈ Sϑ2(σk) and
(ηk, ξk) =
ϑ1∫
0
RkBk(s)wk(s)ds,
ϑ1∫
0
Ak(s)wk(s)ds
=
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
A PURSUIT PROBLEM IN AN INFINITE SYSTEM OF SECOND-ORDER DIFFERENTIAL EQUATIONS 1089
=
ϑ2∫
0
RkBk(s)w̃k(s)ds,
ϑ2∫
0
Ak(s)w̃k(s)ds
∈ Xk(ϑ2, σk).
Hence X(ϑ1, σ) ⊂ X(ϑ2, σ).
Proposition 2. Let ζ = (0, 0, ζ3, ζ4) ∈ R4 satisfy ζ23 + ζ24 > M. Then ζ /∈ Xk(ϑ, σk), k =
= 1, 2, . . . , for all ϑ ≥ 0, where M = supϑ≥0 σ
2m1(ϑ).
Proof. Since
ζ1(ϑ) = 〈ζ, e1(ϑ)〉 = ζ3, ζ2(ϑ) = 〈ζ, e2(ϑ)〉 = ζ4,
then
ζ21 (ϑ) + ζ22 (ϑ) = ζ23 + ζ24 > M.
Hence for all ϑ ≥ 0
(
recall that m1(ϑ) = m2(ϑ) and m3(ϑ) = m4(ϑ)
)
4∑
i=1
ζ2i (ϑ)
σ2kmi
=
ζ21 (ϑ) + ζ22 (ϑ)
σ2km1
+
ζ23 (ϑ) + ζ24 (ϑ)
σ2km3
≥ ζ23 + ζ24
M
+
ζ23 (ϑ) + ζ24 (ϑ)
σ2km3
> 1.
Thus ζ /∈ Xk(ϑ, σk). In other words, the point ζ can not be steered into the origin.
Theorem 1. Let ψ0 = (ψ10, ψ20, . . .), ψk0 = (ηk0,−ξk0), k = 1, 2, . . . . If
ψ0 ∈ X(ϑ, σ) =
⋃
(σ1,σ2,...)
∞∏
k=1
Xk(ϑ, σk),
where union is taken over all the sequences (σ1, σ2, . . .), σi ≥ 0, i = 1, 2, . . . ,
∑∞
k=1
σ2k = σ2.
Then there exists a control w(t) =
(
w1(t), w2(t), . . .
)
, 0 ≤ t ≤ ϑ, such that z(ϑ) = ż(ϑ) = 0, for
the state z(t) of the system (3).
Proof. Since ψ0 = (ψ10, ψ20, . . .) ∈ X(ϑ, σ), then there exists a sequence (σ1, σ2, . . .), σi ≥ 0,
i = 1, 2, . . . ,
∑∞
k=1
σ2k = σ2 such that ψk0 ∈ Xk(ϑ, σk), k = 1, 2, . . . . Hence
ψk0 = (ηk0,−ξk0) =
ϑ∫
0
RkBk(s)wk(s)ds,
ϑ∫
0
Ak(s)wk(s)ds
for some wk(·) ∈ Sϑ(σk). This means
ηk(ϑ) = ηk0 −
ϑ∫
0
RkBk(s)wk(s)ds = 0,
ξk(ϑ) = ξk0 +
ϑ∫
0
Ak(s)wk(s)ds = 0.
Then from (8) and (9) we get zk(ϑ) = żk(ϑ) = 0.
Theorem 1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1090 G. IBRAGIMOV, F. ALLAHABI, A. KUCHKAROV
4. Pursuit differential game. In this section, we study the differential game (2). The following
theorem is true.
Theorem 2. Let
ψ0 ∈ X(ϑ, ρ− σ) =
⋃
(σ1,σ2,...)
∞∏
k=1
Xk(ϑ, σk),
where union is taken over all the sequences (σ1, σ2, . . .), σi ≥ 0, i = 1, 2, . . . ,
∑∞
k=1
σ2k = (ρ−σ)2.
Then pursuit can be completed from the position ψ0 for the time ϑ.
Proof. As ψ0 ∈ X(ϑ, ρ − σ), then there exists a sequence (σ1, σ2, . . .), σi ≥ 0, i = 1, 2, . . . ,∑∞
k=1
σ2k = (ρ − σ)2 such that ψk0 ∈ Xk(ϑ, σk), k = 1, 2, . . . , for some θ. It follows from
Theorem 1 that there exists a control
w0(t) =
(
w0
1(t), w
0
2(t), . . .
