Sa a, Bengheb id; Tou ik, Bou emani; Djamel, Ben e ki
A icle
On Isaac's wa game o a i ion and a ack using dynamic
p og amming app oach
Games
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MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Sa a, Bengheb id; Tou ik, Bou emani; Djamel, Ben e ki (2024) : On Isaac's wa
game o a i ion and a ack using dynamic p og amming app oach, Games, ISSN 2073-4336, MDPI,
Basel, Vol. 15, Iss. 6, pp. 1-18,
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Ci a ion: Sa a, B.; Tou ik, B.; Djamel,
B. On Isaac’s Wa Game o A i ion
and A ack Using Dynamic
P og amming App oach. Games 2024,
15, 35. h ps://doi.o g/10.3390/
g15060035
Academic Edi o : Ul ich Be ge
Recei ed: 14 July 2024
Re ised: 28 Sep embe 2024
Accep ed: 22 Oc obe 2024
Published: 24 Oc obe 2024
Copy igh : © 2024 by he au ho s.
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games
A icle
On Isaac’s Wa Game o A i ion and A ack Using Dynamic
P og amming App oach
Bengheb id Sa a 1,*,†, Bou emani Tou ik 2,† and Ben e ki Djamel 1,†
1Labo a o y o Fundamen al and Nume ical Ma hema ics, Depa men o Ma hema ics, Facul y o Sciences,
Uni e si y o Fe ha Abbas Se i -1, Se i 19000, Alge ia; [email p o ec ed]
2Labo a o y o Applied Ma hema ics, Facul y o Technology, Uni e si y o Fe ha Abbas Se i -1,
Se i 19000, Alge ia; ou ik.bou [email p o ec ed]
*Co espondence: [email p o ec ed]; Tel.: +213-781318405
†These au ho s con ibu ed equally o his wo k.
Abs ac : In his s udy, we use he dynamic p og amming me hod in oduced by Mi ic˘a (2004) o
sol e he well-known wa game o a i ion and a ack as o mula ed by Isaacs (1965). By using
his mode n app oach, we ex end he classical amewo k o explo e op imal s a egies wi hin he
di e en ial game se ing, o e ing a comple e, comp ehensi e and heo e ically obus solu ion.
Addi ionally, he s udy iden i ies and analyzes eedback s a egies, which ep esen a signi ican
ad ancemen o e o he s a egy ypes in game heo y. These s a egies dynamically adap o he
e ol ing s a e o he sys em, p o iding mo e obus solu ions o eal- ime decision-making in con lic
scena ios. This no el con ibu ion enhances he applica ion o game heo y, pa icula ly in he con ex
o wa a e models, and illus a es he p ac ical ad an ages o inco po a ing eedback mechanisms in o
s a egic decision-making. The admissible eedback s a egies and he co esponding alue unc ion
a e cons uc ed h ough a e ined applica ion o Cauchy’s Me hod o cha ac e is ics o s a i ied
Hamil on–Jacobi equa ions. Thei op imali y is p o ed using a sui able Elemen a y Ve i ica ion
Theo em o he associa ed alue unc ion as an a gumen o su icien op imali y condi ions.
Keywo ds: di e en ial game; di e en ial inclusion; eedback s a egies; dynamic p og amming;
Hamil onian low; alue unc ion
1. In oduc ion
Di e en ial games, a b anch o s a ic game heo y, p o ide a amewo k o analyzing
s a egic in e ac ions be ween wo o mo e playe s, in which each playe ’s decisions, in
u n, in luence he dynamics o a sys em. Ru us Isaac’s seminal book Di e en ial Games [
1
],
published in 1965, laid he ounda ion o he s udy o pu sui -e asion games and o e ed
a comp ehensi e amewo k o analyzing di e en ial games in ol ing mul iple playe s.
This wo k laid he g oundwo k o add essing a di e se a ay o con lic scena ios and
inspi ed esea che s o del e u he in o his ield. This explo a ion b ough nume ous
c i icisms o Isaac’s heu is ic app oach o he o e on , including hose highligh ed in [
2
–
8
].
Ul ima ely, he esea ch p omp ed by Isaac’s’ miss eps has le a las ing legacy, mani es ing
in imp o ed me hodologies, in e disciplina y collabo a ions, and a iche unde s anding
o he nuances o he p oblem. Also, he di ide be ween heo e ical igo and p ac ical
applicabili y has p omp ed esea che s o seek a middle g ound. E o s ha e been made o
de elop heo e ical amewo ks ha can accommoda e he in icacies o conc e e examples
(e.g., [
9
–
12
]). This in ol ed e ining exis ing ma hema ical me hods and, in some cases,
in oducing no el app oaches ha main ain a balance be ween heo e ical soundness and
p ac ical ele ance. Since 1982, signi ican de elopmen s ha e occu ed in he heo y
o iscosi y solu ions, ini ia ed by [
13
], which ha e p o ided a ious cha ac e iza ions
o he alue unc ion as a solu ion o he Hamil on–Jacobi-Isaac’s equa ion. Howe e ,
he elabo a ion o he heo y o iscosi y solu ions has no e ec i ely con ibu ed o he
Games 2024,15, 35. h ps://doi.o g/10.3390/g15060035 h ps://www.mdpi.com/jou nal/games
Games 2024,15, 35 2 o 18
accu a e and comple e esolu ion o any conc e e p oblems p oposed in he li e a u e.
