Amino , Bekhzod; Ruziboe , Ma ks
A icle
E asion di e en ial games in he space o squa e
summable sequences
Games
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Amino , Bekhzod; Ruziboe , Ma ks (2024) : E asion di e en ial games in he
space o squa e summable sequences, Games, ISSN 2073-4336, MDPI, Basel, Vol. 15, Iss. 6, pp. 1-7,
h ps://doi.o g/10.3390/g15060038
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Ci a ion: Amino , B.; Ruziboe , M.
E asion Di e en ial Games in he
Space o Squa e Summable Sequences.
Games 2024,15, 38. h ps://doi.o g/
10.3390/g15060038
Academic Edi o : Ul ich Be ge
Recei ed: 20 Oc obe 2024
Re ised: 11 No embe 2024
Accep ed: 18 No embe 2024
Published: 19 No embe 2024
Copy igh : © 2024 by he au ho s.
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A icle
E asion Di e en ial Games in he Space o Squa e
Summable Sequences
Bekhzod Amino and Ma ks Ruziboe *
School o Enginee ing, Cen al Asian Uni e si y, 264 Milliy, bog S , Tashken 111221, Uzbekis an;
[email p o ec ed]
*Co espondence: m. [email p o ec ed]
Abs ac : In his a icle, we conside simple-mo ion pu sui –e asion di e en ial games in he Hilbe
space o squa e summable sequences. We show ha when he playe s ha e he same dynamic
capabili ies, e asion is possible unde some assump ions abou he ini ial posi ions o he playe s.
Keywo ds: Hilbe spaces; basis; pu sue ; e ade ; pu sui –e asion games; e asion s a egy
1. In oduc ion
Di e en ial games we e ini ia ed wi h he pionee ing wo ks o Isaacs [
1
] and Pon ya-
gin [
2
] in he 1960s and ha e since hen been s udied ex ensi ely. Fo example, see [
3
–
9
] and
he e e ences he ein o an ex ensi e his o ical accoun o he opic. Di e en ial games
a e o en di ided in o wo p oblems: he p oblem o pu sui and he p oblem o e asion.
In conside ing hese games, he mos common dynamics a e he ones gi en by simple
mo ion. O en, geome ic o in eg al cons ain s a e imposed on he con ol pa ame e s
o he playe s. In [
10
], i is shown ha in he
n
-dimensional Euclidean ball,
n
lions can
ca ch he man, while he man can escape om
n−
1 lions when he con ol o he playe s
is subjec o geome ic cons ain s. A simila game p oblem was s udied by I ano [
11
]
on any con ex compac se , and an es ima e om he abo e scena io was ob ained o
gua an eed pu sui ime. The con ex in which di e en ial games a e s udied is e y wide.
We e e o [
12
–
17
] o a ious in e es ing esul s o games in unbounded egions as well
as on g aphs.
The e asion p oblem on an in ini e ime in e al was in oduced and s udied in [
18
].
La e , in [
19
], he au ho s sugges ed a new ype o manoeu e o e asion in he game wi h
many pu sue s. A s iking s a egy was sugges ed in [
20
] o s udy an e asion game o one
e ade and se e al pu sue s wi h a s a e cons ain , whe e he e ade was supposed o
mo e in a small neighbo hood in a gi en di ec ion du ing he game. The au ho p o ed
ha i he e ade mo es as e han he pu sue s, hen e asion is possible. This esul
was ex ended in a se ies o wo ks [
21
–
24
] o mo e gene al di e en ial si ua ions. Rela ed
p oblems o e asion om a g oup o pu sue s we e s udied in [25,26].
In [
27
], Pshenichnii conside ed a simple-mo ion di e en ial game wi h many pu sue s
and one e ade in
Rn
, whe e all playe s had he same maximal speed. He p o ed ha i
he ini ial s a e o he e ade is in he in e io o he con ex hull o he pu sue s’ ini ial
s a es, hen he pu sui can be comple ed; o he wise, e asion is possible. Based on his
wo k, Pshenichnii e al. [
28
] de eloped a me hod o esol ing unc ions o sol ing linea
pu sui p oblems wi h many pu sue s. I should be no ed ha he esul s o [
27
] ha e been
ex ended by many esea che s o co e a ious cases in ini e-dimensional spaces [29–31].
