A Differential Game With One
Target and Two Players
Cardaliaguet, P.
IIASA Working Paper
WP-94-068
August 1994
Cardaliaguet, P. (1994) A Differential Game With One Target and Two Players. IIASA Working Paper. IIASA, Laxenburg,
Austria, WP-94-068 Copyright © 1994 by the author(s). http://pure.iiasa.ac.at/4143/
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Working Paper
A Differential Game with one
Target and Two Players
Pierre Cardaliaguet
WP-94-68
August, 1994
FflllASA
h":
International Institute for Applied Systems Analysis
Telephone:
o A-2361 Laxenburg o Austria
+43 2236 71521 o Telex: 079 137 iiasa a o Telefax: +43 2236 71313
A Differential Game with one
Target and Two Players
Pierre Cardaliaguet
WP-94-68
August, 1994
Working Papers are interim reports on work of the International Institute for Applied
Systems Analysis and have received only limited review. Views or opinions expressed
herein do not necessarily represent those of the Institute or of its National Member
Organizations.
IRIllASA
m
...u
International Institute for Applied Systems Analysis
Telephone: +43 2236 71521
Telex: 079 137 iiasa a
A-2361 Laxenburg
Austria
Telefax: +43 2236 71313
A differential game with one target and two
players
Cardaliaguet Pierre
CEREMADE, Uiliversitk Paris-Dauphine
Place du Mar6chal de Lattre de Tassigny
75775 Paris cedex 16
July 5 , 1994
A b s t r a c t : We s t u d y a two players-differential game in which one player wants t h e s t a t e of
t h e system t o reach a n open target and t h e o t h e r player wants t h e s t a t e of t h e system t o avoid
the target. We characterize t h e victory domains of each player as t h e largest set satisfying some
geometric conditions a n d we show a "barrier phenomenon" on t h e boundary of t h e victory domains.
lt6surn6 :
Nous etudions un jeu diffkentiel B deux joueurs d a n s lequel un des joueurs cherche
B faire e n t r e r ]'&tat d u systiime d a n s une cible donnee tandis que I'autre joueur veut q u e ]'&tat d u
systiime Pvite l a cible. Nous caracterisons le domaine d e victoire d e chacun des joueurs comme le
plus grand ensemble verifiant certaines conditions geometriques, puis nous m e t t o n s en evidence un
phenomiine d e barrikre s u r le bord des domaines d e victoire.
ICeywords : Differential Games, Pursuit and Evasion Games, Viability Theory.
A.M.S. classificatioil : 49 J 24, 49 J 52, 90 J 25, 90 J 26.
We study here the target problem: Two players, Ursula and Victor, control the
dynamical system:
xl(t) = f(x(t),u(t), v(t)) for almost every t
u(t) E U , v(t) E V
x(0) = xo
>0
Let R be an open target of D l N . Ursula, acting on u, wants the state of the
system x(.) to reach the target R, while Victor, acting on v, wants the state
of the system x(.) to avoid R. We want to determinate and characterize the
victory domains of each player, i.e., the set of initial positioils xo from where
this player may win whatever does his adversary.
We study this game in the framework of nonanticipative strategies: See ([6],
[7], [8]) If we denote by
U = {u(.) : [0, +co[+ U, measurable application )
(2)
V = {v(.) : [0, +co[+
V, measurable applicatioil )
tlie sets of time-measurable controls, nonanticipative strategies are defined in
tlie following way:
A map a : V + U is a nonanticipative strategy (for Ursula) if it satisfies the
following condition : For any s 0, for any vl(.) and v2(.) of V, such t h a t vl(.)
and v2(.) coiilcide almost everywhere on [0, s], the image a(vl(.)) and a(v2(.))
coincide almost everywhere on [0, s].
Nonanticipative strategies /3 : U --, V (for Victor) are defined in the symetric
way.
>
a
THEVICTORY
DOMAINS
Let us denote by x[xo, u(.), v(.)] the solution of (1)starting from l o E nN,with
u(.) E U and v(.) E V.
We are now ready t o define the victory domains of each player.
- Victor's victory domain is the set of initial positions xo $ R for which Victor
can find a noilanticipative strategy /3 : U --, V such t h a t for any time-measurable
control u(.) E V plaid by Ursula, the solution x[xo, u ( . ) , / ~ ( u ( . ) ) ]avoids R for
any t 0.