This p e ailing anomaly in he s udy o di e en ial games is abou o be co ec ed, due
o he ecen con ibu ions p esen ed in Mi ic˘a’s wo k [
14
–
16
]. The undamen al con en
o his new Dynamic P og amming app oach consis s o su icien op imali y condi ions,
illus a ed by a main heo e ical suppo consis ing o se en e i ica ion heo ems. I
essen ially ex ends o much mo e ealis ic cases, he elemen a y e i ica ion heo em is om
Isaac’s (1965) in [
1
], he only known p e ious one (applied, unjus i iably, o p oblems in
which he alue unc ion is no di e en iable). The cons uc i e aspec o his app oach also
encompasses signi ican ex ensions and gene aliza ions o he cha ac e is ics’ me hod o
non-smoo h Hamil on–Jacobi equa ions, p o iding a igo ous ounda ion o he heu is ic
p ocedu es p oposed by Isaac’s (1965) and a ious o he ela ed wo ks. Unlike p e ious
me hodologies, op imali y is de ined no wi hin he amewo k o saddle poin s [
3
,
17
],
bu a he in he mo e con enien hough os ensibly equi alen class o ela i ely op imal
eedback s a egies. As no ed in [
15
], he only ealis ic app oach o engaging in a di e en ial
game (pa icula ly in he con ex o an op imal con ol p oblem) is o employ eedback
s a egies ha a e calcula ed in ad ance. This pe spec i e is, in ac , sha ed by [
1
,
11
].
One o he classic p oblems in di e en ial games is he Wa o A i ion, a compelling
concep wi h applica ions ac oss a ious ields, including mili a y s a egy and esou ce
managemen . I is ega ded as a well-known example wi hin he domain o di e en ial
games. I models a scena io in which wo o mo e playe s a e engaged in a con es o cap u e
a aluable esou ce, such as e i o y o p ey. Howe e , his con es incu s cos s. Playe s
mus de e mine he du a ion o hei in ol emen in he s uggle, ca e ully weighing he
po en ial bene i s o ic o y agains he cumula i e cos s o sus ained con lic .
The aim o his pape is o apply s ep by s ep manne he heo e ical Dynamic P o-
g amming algo i hm, desc ibed in [
15
,
16
], and o in eg a e hese esul s wi h nume ical
p ocedu es in o de o achie e a mo e igo ous and heo e ically comple e solu ion o he
Wa o A i ion game, which was o mula ed and s udied heu is ically in [
1
] (Sec ion 5.4,
page 96). This model may be conside ed as a wo-playe ze o-sum game, whe e he playe s
ha e comple ely opposi e in e es s namely, a playe ’s gain is an equi alen loss o he
opposing playe . Ou app oach was i s employed o add ess p oblems o excep ionally
high complexi y. Among hese challenges was he well-known Homicidal chau eu game
in [
18
]. Addi ionally, in he explici model illus a ed in [
19
], i was shown ha , only he
maximal alue unc ion is admissible and is associa ed wi h ce ain eedback s a egies. Fu -
he mo e, wi hin he ealm o con lic p oblems, he model p esen ed in [
20
] is pa icula ly
ele an as i p o ides a amewo k ha can be di ec ly compa ed wi h he esul s in [
21
].
The p ima y dis inc ion be ween he wo s udies lies in he me hodological app oaches
employed o analyze wa a e dynamics: dynamic p og amming in ou case, con as ed
wi h he Lanches e equa ion in hei s. The conclusions d awn om his compa ison sug-
ges ha ou app oach o e s a b oade and mo e ealis ic amewo k o modeling wa a e
dynamics, pa icula ly in eal-wo ld scena ios whe e s a egies and condi ions e ol e o e
ime. While he Lanches e equa ion is use ul, i is cons ained by i s s a ic na u e and
limi ed scope, making ou app oach mo e applicable o complex, dynamic con lic s.
U ilizing he Dynamic P og amming me hod o sol e his p oblem p esen s he ad-
an age o allowing us o de e mine all admissible ajec o ies associa ed wi h he p oblem.
Mo eo e , he hypo heses ha need o be e i ied a e signi ican ly mo e na u al and easie
o es ablish, d awing upon elemen s o Hamil on–Jacobi heo y as well as ecen indings
in Non-Smoo h Analysis as e e enced in [15,22–24].
The pape is o ganised as ollows: a e he in oduc ion, we p esen in Sec ion 2 he
o mula ion o he p oblem, i s Dynamic P og amming o mula ion, and he cha ac e -
iza ion o he Hamil onian. Sec ion 3gi es he gene alized s a i ied Hamil onian ield.
In Sec ion 4, we desc ibe he pa ial Hamil onian low whose ajec o ies ha e e minal
segmen s on each o he s a a. Sec ion 5shows he exis ence o he co esponding alue
unc ion which de ines a ce ain pai o admissible and possibly op imal eedback s a e-
Games 2024,15, 35 3 o 18
gies o he conside ed game p oblem. Finally, some concluding ema ks a e p o ided in
Sec ion 6.
2. Fo mula ion o he P oblem
In [
1
], we conside a wa a e game model be ween wo na ions
U
and
V
, engaged in a
p o ac ed wa , ha consis s o op imizing he cos unc ion gi en by he ollowing:
C(u(.), (.)) =ZT
0[(1− ( ))x2( )−(1−u( ))x1( )]d , (1)
and de ined by he wa a e dynamic sys em:
x′=(m1−c1 ( )x2,m2−c2u( )x1,−1),x(0)=x0,
(u( ), ( )) ∈[0, 1]×[0, 1], ∈[0, T],
x0∈R3
+,c1>c2>0,
(2)
he in ol ed unc ions ha e he ollowing signi icance:
•x1( ),x2( ): ep esen he o ce o he na ions Uand V, espec i ely, a ∈[0, T];
•m1,m2: he weapon p oduc ion a e o he na ions Uand V, espec i ely;
•c1
,
c2
: a e he measu e o weapon e ec i eness o
V
e sus
U
and
U
e sus
V
, espec i ely;
•u( )
and
( )
: ep esen he s a egies o he wo na ions (o playe s) in ol ed in he
game. Speci ically,
u( )
is he s a egy chosen by he na ion
U
( he a acke ), which
de e mines he in ensi y o alloca ion o i s mili a y e o s o e ime
. The alue
o
u( )
is cons ained wi hin he in e al
[0, 1]
, whe e
u( ) =
0 ep esen s no a ack,
and
u( ) =
1 ep esen s he maximum possible a ack e o . In ela ion o
( )
i is
he s a egy chosen by he na ion
V
( he de ende ), ep esen ing how much e o
i alloca es o de ending i sel a any ime
. Simila o
u( )
,
( )
lies wi hin
[
0, 1
]
,
wi h
( ) =
0 indica ing no de ense e o and
( ) =
1 ep esen ing he maximum
de ense e o .