Recen ly, con ol p oblems and di e en ial games in in ini e-dimensional spaces ha e
been s udied ac i ely due o hei connec ion o p ocesses desc ibed by pa ial di e en-
ial equa ions a e decomposing he con ol p oblem [
32
], which has also shown some
phenomena [
33
,
34
] in insic o in ini e-dimensional spaces. I is wo h no ing ha in ini e-
dimensional spaces ha e quali a i ely di e en p ope ies [35]. Pu sui –e asion games in
Games 2024,15, 38. h ps://doi.o g/10.3390/g15060038 h ps://www.mdpi.com/jou nal/games
Games 2024,15, 38 2 o 7
in ini e-dimensional spaces a e conside ed in [
36
–
38
]. Fo ecen esul s in his a ea, we
e e o [39–42]
The se ing o he p oblem in [
27
] is also na u al o conside in in ini e-dimensional
spaces. Bu , he pu sui p oblem seems o be challenging in his si ua ion. This is connec ed
o he ac ha he se o coo dina e oc an s in in ini e-dimensional spaces has con inuum
ca dinali y. Using his idea, i was shown in [
43
] ha e asion is possible om any ini ial
condi ion when in eg al cons ain s a e imposed on he con ol pa ame e s o he playe s.
Howe e , a na u al gene aliza ion o [
27
] would be o conside geome ic cons ain s. This
wo k is an a emp o show ha in in ini e-dimensional spaces, he analogs o he heo ems
gi en in [
27
] a e alse in gene al. He e, we p o e ha unde ela i ely mild assump ions
on he dynamics o he sys em and he ini ial posi ions o he playe s, e asion is possible
when he e ade and he pu sue s o he playe s ha e he same ene gy. The es o his
pape is o ganized as ollows: Sec ion 2is de o ed o p elimina ies and con ains s anda d
esul s om unc ional analysis, and Sec ion 3is de o ed o he main esul s.
2. P elimina ies
Le
(E
,
∥·∥E)
be an in ini e-dimensional Banach space o e eal numbe s,
R
, wi h he
ze o elemen
0
. A sys em o elemen s,
{xk}k∈F
(
ca d(F)<∞
o
ca d(F) = ℵ0
), is called
linea ly independen ([
44
], p. 142) i , o any
n∈N
and
{αk}n
k=1⊂R
such ha
n
∑
k=1
αkxk=
0,
i ollows ha
αk=
0 o e e y
k∈1, n
. The sequence
{en}∞
k=1⊂E
is called a (Schaude )
basis i , o any
x∈E
, he e is a unique sequence,
{αn}∞
k=1
, o eal numbe s such ha
x=
∞
∑
n=k
αkek
(i.e., such ha
lim
n→∞∥x−n
∑
k=1
αkek∥E=
0). I is ob ious ha e e y basis o
E
mus be linea ly independen . Recall ha e e y non-ze o subspace,
L
, o a Hilbe space,
H
,
has he ollowing p ope y: o each
z∈ H
, he e a e elemen s
x∈L
and
y∈ H L
such
ha z=x+y. This ac we w i e as H=L⊕L⊥.
Le
be an in eg able unc ion on a segmen
[a
,
b]
. Using he Cauchy–Schwa z
inequali y—∥ ·g∥2≤ ∥ ∥2·∥g∥2— o he unc ions (x)and g(x)≡1, we ob ain ha
b
Z
a
(x)dx
2
≤(b−a)
b
Z
a
( (x))2dx. (1)
Also, om Cauchy–Schwa z inequali y, i ollows ha i
∞
∑
j=1
(αj+βj)2<∞,
∞
∑
j=1
α2
j<∞and
∞
∑
j=1
β2
j<∞,
hen ∞
∑
j=1
(αj+βj)2=
∞
∑
j=1
α2
j+2
∞
∑
j=1
αjβj+
∞
∑
j=1
β2
j, (2)
whe e {αj}∞
j=1,{βj}∞
j=1⊂R.