- Ursula's victory domain is the set of initial positions xo $ R for which Ursula
can find a nonanticipative strategy cr : V -4 U , positive c and T such that,
for any v(.) E U plaid by Victor, the solution x[xo, a ( v ( . ) ) , v(.)] reaches the set
R, := {x I d n c ( ~ ) c) before T.
(We denote by dr<(x) the distance from a point x to a closed set I<.)
>
>
a
ASSUMPTIONS
AND
NOTATIONS
Throughout this paper, we assume that f satisfies:
i)
ii)
iii)
iv)
U and V are metric compact spaces. V E
ndis convex.
f : lRN x U x V + nNis continuous.
f (., u , u) is a t-Lipschitz inap for any u and v.
f is affine in v.
We also always assume that Isaacs' condition is fulfilled:
b'(x,p) E B
~supinf
~ ,< f ( x , u , v ) , p > = infsup < f ( x , u , v ) , p >
u
V
U
We shall denote by B the closed unit ball of the state space n N .
1 - Discrimiilatiilg doinaiils and kernels
We first define, by the mean of geometric conditions, some sets - the discriminating domains and kernels - that shall play a great role in the sequel. For that
purpose, let us first define generalized outward normals for closed sets which
are not regular.
Definition 1 ( P r o x i m a l n o r m a l ) A vector v E lRN is a proximal normal to
a closed set I( at a point x E I( if and only i f : dK(x v) = Ilvll.
W e denote by NPI<(x) the set of proximal normals to I( at x.
+
If I( is a C2 manifolds and x belongs to d K , any outward normal is a proximal
normal up to a positive inultiplicative coefficient.
Definition 2 ( D i s c r i m i n a t i n g d o m a i n ) A closed set D is a discriminating
domain for f if
(4)
Vx E D , Vv E NPD(x), supinf
u
< f ( x , u , v ) , v >< 0
u
If a closed set I( is not a discriminating domain, it contains a largest discriminating domain:
T h e o r e m 1 ( D i s c r i m i n a t i n g kernel) Let I( be a closed subset of l R N . There
is a (may be empty) closed discriminating domain for f contained i n I( which
contains any other discriminating domain for f contained in I(.
This set is called the discriminating kernel of I<for f and is denoted by Discf (I().
DOMAINS A N D VIABILITYTHEORY
DISCRIMINATING
Let us now characterize the discriminating domains for dynamics f in terms of
Viability Theory. T h e definitions and results of Viability Tlieorem can be found
in the monograph ([:I.]).
Proposition 1 A closed set D is a discriminating domain for f if and only if
D is a viability domain for the set-valued maps x --* f (x, u, V) for any u E U.
Note that the discriminating domains for f are actually the discriminating domains defined in ([I.]). The discriminating kernel of a closed set I( can be
computed as intersections of viability domains:
Propositioil 2 ( A l g o r i t h m f o r t h e d i s c r i m i n a t i n g k e r n e l ) Let I( be a closed
subset of l R N . Define the following decreasing sequence of closed sets:
f
Then the intersection of the I(i is equal to ~ i s c(I?).
This results is used to prove Theorem 4 bellow.
2
- Interpretation of the discriminating domaills
WHEN "VICTORHAS A SPY"
T h e following Theorem means that the discriminating domains are the sets in
which Victor can ensure the state of the system to remain as soon as he knows
which control Ursula plays:
Theorem 2 A closed subset D C LRN is a discriminating domain for f if and
only if, for any xo E D l there exists a nonanticipative strategy P : U + V (for
Victor), such that, for any u(.) E U, the solution x[xo,u(.),P(u(.))] remains in
D on [O,+oo).
a WHEN "URSULAHAS A SPY"
If Ursula plays nonanticipative strategies, then discriminating domains are the
sets in which Victor can "almost" ensure the state of the system to remain:
Theorem 3 A closed subset D C lRN is a discriminating domain for f if and
only if, for any xo of D, for any nonanticipative strategy cr : V + U ,for any
positive 6 and for any time T 2 0, there exists a control v(.) E V such that the
solution x[xo, cr(v(.)), v(.)] remains1 in D cB on [O,T].