•
The i s and second equa ion o sys em (2) ep esen , he a e o change in na ion
U
’s
mili a y o ces o e ime ( espec i ely, he a e o change in na ion
V
’s mili a y o ces).
While, he hi d equa ion models he ime e olu ion wi hin he game. I implies ha
ime is dec easing uni o mly, as he con lic p oceeds, om
T
o 0. This nega i e ime
p og ession is a s anda d ea u e in di e en ial games o e lec he coun down owa d
he end o he game.
F om he in ui i e o mula ion o he p oblem in (1) and (2), i is unde s ood ha
he e a e wo na ions (playe s),
U
and,
V
and hey can choose, an op imal s a egy
e
u(.)
,
espec i ely,
e
(.)
o which he dynamic sys em in (2) gene a es a ajec o y
e
x(.)=xe
u,e
(.)
and such ha , he playe
U
ies o minimize he cos unc ional
C(., e
(.))
, while he playe
V ies o maximize he cos unc ional C(e
u(.), .).
2.1. Dynamic P og amming Fo mula ion
In o de o use he Dynamic P og amming app oach in [
15
,
16
], we e o mula e he
p oblem (1) and (2) using s anda d no a ions in game heo y and embedding his p oblem
in a se o p oblems associa ed wi h each ini ial poin in he phase space as in [
18
–
20
]. We
ob ain he ollowing s anda d Lag angian au onomous di e en ial game p oblem which,
in a a he ague o mula ion, may be s a ed as ollows:
P oblem 1. Gi en m1,m2>0, c1>c2>0. Find:
in
u(.)sup
(.)
C(y;u(.), (.)),∀y∈Y0, (3)
subjec o he ollowing:
Games 2024,15, 35 4 o 18
C(y;u(.), (.)) = g(x(T)) + RT
0 0(x( ),u( ), ( ))d ,y∈Y0,
x′( ) = (x( ),u( ), ( )) a.e.(0, T),x(0) = y,
u( )∈U(x( )), ( )∈V(x( )) a.e.(0, T),
x(.)∈Ω,(u(.), (.)) ∈ P, 0(x(.),u(.), (.)) ∈L1([0, T],R),
x( )∈Y0,∀ ∈[0, T),x(T)∈Y1,
(4)
de ined by he ollowing da a:
(x,u, ) = (m1−c1 x2,m2−c2ux1,−1),
0(x,u, ) = (1− )x2−(1−u)x1,
U(x) = U= [0, 1], V(x) = V= [0, 1],g(ξ) = 0, ∀ξ∈Y1,
Y0=R+×R+×[0, T),Y1=R+×R+×{0}.
(5)
whe e
P=U ×V
is he (la ges ) class o measu able admissible con ol unc ions
(u(
.
)
,
(
.
))
and
Ωis he co esponding class o absolu ely con inuous admissible ajec o ies.
2.2. The Hamil onian and he Se o T ans e sely Te minal Poin s
The pseudo-Hamil onian
H(x
,
p
,
u
,
) = ⟨p
,
(x
,
u
,
)⟩+ 0(x
,
u
,
)
is gi en in ou
case by he ollowing:
H(x,p,u, ) = p1m1+p2m2−p3+x2−x1+x1(1−c2p2)u−x2(1+c1p1) , (6)
whe e, p ep esen s Lag ange mul iplie s; using he ac ha :
minu∈U[(1−c2p2)u]=(0 i p2≤1
c2,
1−c2p2i p2>1
c2.
max ∈V[−(1+c1p1) ]=(0 i p1≥ − 1
c1,
−(1+c1p1)i p1<−1
c1.
hence, he co esponding ex eme alue o he con ol pa ame e s is gi en by he o mulas:
b
U(x,p) = b
U(p) =
{0}i p2<1
c2,
{1}i p2>1
c2,
U= [0, 1]i p2=1
c2.
b
V(x,p) = b
V(p) =
{0}i p1>−1
c1,
{1}i p1<−1
c1,
V= [0, 1]i p1=−1
c1.
(7)
The Isaac’s Hamil onian:
H(x,p) = min
u∈Umax
∈VH(x,p,u, ) = max
∈Vmin
u∈UH(x,p,u, ),(x,p)∈Z=dom(H(., .)),
as well as i s domain
Z
a e s a i ied by he s a i ica ion
SH={Z±,±
,
Z±,∓
,
Z0,±
,
Z±,0
,
Z0,0}de ined by he ollowing:
Games 2024,15, 35 5 o 18
Z+,+={(x,p)∈Z:p1>−1
c1,p2>1
c2},
Z+,−={(x,p)∈Z:p1>−1
c1,p2<1
c2},
Z+,0 ={(x,p)∈Z:p1>−1
c1,p2=1
c2},
Z−,+={(x,p)∈Z:p1<−1
c1,p2>1
c2},
Z−,−={(x,p)∈Z:p1<−1
c1,p2<1
c2},
Z−,0 ={(x,p)∈Z:p1<−1
c1,p2=1
c2},
Z0,+={(x,p)∈Z:p1=−1
c1,p2>1
c2},
Z0,−={(x,p)∈Z:p1=−1
c1,p2<1
c2},
Z0,0 ={(x,p)∈Z:p1=−1
c1,p2=1
c2}.
(8)
I we deno e by
H±,±(
., .
) = H(
., .
)|Z±,±
,
H±,∓(
., .
) = H(
., .
)|Z±,∓
,
H±,0(
., .