3. Resul s
Le
(ℓ2
,
∥ · ∥)
be a sepa able Hilbe space o e eal numbe s,
R
, wi h he usual
scala p oduc
⟨·
,
·⟩
. Le
F
be a ini e o coun able subse o
N
. In he space
ℓ2
, conside
a di e en ial game wi h
F
deno ing pu sue s whose coo dina es a ime
a e gi en by
xk( ) = (xk1( )
,
xk2( )
,
. . .)
,
k∈F
, and one e ade whose coo dina es a his ime a e gi en
by
y( ) = (y1( )
,
y2( )
,
. . .)
. Le
˙y( ) = ( )
and
˙
xk( ) = uk( )
o all
k∈F
. Fo
>
0,
we assume ha
∥ ( )∥ ≤
1 and
∥uk( )∥ ≤
1 o all
k∈F
. As usual,
y(
0
)=xk(
0
)
o
e e y
k∈F
; i.e., he ini ial posi ion o he e ade does no coincide wi h ini ial posi ion o
any pu sue .
Games 2024,15, 38 3 o 7
Lemma 1. I he e exis s a non-ze o ec o
s∈ℓ2
such ha
⟨s
,
y(
0
)−xk(
0
)⟩ ≥
0 o all
k∈F
,
hen a oidance o con ac is possible.
P oo .
Conside a uni ec o
w=s
∥s∥
, which has he same di ec ion as
s
. Se
( ) = w
o
>0. Hence, y( ) = y(0) + R
0 (s)ds = (y1(0) + w1 ,y2(0) + w2 , . . .)∈ℓ2. We will show
ha
xk( )=y( )
o all
k∈F
and
>
0. Le us assume he opposi e; i.e., suppose ha he e
exis n∈Fand τ>0 such ha y(τ) = xn(τ). Since
xn(τ) = (xn1(0) +
τ
Z
0
un1(s)ds,xn2(0) +
τ
Z
0
un2(s)ds, . . .)
and
y(τ) = y(0) + Zτ
0 (s)ds = (y1(0) + w1τ,y2(0) + w2τ, . . .),
xnk(0) +
τ
R0
unk(s)ds =yk(0) + wkτ o e e y k∈N. The e o e,
A:=
τ
Z
0
un1(s)ds
2
+
τ
Z
0
un2(s)ds
2
+. . . =
= (y1(0)−xn1(0) + w1τ)2+ (y2(0)−xn2(0) + w2τ)2+. . . =
(Using Equali y (2), we ob ain)
=
∞
∑
j=1
(yj(0)−xnj(0))2+2τ
∞
∑
j=1
(yj(0)−xnj(0))wj+τ2
∞
∑
j=1
w2
j=
=
∞
∑
j=1
(yj(0)−xnj(0))2+2τ
∥s∥⟨y(0)−xn(0),s⟩+τ2∥w∥2
Since
∞
∑
j=1
(yj(0)−xnj(0))2>0, τ>0, ∥w∥=1 and ⟨y(0)−xn(0),s⟩ ≥ 0, A>τ2.
On he o he hand, aking in o accoun ha
∥u( )∥ ≤
1 o e e y
>
0 and applying
(1) o e e y e m, we ha e
A=
τ
Z
0
un1(s)ds
2
+
τ
Z
0
un2(s)ds
2
+. . . ≤
≤τZτ
0u2
n1(s)ds +τZτ
0u2
n2(s)ds +. . . =τZτ
0
∞
∑
j=1
u2
nj(s)ds ≤τZτ
01ds =τ2
Hence,
A≤τ2
. The ob ained con adic ion shows ha
xk( )=y( )
o all
k∈F
and
>0.
Theo em 1. Le
{xk(
0
)}∞
k=1
be a basis o
ℓ2
and
y(
0
)=
∞
∑
k=1
βkxk(
0
)
, whe e
∞
∑
j=1
βj=
1. Then,
a oidance o con ac is possible.