+
a
APPLICATIONS
TO
T H E TARGET PROBLEM
Theorem 4 ( A l t e r n a t i v e ) Set I( := lRN\C2. Then:
- Victor's victory domain is equal to Discj(I<).
- Ursula's victory domain is equal to I(\Discj(Ir').
Note that the victory domains of the two players form a partition of the closed
set I<. A similar Alternative Theorem have been obtained by Krasovskii &
Subbotin in the framework of the positional strategies (See [lo]). Our results
are thinest because there is no need t o introduce the notion of "constructive
motions" as in ([lo]).
Moreover, we characterize the victory domains by means of geometrical conditions (it is the discriminating kernel of a closed set). This characterization is
used in the joint work with M. Quincampoix & P. Saint-Pierre ([dl) to compute
numerically the victory domains (see also [12] in the framework of control theory).
- Semi-permeable surfaces
3
Theorem 5 Let I( be a closed subset of lRN and x belong to the boundary
of Discj(I() and to the interior of I(. Then, for any proximal normal v to
l R N \ ~ i s c j ( l < )at x, one has:
(6)
supinf
< f ( x , u , v ) , - v > 2 0,
v
U
Since Discj(I() is a domain for f , it also satisfies (4) with D := Discj(I().
In particular, if Discj(I() is a smooth (say C2) manifold in the neighbourhood
of some point t of dDiscj(I<)\dI(, the outward normal v of the boundary of
Discj(I<) satisfies the so-called Isaacs' equation:
(7)
supinf
U
'We denote by D
< f(x,u,v),v > = 0
v
+ cB the set of points x such that d o ( x ) 5 c.
thanks to (4) and (6). So, combining (4) and (6) gives a generalizalized solution
of equation (7).
Sinooth surfaces satisfying (7) are called semi-permeable surfaces by Isaacs. The
reason is that each player can avoid the state of the system to cross this surface
in one sense. In general, the boundary of the victory domains are not smooth
and Isaacs' methods do not work any more. But we prove bellow that, in any
case, the boundary of the victory domains is "almost semi-permeable". Similar
results have been obtained by M. Quincampoix in the framework of Control
Theory (See [ll]).
Theorem 6 Let I( be a closed subset of RlN1 zo btlong to dDiscj(I<) but not
to d K .
If p : U + V is a winning nonanticipative strategy2 for Victor, there is a time
T > 0 such that, for any positive 6, Ursula can find a control u(.) E V such that
the solution z[zo, u(.), P(u(.))] remains in dDiscj(I<) + C B on [0, T ] .
This result means that Ursula can "aln~ost"ensure the state of the system to
remain in dDiscj (I<). A similar result holds true if "Ursula has a spy".
References
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[2] AUBIN J.-P.& FRANKOWSKA H. (1990) SET-VALUED
ANALYSIS.
Birkhauser.
[3] BERNHARD P. (1979) Contribution d / ' d u d e des Jeux Diffirentiels
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[4] CARDALIAGUET P., QUINCAMPOIX M. & SAINT-PIERRE P.
(1994) Some algorithms for a game with two-players and one target,
M2AN, Vol. 28, N. 4.
[5] CARDALIAGUET P. (1994) Domaines discriminants en jeux
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differential games Mem. Amer. Math. Soc., 126.
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representation formulas for solutions of Hamilton-Jacobi Equations
Tkansactions of A.M.S., 282, pp. 487-502.
'A winning nonanticipative strategy for Victor is a strategy which ensures the state of the
system to remain in Disc! (Ii)forever.
[8] FRANKOWSICA H. & QUINCAMPOIX M. (1992) Isaacs' equations for value-functions of differential games International Series
of Numerical Mathematics, Vol. 107, Birkhauser Verlag Basel.
[9] ISAACS R. (1965) DIFFERENTIAL
GAMESWiley, New York.
[lo] KRASOVSKII N.N. & SUBBOTIN A.1. (1988) GAME-THEORICAL
Springer-Verlag, New-York.
CONTROLPROBLEMS
[ l l ] QUINCAMPOIX M. (1992) Differential Inclusion and Target Problem SIAM J . Control and Optimization, Vol. 30, No 2, pp. 324-335.
[12] SAINT-PIERRE P. (1992) Approximation of the Viability Iiernel,
Appl Math Optim, 29: 187-209.