) =
H(., .)|Z±,0 ,H0,±(., .) = H(., .)|Z0,±,H0,0(., .) = H(., .)|Z0,0 we ob ain:
H+,+(x,p) = p1m1+p2(m2−c2x1)−p3+x2,
H+,−(x,p) = p1m1+p2m2−p3+x2−x1,
H+,0(x,p) = p1m1+m2
c2−p3+x2−x1,
H−,+(x,p) = p1(m1−c1x2) + p2(m2−c2x1)−p3,
H−,−(x,p) = p1(m1−c1x2) + p2m2−p3−x1,
H−,0(x,p) = p1(m1−c1x2) + m2
c2−p3−x1,
H0,+(x,p) = −m1
c1+p2(m2−c2x1)−p3+x2,
H0,−(x,p) = −m1
c1+p2m2−p3+x2−x1,
H0,0(x,p) = −m1
c1+m2
c2−p3+x2−x1.
(9)
Nex , we need o compu e he se o e minal ans e sali y alues de ined in he
gene al case by he ollowing:
Z∗
+,−={(ξ,q)∈Y1×R3:H(ξ,q) = 0, ⟨q,¯
ξ⟩=Dg(ξ)¯
ξ,∀¯
ξ∈TξY1},
Dg(ξ)¯
ξ=∂g
∂ξ (ξ)¯
ξ.(10)
Lemma 1. The se o e minal ans e sali y alues, Z∗, in ou case is gi en by he ollowing:
Z∗={((s1,s2, 0),(0, 0, s2−s1));s1,s2≥0} ⊂ Z+,−. (11)
P oo o Lemma 1.
Since,
g(ξ)=
0 and he angen space
TξY1=R×R×{
0
}
hen, i
ollows om (10) ha , q1ξ1+q2ξ2+q3ξ3=0, ∀ξ1,ξ2∈R,ξ3=0 and, he e o e
q1=q2=0, q3∈R.
S a ing om he ac ha , o
ξ= (s1
,
s2
, 0
)∈Y1
,
q= (
0, 0,
q3)
,
q3∈R
. I
z= (ξ
,
q)∈
Z+,+∪Z−,±
hen, we ob ain he ollowing con adic ions,
q2=
0
≯1
c2
,
q1=
0
≮−1
c1
, and
i
z=(ξ,q)∈Z±,0 ∪Z0,±∪Z0,0
we ob ain
q2=1
c2=
0 and
q1=−1
c1=
0. The e o e, he
only admissible ajec o ies a e he ones which ha e segmen s on he s a um
Z+,−
because,
q1=
0
>−1
c1
,
q2=
0
<1
c2
. Besides, using he ac ha ,
H+,−(ξ
,
q) = −q3+s2−s1=
0
hence, q3=s2−s1.
3. Gene alized Hamil onian and Cha ac e is ic Flow
The i s main compu a ional ope a ion consis s o he backwa d in eg a ion o
≤
0,
o he Hamil onian inclusion:
(´
x,´
p)∈d♯
SH(x,p),(x(0),p(0)) = z= (ξ,q)∈Z∗, (12)
de ined by he gene alized Hamil onian o ien o ield d♯
SH(., .):
Games 2024,15, 35 6 o 18
d♯
SH(x,p) = n(´
x,´
p)∈T(x,p)Z;´
x∈ (x,b
U(x,p),b
V(x,p)),
⟨´
x,¯
p⟩−⟨´
p,¯
x⟩=DH(x,p)( ¯
x,¯
p),∀(¯
x,¯
p)∈T(x,p)Zo,(13)
whe e,
DH(x,p)(x,p)
deno es he di ec ional de i a i e o Hamil onian unc ion
H(., .)
a
he poin (x,p)∈Zin he di ec ion (x,p)∈T(x,p)Zand is desc ibed as ollows:
DH(x,p)(x,p)=∂H
∂x(x,p)x+∂H
∂p(x,p)p.
As speci ied in he Algo i hm in [
15
,
16
], o each e minal poin
z= (ξ
,
q)∈Z∗
one
should iden i y he maximal solu ions: X∗(.)=(X(.),P(.)) :I(z)=( −(z), 0]→Z, o he
Hamil onian inclusion in (12) ha sa is ies he ollowing condi ions:
X( )∈Y0∀ ∈I0(z) = ( −(z), 0),
H(X( ),P( )) = 0, ∀ ∈I(z),
X′( ) = (X( ),u( ), ( )) a.e. I0(z),
u( )∈b
U(X∗( )), ( )∈b
V(X∗( )), a.e. I0(z).
(14)
I he e a e se e al solu ions o he same e minal poin
z= (ξ
,
q)∈Z∗
, i is necessa y
o pa ame e ize all hese solu ions by
λ∈Λ(z)
in o de o ob ain he gene alized Hamil-
onian low
X∗(
., .
)=(X(
., .
)
,
P(
., .
)) :B={( ,a), ∈I(a)a∈A}→Z
;
A=g aph(Λ(.))
,
a=(z,λ)
. We ecall also, he ac ha , o each
( ,a)∈B0={( ,a)∈B, =0}
he
Hamil onian low X∗(., .)de ines he con ols and, espec i ely, he ajec o ies:
u ,a(s)=ua( +s), ,a(s)= a( +s),s∈[0, − ],
x ,a(s)=X( +s,a),(15)
which a e admissible wi h espec o he ini ial poin
y=X( ,a)∈Y0
, and o which he
alue o he cos unc ional in (4)is gi en by he unc ion V(., .)de ined by he ollowing:
V( ,a)=g(ξ) +
Z0⟨P(σ,a),X′(σ,a)⟩dσ,a=(z,λ), (16)
and which, oge he wi h he Hamil onian low
X∗(., .)=(X(., .),P(., .))
de ines he
gene alized cha ac e is ic low
C∗(., .)=(X∗(., .),V(., .))
; using he de ini ion o he
Hamil onian
H(., .)
and he second condi ion in
(14)
one has
<P(σ,a)
,
X′(σ,a)>=
− 0(X(σ,a),b
u(X∗(σ,a)),b
(X∗(σ,a)))
, i ollows om
(3)
ha , he unc ion
V(
., .
)
ha ing
as o mula:
V( ,a)=Z
0((1−b
u(X∗(σ,a)))X1(σ,a)−(1−b
(X∗(σ,a)))X2(σ,a))dσ, (17)
he e o e, i ollows om
(8)
and
(9)
ha , he Hamil onian o ien ed ield
d♯
SH(
., .