P oo . Le zk=y(0)−xk(0) o e e y k∈N. The e a e wo cases:
Case 1. The sequence {zk}∞
k=1is a basis o ℓ2.
Games 2024,15, 38 4 o 7
Le
L
be he closed linea span o he ec o s
{zk}∞
k=2
; hence,
L=ℓ2
. Since
ℓ2
is a
Hilbe space,
ℓ2=L⊕L⊥
, whe e
L⊥
is he o hogonal complemen o
L
and
L⊥={0}
.
Le ˜
w∈L⊥be a non-ze o ec o . We de ine he ec o w∈ℓ2in he ollowing way:
w=(˜
wi ⟨˜
w,z1⟩ ≥ 0
−˜
wi ⟨˜
w,z1⟩<0
The e o e,
⟨w
,
zk⟩ ≥
0 o e e y
k∈N
, and hence, due o lemma 1, he a oidance o
con ac is possible.
Case 2. The sequence {zk}∞
k=1is no a basis o ℓ2.
Suppose ha
L=ℓ2
. Since he sequence
{zk}∞
k=1
is no a basis o
ℓ2
, he e exis s a ec o
∈ℓ2
such ha
=
∞
∑
k=1
αkzk=
∞
∑
k=1
βkzk
, whe e
{αj}∞
j=1
,
{βj}∞
j=1⊂R
and
αj=βj
o a
leas one
j∈N
. Hence,
∞
∑
k=1
γkzk=0
, whe e
γk=αk−βk
(
∀k∈N
) and
∞
∑
j=1
γ2
j=
0. Taking
in o accoun ha
zk=y(
0
)−xk(
0
)
(
∀k∈N
), we ob ain ha
∞
∑
k=1
γky(
0
) =
∞
∑
k=1
γkxk(
0
)
.
I
∞
∑
j=1
γj=
0, hen
∞
∑
k=1
γkxk(
0
) = 0
, which con adic s he assump ion ha
{xk(
0
)}∞
k=1
is a
basis o
ℓ2
. The e o e,
∞
∑
j=1
γj=
0. I o each
j∈N
, we de ine
˜
γj=γj
∞
∑
j=1
γj
, hen
∞
∑
j=1
˜
γj=
1 and
y(
0
) =
∞
∑
k=1
˜
γkxk(
0
)
, which also con adic he condi ion o he heo em. Thus, he equali y
L=ℓ2
does no hold. Le
L
be he closed linea span o he ec o s
{zk}∞
k=1
. I
L=ℓ2
,
hen, acco ding o he abo e a gumen , he e exis s a non-ze o ec o
˜
w∈ℓ2
such ha
⟨˜
w
,
zk⟩=
0, o e e y
k∈N
, and hence, due o Lemma 1, he a oidance o con ac is
possible. In o de o inish he p oo , i is enough o show ha he equali y
L=ℓ2
does no
hold.
Since each basis o a Banach space ze o elemen may ha e 0 coe icien s, om he
abo e heo em, we ob ain he ollowing:
Co olla y 1. Le
{xk(
0
)}∞
k=1
be a basis o
ℓ2
and
y(
0
) = 0
. Then, a oidance o con ac is possible.
The ollowing p oposi ion shows ha he condi ions o Theo em 1a e su icien bu
no necessa y.
Le
E( )
be he se o poin s o
ℓ2
ha e ade
E
can each du ing he ime
>
0 om
he s a ing posi ion. The se Pk( )has a simila meaning o each k∈N.
P oposi ion 1. Le
y(
0
) = 0
and
x2k(
0
) = ek
,
x2k−1(
0
) = −ek
o each
k∈N
. Then, he se
E( )is no a subse o he se Sk∈NPk( ).
P oo .