)
, 0 ≤ t ≤ ϑ,
∞∑
k=1
λrk
T∫
0
∣∣w0
k(s)
∣∣2ds ≤ (ρ− σ)2,
such that z(ϑ) = ż(ϑ) = 0 in (3). We show that pursuit can be completed for the time ϑ. To this end
we offer to the pursuer the following strategy:
uk(t, v) = vk(t)− w0
k(t), k = 1, 2, . . . , (15)
where v(·) is any admissible control of the evader. Then it is clear that z(ϑ) = ż(ϑ) = 0 for the
system (2) (see the proof of Theorem 1).
What is left is to show the admissibility of the strategy (15). It can be shown by using the
Minkowski inequality as follows: ∞∑
k=1
λrk
ϑ∫
0
|uk(t, v)|2dt
1/2 =
∞∑
k=1
λrk
ϑ∫
0
|vk(t)− w0
k(t)|2dt
1/2 ≤
≤
∞∑
k=1
λrk
ϑ∫
0
|vk(t)|2dt
1/2 +
∞∑
k=1
λrk
ϑ∫
0
|w0
k(t)|2dt
1/2 ≤ σ + ρ− σ = ρ.
Theorem 2 is proved.
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ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
|
| id | umjimathkievua-article-2491 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:24:29Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ef/ee709f1d92ad3910d97bb89e5426a4ef.pdf |
| spelling | umjimathkievua-article-24912020-03-18T19:16:44Z A Pursuit Problem in an Infinite System of Second-Order Differential Equations Проблема переслідування в нескінченній системі диференціальних рівнянь другого порядку Allahabi, F. Ibragimov, G. I. Kuchkarov, A. Аллахабі, Ф. Ібрагімов, Г. І. Кучкаров, А. We study a pursuit differential game problem for an infinite system of second-order differential equations. The control functions of players, i.e., a pursuer and an evader are subject to integral constraints. The pursuit is completed if z(τ) = \( \dot{z} \) (τ) = 0 at some τ > 0, where z(t) is the state of the system. The pursuer tries to complete the pursuit and the evader tries to avoid this. A sufficient condition is obtained for completing the pursuit in the differential game when the control recourse of the pursuer is greater than the control recourse of the evader. To construct the strategy of the pursuer, we assume that the instantaneous control used by the evader is known to the pursuer. Вивчається проблема переслідування в диференціальній rpi для нескінченної системи диференціальних рівнянь другого порядку. Керівні функції гравців, тобто переслідувача та переслідуваного, мають деякі обмеження. Переслідування завершується, коли z(τ) = \( \dot{z} \) (τ) = 0 для деякого τ > 0, де z(t) — стан системи. Переслідувач намагається завершити переслідування, а переслідуваний намагається цього уникнути. Встановлено достатню умову завершення переслідування в диференціальній грі, коли зворотнє управління для переслідувача більше, ніж для переслідуваного. Для побудови стратегії переслідувача вважаємо, що миттєве керування, застосоване переслідуваним, є відомим переслідувачу. Institute of Mathematics, NAS of Ukraine 2013-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2491 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 8 (2013); 1080–1091 Український математичний журнал; Том 65 № 8 (2013); 1080–1091 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2491/1748 https://umj.imath.kiev.ua/index.php/umj/article/view/2491/1749 Copyright (c) 2013 Allahabi F.; Ibragimov G. I.; Kuchkarov A. |
| spellingShingle | Allahabi, F. Ibragimov, G. I. Kuchkarov, A. Аллахабі, Ф. Ібрагімов, Г. І. Кучкаров, А. A Pursuit Problem in an Infinite System of Second-Order Differential Equations |
| title | A Pursuit Problem in an Infinite System of Second-Order Differential Equations |
| title_alt | Проблема переслідування в нескінченній системі диференціальних рівнянь другого порядку |
| title_full | A Pursuit Problem in an Infinite System of Second-Order Differential Equations |
| title_fullStr | A Pursuit Problem in an Infinite System of Second-Order Differential Equations |
| title_full_unstemmed | A Pursuit Problem in an Infinite System of Second-Order Differential Equations |
| title_short | A Pursuit Problem in an Infinite System of Second-Order Differential Equations |
| title_sort | pursuit problem in an infinite system of second-order differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2491 |
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