)
is gi en
by he o mulas:
d♯
SH(x,p) =
d♯
SH±,±(x,p)i (x,p)∈Z±,±,
d♯
SH±,∓(x,p)i (x,p)∈Z±,∓,
d♯
SH±,0(x,p)i (x,p)∈Z±,0,
d♯
SH0,±(x,p)i (x,p)∈Z0,±,
d♯
SH0,0(x,p)i (x,p)∈Z0,0.
(18)
Games 2024,15, 35 7 o 18
Since he mani olds
Z±,±
,
Z±,∓⊂Z
a e open subse s, he Hamil onian o ien ed ields
d♯
SH±,±(., .)and d♯
SH±,∓(., .)in (13)coincide wi h classical Hamil onian ec o ields:
d♯
SH±,±(x,p) = n(∂H±,±
∂p(x,p),−∂H±,±
∂x(x,p)o,
d♯
SH±,∓(x,p) = n(∂H±,∓
∂p(x,p),−∂H±,∓
∂x(x,p)o,(19)
which a e easy o calcula e and will be desc ibed and s udied la e . While, on he sin-
gula s a um
e
Z∈Z±,0,Z0,±,Z0,0
he co esponding Hamil onian ield
d♯
Se
H(
., .
)∈
nd♯
SH±,0(., .),d♯
SH0,±(., .),d♯
SH0,0(., .)ois cha ac e ized by he ollowing esul .
Lemma 2. Fo any (x,p)∈e
Z one has he ollowing:
d♯
Se
H(x,p) = ∅. (20)
P oo o Lemma 2.
I
(x,p)∈Z±,0
, in o de o compu e he gene alized Hamil onian ield
d♯
SH±,0(
., .
)
, we no e i s ha , acco ding o some classical esul s as in [
15
], he angen
space o he i e-dimensional mani olds Z±,0 is gi en by he ollowing:
T(x,p)Z±,0 ={(¯
x,¯
p)∈R3×R3;¯
p2=0}, (21)
and
DH+,0(x
,
p)( ¯
x
,
¯
p) = −¯
x1+¯
x2+m1¯
p1−¯
p3
. The e o e, he condi ion
⟨´
x
,
¯
p⟩−⟨´
p
,
¯
x⟩=
DH+,0(x,p)( ¯
x,¯
p)is ully cha ac e ized by he exp ession:
(p′
1−1)¯
x1+ (1+p′
2)¯
x2+p′
3¯
x3+ (m1−x′
1)¯
p1−x′
2¯
p2−(x′
3+1)¯
p3=0,
∀¯
xi,¯
pi∈Ri=1, 2, 3. (22)
I ollows ha , a each poin (x,p)∈Z+,0 one has he ollowing:
x′
1=m1,x′
2=0, x′
3=−1, p′
1=1, p′
2=−1, p′
3=0, (23)
since (x′,p′)∈T(x,p)Z+,0 hen, p′
2=0, his con adic s he ac ha p′
2=−1.
Symme ically, on he s a um Z−,0 wo king as in he p e ious case we ob ain:
x′
1=m1−c1x2,x′
2=0, x′
3=−1, p′
1=1, p′
2=c1p1,p′
3=0, (24)
since
(x′,p′)∈T(x,p)Z−,0
hen,
p′
2=
0. While, om
(24)
i ollows ha ,
p1=
0 ha
con adic s he ac ha
(x,p)∈Z−,0
. Conce ning he s a a
Z0,±
, he p oo is conduc ed in
he same way as in he p e ious cases.
Nex , on he s a um
Z0,0
, using he same ype o compu a ions and a gumen s as in
abo e, we ob ain:
T(x,p)Z0,0 ={(x,p)∈R2×R2;(p1,p2)=(0, 0)},
DH0,0(x,p)(x,p)=−x1+x2−p3.(25)
While he condi ion
⟨x′,p⟩−⟨p′,x⟩=DH0,0(x,p)(x,p)
is cha ac e ized by he exp ession:
p′
1−1x1+p′
2+1x2−x′
3+1p3+p′
3x3=0, ∀xi,p3∈R,
om he e we deduce ha a each poin (x,p)∈Z0,0 one has he ollowing:
x′
1,x′
2∈R2,x′
3=−1, p′
1=1, p′
2=−1, p′
3=0,
he ac ha
(x′,p′)∈T(x,p)Z0,0
gi es
p′
1=
0
=
1 and
p′
2=
0
=−
1, which leads o a
con adic ion.
Games 2024,15, 35 8 o 18
3.1. The Hamil onian Sys em on he Open S a um Z+,+
On he open s a um
Z+,+
o which,
p1>−1
c1
and
p2>1
c2
he di e en ial inclusion
in (19)coincides wi h he smoo h Hamil onian sys em:
(x′= (m1,m2−c2x1,−1),
p′= (p2c2,−1, 0).(26)
S anda d esul s om di e en ial equa ions heo y show ha he gene al solu ion o
he sys em (26)is desc ibed by he o mulas:
x+,+( ) = (m1 +k1,−c2m1
2 2+ (m2−c2k1) +k2,− +k3), <0,
p+,+( ) = (−c2
2 2+k4c2 +k5,− +k4,k6),ki∈R,i=1, . . . , 6. (27)
3.2. The Hamil onian Sys em on he Open S a um Z+,−
On he open s a um
Z+,−
o which
p1>−1
c1
and
p2<1
c2
he di e en ial inclusion
in (19)coincides wi h he Hamil onian sys em:
(x′= (m1,m2,−1),
p′= (1, −1, 0),(28)
i s gene al solu ion is desc ibed by he ollowing:
x+,−( ) = (m1 +k1,m2 +k2,− +k3), <0,
p+,−( ) = ( +k4,− +k5,k6),ki∈R,i=1, . . . , 6. (29)
3.3. The Hamil onian Sys em on he Open S a um Z−,+
On he s a um
Z−,+
o which
p1<−1
c1
and
p2>1
c2
di e en ial inclusion
(
19
)
coincides wi h he Hamil onian sys em:
(x′= (−c1x2+m1,−c2x1+m2,−1),
p′= (c2p2,c1p1, 0),(30)
which has as a gene al solu ion:
x−,+
1( ) = k1e−√c1c2 +k2e√c1c2 +m2
c2, <0,
x−,+
2( ) = k1qc2
c1e−√c1c2 −k2qc2
c1e√c1c2 +m1
c1,
x−,+
3( ) = − +k3,
p−,+
1( ) = k4e−√c1c2 +k5e√c1c2 ,
p−,+
2( ) = −k4qc1
c2e−√c1c2 +k5qc1
c2e√c1c2 ,
p−,+
3( ) = k6,ki∈R,i=1, . . . , 6.