Fix any
τ>
0. Since
∥˙y( )∥ ≤
1,
E(τ) = B(0
,
τ)
. Fo he same eason,
Pk(τ) =
B(xk(
0
)
,
τ)
o each
k∈N
. We will show ha
B(0
,
τ)⊂ Sk∈NB(xk(
0
)
,
τ)
. Le
n∈N
be
such ha n>4τ2. Cons uc he ec o
zτ= ( τ
√n,τ
√n, . . . , τ
√n, 0, 0, . . .)∈ℓ2
whe e 0 s a s om he
(n+
1
)
- h coo dina e. I is ob ious ha
∥zτ∥=τ
, and hence,
zτ∈B(0,τ). I k≥n+1, hen
∥zτ±ek(0)∥=pτ2+1>τ;
Games 2024,15, 38 5 o 7
i.e., zτ/∈B(±ek(0),τ) o any k≥n+1. I k≤n, hen
∥zτ±ek(0)∥= n·τ2
n±2τ
n+1.
Since τ>0 and n>4τ2,±2τ
n+1>0. The e o e,
n·τ2
n±2τ
n+1>τ,
i.e., zτ/∈B(±ek(0),τ) o any k≤n. Hence, zτ/∈Sk∈NB(xk(0),τ). The e o e,
B(0,τ) [
k∈N
B(xk(0),τ)=∅,
and his di ec ly implies ha E( )⊂ Sk∈NPk( ).
4. Discussion
In his wo k, we conside ed simple-mo ion di e en ial games in
ℓ2
whe e all he
playe s ha e he same dynamic capabili ies. Unde some assump ions on he ini ial
posi ions o he playe s, we showed ha e asion is possible om any ini ial posi ion.
Fo he momen , i is no clea o us how o gene alize he esul s o his pape o
ℓp
spaces
wi h 1
≤p≤+∞
. I should be possible wi h sui able changes o Lemma 1, which uses he
scala p oduc in an essen ial way. This wo k is an a emp o show ha he esul s o [
27
]
a e alse in in ini e-dimensional spaces. We cons uc ed a piecewise-cons an s a egy o
he e ade , which gua an ees e asion. To s eng hen ou esul s, one has o eso o a
mo e wide class o s a egies. I would also be in e es ing o conside a simila p oblem in
an a bi a y Banach space. I would also be in e es ing o conside he p oblem when he
con ol pa ame e s o he playe s sa is y di e en ypes o cons ain s.
Au ho Con ibu ions: In es iga ion, B.A. and M.R.; me hodology, B.A. and M.R.; p ojec adminis a ion,
B.A. and M.R.; alida ion, B.A. and M.R.; isualiza ion, B.A. and M.R.; w i ing—o iginal d a p epa a ion,
B.A. and M.R.; w i ing— e iew and edi ing, B.A. and M.R. All au ho s ha e con ibu ed equally a all
s ages o he esea ch. All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding: This esea ch ecei ed no ex e nal unding.
Da a A ailabili y S a emen : The o iginal con ibu ions p esen ed in he s udy a e included in he
a icle, u he inqui ies can be di ec ed o he co esponding au ho .
Acknowledgmen s: The au ho s hank Ga u jan Ib agimo o sugges ing his di ec ion o esea ch
and he anonymous e iewe s o hei use ul commen s.
Con lic s o In e es : The au ho s decla e no con lic s o in e es .
Re e ences
1. Isaacs, R. Di e en ial Games; John Wiley & Sons: New Yo k, NY, USA, 1965.
2. Pon yagin, L.S. On he heo y o di e en ial games Russian Ma h. Su eys 1966,21, 193–246.
3.
Ba¸sa T.; Olsde , G. Dynamic Noncoope a i e Game Theo y; Rep in o he second (1995) edi ion; Classics in Applied Ma hema ics;
SIAM: Philadelphia, PA, USA, 1999; Volume 23 , pp. x i+519.
4. Ba is ini, S. A S ochas ic Cha ac e iza ion o he Cap u e Zone in Pu sui -E asion Games. Games 2020,11, 54. [C ossRe ]
5.
Buckdahn, R.; Ca daliague , P.; Quincampoix, M. Some ecen aspec s o di e en ial game heo y. Dyn. Games Appl. 2011,1,
74–114. [C ossRe ]
6. F iedman, A. Di e en ial Games; John Wiley and Sons: New Yo k, NY, USA, 1971.
7.
Hajek O. Pu sui Games: An In oduc ion o The heo y and Applica ions o Di e en ial Games o Pu sui and E asion; Ma hema ics in
Science and Enginee ing; Academic P ess: New Yo k, NY, USA, 1975; Volume 120.
Games 2024,15, 38 6 o 7
8. K aso skii, N.N.; Subbo in, A.I. Game-Theo e ical Con ol P oblems; Sp inge : New Yo k, NY, USA, 1988.
9. Pe osyan, L.A. Di e en ial Games o Pu sui ; Wo ld Scien i ic: Singapo e; London, UK, 1993.
10. C o , H.T. Lion and man: A pos sc ip . J. Lond. Ma h. Soc. 1964,39, 385–390. [C ossRe ]
11. I ano , R.P. Simple pu sui -e asion on a compac con ex se . Dokl. Akad. Nauk. SSSR 1980,254, 1318–1321.
12. Be ko i z, L.D. Di e en ial game o gene alized pu sui and e asion. SIAM J. Con . 1986,24, 361–373. [C ossRe ]
13.
Bha acha ya, S.; Ba¸sa , T.; Ho akimyan, N. A isibili y-based pu sui -e asion game wi h a ci cula obs acle. J. Op im. Theo y
Appl. 2016,171, 1071–1082. [C ossRe ]
14. Kons an inidis G., Kehagias, A. Simul aneously mo ing cops and obbe s. Theo e . Compu . Sci. 2016,645, 48–59. [C ossRe ]
15.
Sco , W.L.; Leona d, N.E. Op imal e asi e s a egies o mul iple in e ac ing agen s wi h mo ion cons ain s. Au om. J. IFAC 2018,
94, 26–34. [C ossRe ]
16.
Sun, W.; Tsio as, P. An op imal e ade s a egy in a wo-pu sue one-e ade p oblem. In P oceedings o he 53 d IEEE Con e ence
on Decision and Con ol, Los Angeles, CA, USA, 15–17 Decembe 2014; pp. 4266–4271.
17.
Alexande , S.; Bishop, R.; Ch is , R. Cap u e pu sui games on unbounded domain. Lënseignemen Ma hëma ique 2009,55, 103–125.
[C ossRe ]
18. Pon yagin, L.S.; Mischenko, Y.F. The p oblem o e asion in linea di e en ial games. Di e . U a n. 1971,7, 436-445.
19.
Mishchenko, E.F.; Nikol’skii, M.S.; Sa imo , N.Y. A oidance encoun e p oblem in di e en ial games o many pe sons. T udy
MIAN USSR 1977,143, 105–128.
20. Che nous’ko, F.L. A P oblem o E asion o Se e al Pu sue s. P ikl. Ma . Mekh. 1976,40, 14–24.
21.
Che nous’ko, F.L.; Zak, V.L. On di e en ial games o e asion om many pu sue s. J. Op im. Theo y Appl. 1985,46, 461–470.
[C ossRe ]
22. Zak, V.L. On a p oblem o e ading many pu sue s. J. Appl. Ma hs Mekhs 1978,43, 492–501. [C ossRe ]
23.
Zak, V.L. P oblem o E asion om Many Pu sue s Con olled by Accele a ion. Iz . Akad. Nauk. SSSR Tekhnicheskaya Kibe n. 1981,
2, 57–71.
24. Zak, V.L. The P oblem o E ading Se e al Pu sue s in he P esence o a S a e Cons ain . So . Ma h. Dokl. 1982,26, 190–193.
25.
Bo owko, P.; Rzymowski, W.; S achu a, A. E asion om many pu sue s in he simple mo ion case. J. Ma h. Anal. Appl. 1988,135,
75–80. [C ossRe ]
26. Chodun, W. Di e en ial games o e asion wi h many pu sue s. J. Ma h. Anal. Appl. 1986,142, 370–389. [C ossRe ]
27. Pshenichnii, B.N. Simple pu sui by se e al objec s. Cybe n. Sys . Anal. 1976,12, 145–146. [C ossRe ]
28.
Pshenichnii, B.N.; Chik ii, A.A.; Rappopo , J.S. An e icien me hod o sol ing di e en ial games wi h many pu sue s. Dokl.