(31)
3.4. The Hamil onian Sys em on he Open S a um Z−,−
On he open s a um
Z−,−
o which
p1<−1
c1
and
p2<1
c2
he di e en ial inclusion
in (19)coincides wi h he smoo h Hamil onian sys em:
(x′= (m1−c1x2,m2,−1),
p′= (1, c1p1, 0),(32)
which, in u n, has he gene al solu ion:
x−,−( ) = (−1
2c1m2 2+ (m1−c1k1) +k2,m2 +k1,− +k3), <0,
p−,−( ) = ( +k4,1
2c1 2+c1k4 +k5,k6),ki∈R,i=1, . . . , 6. (33)
Games 2024,15, 35 15 o 18
b
B−,−(x)=b −,−(x),bs−,−
1(x),bs−,−
2(x),x∈Y−,−
0,
b −,−(x)=−x3,
bs−,−
1(x)=x1+x2+(m1+m2)x3−c1x2x3−1
2c1m2x2
3−m2
2c1,
bs−,−
2(x)=x2+m2x3.
(67)
P oo o Lemma 3. (1)
I
x=(x1,x2,x3)∈Y+,−
0
hen, i ollows om
(34)
ha , a
poin
( ,s1,s2)∈B+,−
o which
X+,−( ,s1,s2)=x
is cha ac e ized by he exp essions,
x1=m1 +s1
,
x2=m2 +s2
,
x3=−
. Hence, he exis ence and uniqueness o he
unc ions =b +,−(x)<0, s1=bs+,−
1(x)and s2=bs+,−
2(x)ha ing he o mulas as in (66).
(2)
In o de o p o e he second s a emen , use he same ype o compu a ion and
a gumen s as in he p e ious case. Thus, i ollows easily om
(44)
ha , he e exis
=b −,−(x),s1=bs−,−
1(x)and s2=bs−,−
2(x)o he o m as in (67).
The esul s in Lemma 3 show ha he cha ac e is ic lows
C∗
+,−(., .)=X∗
+,−(., .),V(., .)
and
C∗
−,−(., .)=X∗
−,−(., .),V(., .)
desc ibed, espec i ely, in
(17)
,
(34)
and
(44)
a e in e -
ible in he sense o (62)and gene a e he smoo h pa ial p ope alue unc ion:
W0(x)=W+,−
0(x)=1
2(m2−m1)x2
3+(x2−x1)x3,x∈Y+,−
0,
W−,−
0(x)=1
6c1m2x3
3−1
2m1x2
3+1
2c1x2x2
3−x1x3,x∈Y−,−
0,(68)
which is o class C1and may be na u ally ex ended by W(ξ)=g(ξ)=0, ∀ξ∈Y1.
While, om
(7)
and
(64)
, we deduce ha he co esponding admissible eedback
s a egies a e gi en by he ollowing:
e
U(x)×e
V(x)={(e
u+,−(x),e
+,−(x))}={(0, 0)},x∈Y+,−
0,
{(e
u−,−(x),e
−,−(x))}={(0, 1)},x∈Y−,−
0.(69)
The main esul in his sec ion is he ollowing.
Theo em 1. The ollowing s a emen s a e ue:
1.
The unc ion
W0(.)
de ined in
(68)
is a solu ion o Isaac’s equa ion de ined in
(65)
on he
co esponding open domain
Y+,−
0∪Y−,−
0
. Mo eo e , each o hem is he alue unc ion in he
sense o (60)o he co esponding admissible eedback s a egies gi en in (69).
2.
The eedback s a egies
e
U(.),e
V(.)
desc ibed in
(69)
a e op imal o he es ic ion on hei
open domain Y+,−
0∪Y−,−
0.
P oo o Theo em 1.
Fo
(1)
, om
(9)
,
(14)
,
(63)
,
(65)
and
(69)
i ollows ha , i
x∈Y+,−
0
hen:
minu∈U(x)max ∈V(x)[DW+,−
0(x). (x,u, )+ 0(x,u, )]
=min
u∈U(x)max
∈V(x)H(x,e
P+,−(x),u, ) = H(x,e
P+,−(x),e
u+,−(x),e
+,−(x))
=H+,−(X+,−(b
B+,−(x)),e
P+,−(x)) = 0,
while, i x∈Y−,−
0we ob ain:
minu∈U(x)max ∈V(x)[DW−,−
0(x). (x,u, )+ 0(x,u, )]
=min
u∈U(x)max
∈V(x)H(x,e
P−,−(x),u, ) = H(x,e
P−,−(x),e
u−,−(x),e
−,−(x))
=H−,−(X−,−(b
B−,−(x)),e
P−,−(x)) = 0,
hence, W0(.)de ined in (68)is a solu ion o Isaac’s’ Equa ion (65).
(2)
. Since he alue unc ion
W0(.)
in
(68)
is o class
C1
hen, in o de o p o e he
op imal o he pai o eedback s a egies in
(69)
, we use he well-known Elemen a y Ve i ica-
Games 2024,15, 35 16 o 18
ion Theo em [
1
,
12
,
15
] acco ding o which, a su icien op imal condi ion o he admissible
eedback s a egies e
U(.),e
V(.)is o check he ollowing di e en ial inequali ies:
in u∈U,¯
∈˜
V(x)[DW0(x) (x,u,¯
)+ 0(x,u,¯
)] ≥0,
sup¯
u∈˜
U(x), ∈V[DW0(x) (x,¯
u, )+ 0(x,¯
u, )] ≤0. (70)
Fi s , i x∈Y+,−
0 hen, i ollows om (5),(68)and (69) ha :
DW+,−
0(x)=(−x3,x3,−x1+x2+m2x3−m1x3)
(x,u,¯
) = (m1,m2−c2ux1,−1), 0(x,u,¯
)=x2−(1−u)x1
(x,¯
u, ) = (m1−c1 x2,m2,−1), 0(x,¯
u, ) = (1− )x2−x1
u=e
u+,−(x)=0, =e
+,−(x)=0,
(71)
and, he e o e:
in
u∈U,¯
∈˜
V+,−(x)hDW+,−
0(x) (x,u,¯
)+ 0(x,u,¯
)i=in
u∈U[(1−c2x3)ux1],
since c1>c2and x3∈[0, 1
c1)we deduce ha :
in
u∈U[(1−c2x3)ux1]=0,
o he second inequali y, i ollows om (71) ha :
sup
¯
u∈˜
U+,−(x), ∈VhDW+,−
0(x) (x,¯
u, )+ 0(x,¯
u, )i=sup
∈V
[(−1+c1x3)x2 ]=0.
Nex , i
x∈Y−,−
0
we use he same ype o compu a ion and a gumen s as in p e ious
case; hus, i ollows om (68)and (69) ha :
DW−,−
0(x)=−x3,1
2c1x2
3,−x1+c1x2x3−m1x3+1
2c1m2x2
3,
(x,u,¯
) = (m1−c1x2,m2−c2ux1,−1), 0(x,u,¯
)= (u−1)x1,
(x,¯
u, ) = (m1−c1 x2,m2,−1), 0(x,¯
u, ) = (1− )x2−x1,
u=e
u−,−(x)=0, =e
−,−(x)=1,
(72)
and we also ind:
in
u∈U,¯
∈˜
V−,−(x)hDW−,−
0(x) (x,u,¯
)+ 0(x,u,¯
)i=in
u∈U(1−1
2c1c2x2
3)x1u,
om he e, we can ex ac wo cases:
Case 1 : I x3∈1
c1,1
c1q2c1
c2−1 hen, 1 −1
2c1c2x2
3>0 hence:
in
u∈U1−1
2c1c2x2
3x1u=0.
Case 2
:
I
x3∈1
c1,− 1(s1,s2)
,
s1∈m1
c1,m1
c1h1+m1
2m2i
,
s2∈˜
s2,e˜
s2
hen, om
(56)
we ob ain,
x3<− 1(s1,s2)<1
c1q2c1
c2−1
and he es o he p oo is conduc ed in he same
way as in he p e ious case; he e o e, he i s inequali y in (70)is e i ied in bo h cases.
Fo he second inequali y, one has he ollowing:
sup
¯
u∈˜
U−,−(x), ∈VhDW−,−
0(x) (x,¯
u, )+ 0(x,¯
u, )i=sup
∈V
[(1−c1x3)x2(1− )] =0,
Games 2024,15, 35 17 o 18
which p o es inequali ies
(70)
and hence he op imali y o he admissible eedback s a e-
gies e
U(.),e
V(.)holds.
6. Conclusions
Finally, we a e now in a posi ion o de i e se e al key indings, o which he ollowing
a e no ewo hy:
1. Ou analysis builds upon Isaacs’ ounda ional amewo k o he wa game o a i ion
and a ack, he eby enhancing he unde s anding o he s a egic in e ac ions be ween
na ions. We demons a e ha he ex emi ies o he maximal in e al o ajec o ies
p o ide a mo e nuanced pe spec i e han p e iously a icula ed, e ealing mul iple
pa hways o con lic esolu ion ha ex end beyond Isaacs’ o iginal single-pa h analysis.
2.
The in oduc ion o newly ex ended ajec o ies subs an ially modi ies he dynamics
o he a i ion model. In con as o Isaacs’ ini ial conside a ions, ou indings sugges
ha na ions can employ s a egies designed o deple e he opponen ’s esou ces
g adually, he eby p olonging he du a ion o he con lic . This shi unde sco es he
necessi y o conside ing a ange o s a egic esponses in ex ended engagemen s.
3.
Ou esul s unde sco e ha speci ic selec ions ega ding he ex emi ies o ajec o ies,
pa icula ly hose a icula ed in equa ions
(40)
, can yield ixed du a ions o wa a e.
This insigh highligh s he c i ical impo ance o s a egic decision-making in shaping
con lic ou comes, whe eby he absence o a i ion may a ise unde ce ain condi ions,
ul ima ely in luencing he wa ’s ajec o y.
4.
The analysis e eals signi ican di e ences in how na ions, ep esen ed by
U
and
V
,
conduc hei mili a y engagemen s. Ou indings indica e ha while one na ion may
encoun e diminishing e u ns in s eng h, he o he can achie e a s eady inc ease in
powe . This asymme y sugges s ha s a egic ad an ages may luc ua e o e ime,
a o ing he na ion ha e ec i ely manages a i ion.
5.
We demons a e ha while na ion
V
is capable o sus aining p olonged mili a y ope a-
ions wi h a g adual inc ease in s eng h, na ion
U
aces a signi ican decline in esou ces,
ul ima ely esul ing in i s disad an age. This asymme y sugges s ha he balance o
powe p og essi ely shi s in a o o
V
, wi h subs an ial implica ions o op imal s a e-
gies and con lic ou comes. By con as ing ou indings wi h Isaac’s’ o iginal analyses,
we illumina e p e iously unexplo ed possibili ies wi hin he s uc u e o he maximal
in e al o ajec o ies and hei in luence on he dynamics o wa a e. The conclusions
p o ide a nuanced pe spec i e on he impac o s a egic choices on a i ion and wa a e
du a ion, indica ing he likelihood o V’s e en ual iumph o e U.