Akad. Nauk SSSR 1981,256, 530–535. (In Russian)
29.
Banniko , A.S.; Pe o , N.N. To non-s a iona y g oup pu sui p oblem. T . Ins . Ma h. Mech. U O RAN 2010,16, 40–51. [C ossRe ]
30.
Ib agimo , G.; Fe a a, M.; Kuchka o , A.; Panse a, B.A. Simple mo ion e asion di e en ial game o many pu sue s and e ade s
wi h in eg al cons ain s. Dyn. Games Appl. 2018,8, 352–378. [C ossRe ]
31.
Vagin, D.; Pe o , N. The p oblem o he pu sui o a g oup o igidly coo dina ed e ade s. J. Compu . Sys . Sci. In . 2001,40,
749–753.
32.
Azamo , A.A.; Bakh amo , J.A.; Akhmedo , O.S. On he Che nousko ime-op imal p oblem o he equa ion o hea conduc i i y
in a od. U al. Ma h. J. 2019,5, 13–23. [C ossRe ]
33.
Azamo , A.; Ib agimo , G.; Mamayusupo , K.; Ruziboe , M. On he s abili y and null-con ollabili y o an in ini e sys em o
linea di e en ial equa ions. J. Dyn. Con ol Sys . 2021,29, 595–605. [C ossRe ]
34.
Ruziboe , M.; Ib agimo , G.; Mamayusupo , K.; Khai me o , A.; Panse a, B.A. On a linea di e en ial game in he Hilbe Space
ℓ2.Ma hema ics 2023,11, 49–87. [C ossRe ]
35.
Amino , B.R.; Chilin, V.I. Isome ies o eal subspaces o sel -adjoin ope a o s in Banach symme ic ideals. Vladika kaz Ma h. J.
2019,21, 11–24.
36.
Ib agimo , G.I.; Salimi, M.; Amini, M. E asion om many pu sue s in simple mo ion di e en ial game wi h in eg al cons ain s.
Eu . J. Ope . Res. 2012,218, 505–511. [C ossRe ]
37.
Ib agimo , G.; Ruziboe , M.; Zaynabiddino , I.; Panse a, B.A. E asion di e en ial game o mul iple pu sue s and a single e ade
wi h geome ic cons ain s in ℓ2.Games 2023,14, 52. [C ossRe ]
38.
Alias, I.A.; Ib agimo , G.; Rakhmano , A. E asion Di e en ial Game o In ini ely Many E ade s om In ini ely Many Pu sue s in
Hilbe Space. Dyn. Games Appl. 2017,7, 347–359. [C ossRe ]
39. Jia, Z.; Liu, X. Op imal con ol o mul i ac o unce ain sys em wi h jumps. In . J. Con ol 2022,96, 1272–1287. [C ossRe ]
40.
Jia, Z.; Liu, X. Op imis ic alue models o saddle poin equilib ium con ol p oblems o unce ain jump sys ems. Op imiza ion
2024 , 1–34. [C ossRe ]
41.
Jia, Z.; Liu, X. Unce ain s ochas ic hyb id di e en ial game sys em wi h V-n jumps:Saddle poin equilib ium, s a egies and
applica ion o ad e ising duopoly game. Chaos. Soli on. F ac 2023,171, 113490. [C ossRe ]
42.
Kumko , S.S.; Le Ménec, S.; Pa sko, V.S. Ze o-sum pu sui -e asion di e en ial games wi h many objec s: Su ey o publica ions.
Dyn. Games Appl. 2017,7, 609–633. [C ossRe ]
Games 2024,15, 38 7 o 7
43.
Ib agimo , G.I.; Hasim, R.M. Pu sui and e asion di e en ial games in Hilbe space. In . Game Theo y Re . 2010,12, 239–251.
[C ossRe ]
44. B ezis, H. Func ional Analysis, Sobole Spaces and Pa ial Di e en ial Equa ions; Sp inge : Be lin/Heidelbe g, Ge many, 2011.
Disclaime /Publishe ’s No e: The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual
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