Howe e , wo king heu is ically, Isaacs in [
1
] ies o iden i y ce ain geome ic concep s
such as, he dispe sal line, equi ocal line and singula su ace,... , e c. Un o una ely, he
signi icance o op imali y was no speci ied. To add ess his aspec , in wo ks like [
3
,
9
,
12
,
17
],
op imali y is examined h ough he saddle poin condi ion o he cos unc ion
C(., .)
, in
he sense ha :
Ce
U(.),V(.)≤ Ce
U(.),e
V(.)≤ CU(.),e
V(.),∀(U(.),V(.)) ∈ P. (73)
In ela ion o ou app oach, he op imali y o a pai o admissible eedback s a egies
e
U(.),e
V(.)
is con i med h ough he e i ica ion o he weake condi ions in
(70)
which
a e easie o e i y, and much mo e e icien because do no equi e he p esence o all pai s
o admissible s a egies,
e
U(.),V(.)
and
U(.),e
V(.)
which in e u n, a e necessa y when
checking he saddle poin op imali y condi ion in (73).
In summa y, he cu en s udy encompasses con ibu ions om he au ho s in he
ollowing di ec ions:
1.
The use o some ecen concep s and esul s om Non-Smoo h Analysis and ele an
applica ions in he di e en ial games heo y, as well as employing he syn hesis o
Games 2024,15, 35 18 o 18
he e y ecen heo y in [
14
–
16
] ega ding he igo ous app oach and cons uc i e o
di e en ial game p oblems;
2.
The iden i ica ion o a pai o eedback s a egies, as well as he co esponding com-
ple e solu ion and he igo ous demons a ion o i s op imali y;
3.
The de elopmen o he implemen a ion wi h MATLAB 2018-so wa e, has aced he
e olu ions o he s a e’s cons ain s conside ed in he p oblem. The esul s ound show
ha , Dynamic P og amming is he mos e ec i e ool o he comple e esolu ion o
conc e e p oblems and p o ides accu a e esul s.
Au ho Con ibu ions: Concep ualiza ion, B.S. and B.T.; me hodology, B.S. and B.T.; so wa e, B.S.
and B.D.; alida ion, B.S., B.T. and B.D.; o mal analysis, B.S. and B.D.; w i ing—o iginal d a
p epa a ion, B.S. and B.T.; w i ing— e iew and edi ing, B.S., B.T. and B.D.; supe ision, B.T. and B.D.
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 da a p esen ed in he pape a e a ailable upon eques .
Con lic s o In e es : The au ho s decla e ha hey ha e no con lic s o in e es .
Re e ences
1. Isaacs, R. Di e en ial Games; Wiley: New Yo k, NY, USA, 1965.
2.
Ba di, M.; Capuzzo Dolce a, I. Op imal Con ol and Viscosi y Solu ions o Hamil on–Jacobi-Bellman Equa ions; Bi khuse : Be lin,
Ge many, 1997.
3. Ba¸sa , T.; Olsde , G. Dynamic Noncoope a i e Game Theo y, 2nd ed.; Academic P ess: London, UK, 1995.
4. Blaquiè e, A.; Lei mann, G. Jeux Quan i a i s; Gau hie -Villa s: Pa is, F ance, 1969.
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Kamne a, L.V.; Pa sko, V.S.; Tu o a, V.L. Analy ical and nume ical s udy o he Dolichob achis och one p oblem. In P oceedings
o he Analysis and Con ol o De e minis ic and S ochas ic E olu ion Equa ion, B essanone-B ixen, I aly, 3–7 July 2000.
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8.
Pa sko, V.S.; Tu o a, V.L. Le el se s o he alue unc ion in di e en ial games wi h he Homicidal chau eu dynamics. In . Games
Theo y Re . 2001,3, 67–112. [C ossRe ]
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13.
C andall, M.G.; E ans, L.C.; Lions, P.L. Some p ope ies o iscosi y solu ions o Hamil on–Iacobi–Bellman equa ions. T ans.
AMS 1984,282, 487–502. [C ossRe ]
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Mi ic˘a, ¸S. Ve i ica ion heo ems o op imal eedback s a egies in di e en ial games. In . Game Theo y Re . 2003,5, 167–189. [C ossRe ]
15.
Mi ic˘a, ¸S. Cons uc i e Dynamic P og amming in Op imal Con ol Au onomous P oblems; Edi u a Academiei Rom
ˆ
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ne: Bucha es ,
Romania, 2004.
16.
Mi ic˘a, ¸S. Use ’s Guide on Dynamic P og amming o au onomous di e en ial games and op imal con ol p oblems. Re . Romaine
Ma h. Pu es Appl. 2004,49, 501–529.
17. Ellio , R.J. Viscosi y Solu ions and Op imal Con ol; Pi man Resea ch No es in Ma hema ics; Longman: Ha low, UK, 1987.
18. Mi ic˘a, ¸S.; Bou emani, T. On he solu ion o he homicidal chau eu di e en ial game. Ma h. Rep. 2006,8, 53–81.
19.
Bou emani, T.; Mi ic˘a, ¸S. On he solu ion o a simple di e en ial game wi h a singula ocal line. Bull. Ma h. Soc. Sci. Ma h. Roum.
2006,49, 113–139.
20.
Bou emani, T.; Slimani, Y. S udy o a wa a e di e en ial game ia Dynamic P og amming app oach. Dyn. Games Appl. 2024,14,
733–750. [C ossRe ]
21.
Chen, X.; Qiu, J. Di e en ial Game o a Class o Wa a e Dynamic Sys ems wi h Rein o cemen Based on Lanches e Equa ion.
Abs . Appl. Anal. 2014,2014, 1–8. [C ossRe ]
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Mi ic˘a, ¸S. Hamil on–Jacobi equa ions on possibly non-symplec ic di e en iable mani olds. Bull. Ma h. Soc. Sci. Ma h. Roumanie
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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
au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o
people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .
