- Email: [email protected]
On the Nash-M equilibrium in differential game under uncertainty
Copyright © IFAC Control Applications of Optirnisation. Visegrad. Hungary. 2003
ELSEVIER
IFAC PUBLICATIONS www.elsevier.comllocatelifac
On the Nash-M equilibrium in differential game under uncertainty L.CHERFI ·and M.S.RADJEF t Labor-atory LAMOS, Univenity of Bejai •. Algeria
Abstract
out uncertainty, the sufficient condition for existence of M equilibrium depend on the system of matrix differential equation. We note that as far as we know there is no general results for existence of equilibrium for game under uncertainty.
For a linear quadratic differential game with uncertainty, we give sufficient conditions for existence of a Nash equilibrium, These conditions are given under matrix differential Riccati form. Copyright © 2003 IFAC
1. Introduction
n.
This paper is concerned with linear differential game (LQ) problem in a finite time horiwn with additive uncertainty both in the state and in the objective function. A distinctive feature of the problem under consideration is that the uncertainty is independent of command action of each player. It is well know that there exits a rather extensive literature about Nash equilibrium existence and their corresponding Riccati equation see [4, 5, 3] for existence and [1] for an analytic solution of the matrix Riccati equation. There are few papers which concern equilibrium situation under uncertainty other than stochastic one, some exception may be found in multicriteria decision making field [6, 7,2]. The object of this paper is to show how we can formulate an equilibrium when there are uncontrollable parameter in the dynamic and objective function of the game. As an alternative of equilibrium we cite the paper of (Jank and Kun [2]) where the uncertainty depend on the Nash strategy of both player. According to our assumption (the uncertainty do not dependent on the command action of each player) their formulation may be not adequate for our game. For that we propose a new concept called the Nash-M equilibrium. This concept is defined like the players use Nash equilibrium for the formulation of their strategy unless they use M minimal optimality to minimize the uncertainty influence. We get analogous properties to other differential game with-
Problem statement.
We consider the linear quadratic game with N = 2 players ( N, E, {U;}i=1.2, Z, {Fi(Ul, U2, Z)}i=1.2 ), (1) where
with x E Rn is the phase vector, Ui E Rn is the i-th player's command action, (i = 1,2), z E Rn is the uncertainty, (to, xo) E [0, t f) x Rn is an initial position of the game. The payoff function of each player is defined for (i = 1,2) as following Fi
+ +
(Ul' U2, z) = x'(tf )Kifx(tf)
it! {u1Ri1Ul , + ' } dt to it! {Z,Piz + X'} QiX dt, to U2~2U2
(3)
where Kif, ~j, Pi, Qi are constant symmetric matrix in order n x n. Let M = (Mij) E M 2x2 be a matrix with positif constant coefficients. We assume that both players use the principle of Nash-M equilibrium to formulate the solution of the game (1).
Definition 1. A triplet (ui, u~, zM) is called Nash-M equilibrium in the differential game (1) with the initial position (to, x o) E [0, t f) x Rn if:
• E-mail [email protected] tE-mail [email protected]
207
(1) the situation u e = (u!, u~) is a Nash equilibrium in the game with two players
( {N}, E, {F;(u, zM)};=1,2)'
(2) Vu; E
jRn,
W 1(U N\i>U;,zM,V;)
(4)
0,
$
(ll)
(3) Vz E jRn, there exists io E {1,2} such that 2
L miokWk(Ue, z, Vk ) ~ 0,
(12)
k=1
(2) the uncertainty zM is Slater-minimal in problem with two criteria:
then the triplet (u!, u~, zM) is the Nash M equilibrium in the game (1) for any initial position:
2
( ]Rn,
{Lm;kFk(ue,Z)} ). k=1 ;=1,2
[6] If for every (u1(t, x), u~(t, x), z(t, x)) there exists io E {I, 2} such that Vz E ]Rn
Proposition
1.
2
11
(6)
2
1110. (13)
Stage 1. Let (to, xo) E [0, tf) x jRn an initial position of the game and x e (t), to $ t $ t f be the solution of differential system (2) with
z = zM(t,x) = pM(t)X, Ui (i = 1,2), i.e.
LmiokFk(Ue,zM) $ LmiokFk(Ue,Z), (7) k=1 k=1 then the uncertainty zM is Slater minimal in the problem with two criteria (6).
Xo
dxe(t) = [A(t) dt
= ui(t,x) = G~(t)x
+ GHt) + G~(t) + pM (t)]xe(t).
(14) Substituting x = xe(t), to $ t $ tf in (10) we obtain for all t E [to, tf]:
Ill. Sufficient condition In this section, we present sufficient conditions for existence of the Nash-M equilibrium in the game (1). For (i = 1,2) we introduce the following matrix functions.
aV;
= {it
Wi
(u, Z, Vi)
+
ax' [A(t)x + f; Uj + z1 [all.]'
Integrating this relation both to and t f ' using (14) and (9) we get (15)
(8) So
2
o
2
Z' Pi Z
ie V(to,xo) E [to,tf) x
Theorem 1. Assume that the following conditions are satisfied for the game (1): there exists a continuously differentiable functions of Lyapunov-Bellman V;(t,x) (i = 1,2), the strategies u~(t,x) = GHt)x (i = 1,2) and an uncertainty zM (t, x) = pM (t)x, where GH')' pM (.) E Cnxn[to, tf]' such that
jRn,
(i
= 1,2). (17)
Stage 2. In this section we use (11) to show that the situation u e is a Nash situation in the game (4). For (i = 1) we have: Wl(Ul,U~,ZM, VI) $ O.
(18)
Integrating this relation both to and t fusing (ll) and (17) we obtain
(9)
=
(16)
Fi(U e , zM, to, xo) = V;(to, XO).
(i = 1,2);
l1) for (i
Wt(t)dt =
+ x' Qi X .
j=1
V(t,x) E [to,tf] x
i
tl
to
L u~R;juj +
+
=
jRn:
o ~
1,2) 0,
(10)
208
tl W (t)dt = i to 1
(19)
i.e
Corollary 1. It comes from the proposition 1 that
It comes from the relation (20) that
'v'U1
E !Rn. (26) There exists io E {I, 2} such that
Using the equality (17) and the inequality (21) we obtain 'v'U1 E !Rn :
[~miOkWk(t,x'Ue(t,x),Z' Vk)]
=
min ~
zElRn
k=l
2
L
In analogous way we use the relation ref to prove that 'v'U2 E !Rn
(27)
miok Wk(t, x, ue(t, x), zM, Vk).
k=l
(23)
So the Lyapunov function are the solution of partial differential equation (26) and (27) .
So the situation (ui,u~,zM) is a Nash equilibrium in the problem 4. Stage 3 In this step we show that zM is M minimal in the problem with two criteria (6) . Let io = 1 such that (12) is satisfied and i(t) , to :S t :S t f be the solution of the differential system (2) with z(t,x) = Pi, ui(t, x) = Gii, (i = 1,2). Substituting x = i(t) into (12), we obtain after integration of this relation from to to tf :
Corollary 2. It comes from the proposition (1) that \f(to , xo) E [to, t f) x Rn , the optimal value of the game (1) is
i
t!
to
tmlkWk(t,i,Ue , Z, Vk)
~ 0,
'v'z E
Fi(ue, zM,to , xo)
jRn;
k=l
m1k Vi(to,
1, 2) .
Proposition 2. Assume that the following conditions are satisfied for the game (1):
2
e m1kFk( U , Z, to , XO) ~ L
=
In this section, we show that the existence of strategy of each player and of the uncertainty in the Nash M equilibrium is given under the solution of matrix differential Riccati equation.
(24)
L
(i
IV. Riccati equation
k=l
2
= Vi(to,xo)
XO).
a)
k=l
Ru
Using that
< 0,
R22
< 0;
(28)
b) there exists io E {1 , 2} such that 2
so
P(mio) = L 2
miokPk
> 0;
(29)
k=l
2
Lm1kFk(ue,Z) ~ LmlkFk(ue,zM). (25) k=l
c) the solution (K 1 (t),K 2 (t)) of the system of
k=l
differential Riccati equation
So the uncertainty problem
zM
is Slater minimal in the K1
+
2
( Rn, {LmikFk(ue,z,to,xo)} k=l
).
209
-K1A(t) - A'(t)K1
+
K1Rl/ K1 K2R221 K1
+ +
K2R2l R12R221 K2 - Q1 K( mio)p- l( mi o)K1 K1P- 1(mio )K( mio )
+
K( mio )?(mio1)K (mio) ;
i=1 ,2
Remark 1. We note that we can do the same procedure if (12) is satisfied for io = 2 to show the existence of M uncertainty situation.
=
+ K1R221 K2
K2
= -K2A(t) - A'(t)K2
+ +
K2R2lK2 KlR"]} K2 + K2R"]} Kl KlR"]}R2lR"]}Kl - Q2 K (miO )p-l (miO)K2 K2P-l( mio)K( mio) K(mio)i'(mi02)K(mio)
+ +
and
which hold the realization of (26). We consider now the function
Wet, x, uHt, x), U2(t, x), zM (t, x» 2
(30)
L miokWk(t,x, ub U2, z, Vk(t,X». k=l
where 2
K(mio)
Substituting (32) and (33) in precedent equation, we obtain Vt E [to, t f], Vx E Rn.
= L miok.Kk k=l
W(t,x,Ui(t,X),U2(t,X),ZM(t,X» = O. (34) We have
and
[aa~ (t,X,U H t,x),U 2(t,X),Z(t,x»]
=
2
exists on [to, t f].
2K(mio»X + 2{L miokPdz
Then, for any initial position, (to, xo) E [0, t f) x (]Rn \ On), the Nash-M equilibrium (u e , zM) in the game (1) is given by
uHt, x) = -R;/ Ki(t)X
(i = 1,2),
(31)
zM(t,x) = {- LmiokPd-lK(mio)x.
(32)
k=l
and
[a;;!' (t,x,Ui(t,X),U2(t,X),Z(t,x»] =
2
k=l
Proof. We use the proposition 1. Let \ti(t, x) = x' (t)Ki(t)X(t), (i = 1,2) where (K 1 (t), K 2 (t» are the solutions of the differential system (30). Substituting (31) and (32) in (10) we obtain
W i (ui,u 2,zM, V;)
=0
Vx
E]Rn
(i
min W(t,x,ui(t,x),u2(t,x),z(t, x»
zERn
W(t,x,ui(t,X),U2(t,x),zM(t,x» = 0,
= 1,2). (33)
We have
[~:i(t'X,Ui,U:;"'\i'Z' Vi)] = 2RiiKi(t)x+2~iUi' And
Since
which hold the realization of (27). Finally it comes from the proposition 1 that uHt, x), (i = 1,2) and zM (t, x) given by the relation (31) and (32) define Nash-M equilibrium in the game (1).
Corollary 3. It comes from the proposition 2 and the corollary 2 that the optimal value of the game is ~i
< 0, then
Fi(ue,zM,to,XO) = X~Ki(tO»xo
max W 1 (t,X,Ul,U2,zM, Vd "I
ERn
210
(i = 1,2). (35)
V. Conclusion In this paper we use Bellman approach to construit sufficient conditions for existence of situation of the Nash-M equilibrium in differential game under uncertainty. As in the undisturbed game these conditions depend on the existence of coupled matrix differential equation. We also satisfy some other conditions which are related to eigenvalues of matrix coefficient arising in payoff function which is the main inconvenient condition since the spectrum of the sum of matrix is not well knew from the literature. This inconvenient may be surmounted by an adequate choice of matrix M. We give also an expression of an analytic solution of the game as an expression of the solution of differential matrix Riccati equation. We note that the Nash M equilibrium may be extended to any game with N > 2 players.
References [1] H Abou-Kandil, Analytic solution for a class of linear quadratic open loop N ash games, INT.J.CONTROL vol. 43 (1986), no. no 3, 997-1002. [2] G.Jank and G.Kun, Optimal control of disturbed linear quadratic differential game, European Journal of Control vol. 08 (2002), 152-162. [3] J.C.Engwerda, On the open loop Nash equilibrium in LQ games, Journal of Economic Dynamics and Control vol. 22 (1998), 729762. [4] T.Bazar and G.Olsder, Dynamic non cooperative game theory, Academic Press, 1995. [5] T.Eisele, Non existence and no uniqueness of open loop equilibria in linear quadratic game, Journal of Optimization Theory and its Applications vol. 37 (1982), no. 4, 443468. [6] V.I.Zhukovskiy and M.E.Salukvadze, The vector valued maximin, Academic Press, Inc.N.Y, 1993. [7] V.I.Zhukovskiy and M.S.Radjef, Objection and counter objection in differential game under uncertainty, Optimization, Vol 50 pp.459-475, 2001.
211
ELSEVIER
IFAC PUBLICATIONS www.elsevier.comllocatelifac
On the Nash-M equilibrium in differential game under uncertainty L.CHERFI ·and M.S.RADJEF t Labor-atory LAMOS, Univenity of Bejai •. Algeria
Abstract
out uncertainty, the sufficient condition for existence of M equilibrium depend on the system of matrix differential equation. We note that as far as we know there is no general results for existence of equilibrium for game under uncertainty.
For a linear quadratic differential game with uncertainty, we give sufficient conditions for existence of a Nash equilibrium, These conditions are given under matrix differential Riccati form. Copyright © 2003 IFAC
1. Introduction
n.
This paper is concerned with linear differential game (LQ) problem in a finite time horiwn with additive uncertainty both in the state and in the objective function. A distinctive feature of the problem under consideration is that the uncertainty is independent of command action of each player. It is well know that there exits a rather extensive literature about Nash equilibrium existence and their corresponding Riccati equation see [4, 5, 3] for existence and [1] for an analytic solution of the matrix Riccati equation. There are few papers which concern equilibrium situation under uncertainty other than stochastic one, some exception may be found in multicriteria decision making field [6, 7,2]. The object of this paper is to show how we can formulate an equilibrium when there are uncontrollable parameter in the dynamic and objective function of the game. As an alternative of equilibrium we cite the paper of (Jank and Kun [2]) where the uncertainty depend on the Nash strategy of both player. According to our assumption (the uncertainty do not dependent on the command action of each player) their formulation may be not adequate for our game. For that we propose a new concept called the Nash-M equilibrium. This concept is defined like the players use Nash equilibrium for the formulation of their strategy unless they use M minimal optimality to minimize the uncertainty influence. We get analogous properties to other differential game with-
Problem statement.
We consider the linear quadratic game with N = 2 players ( N, E, {U;}i=1.2, Z, {Fi(Ul, U2, Z)}i=1.2 ), (1) where
with x E Rn is the phase vector, Ui E Rn is the i-th player's command action, (i = 1,2), z E Rn is the uncertainty, (to, xo) E [0, t f) x Rn is an initial position of the game. The payoff function of each player is defined for (i = 1,2) as following Fi
+ +
(Ul' U2, z) = x'(tf )Kifx(tf)
it! {u1Ri1Ul , + ' } dt to it! {Z,Piz + X'} QiX dt, to U2~2U2
(3)
where Kif, ~j, Pi, Qi are constant symmetric matrix in order n x n. Let M = (Mij) E M 2x2 be a matrix with positif constant coefficients. We assume that both players use the principle of Nash-M equilibrium to formulate the solution of the game (1).
Definition 1. A triplet (ui, u~, zM) is called Nash-M equilibrium in the differential game (1) with the initial position (to, x o) E [0, t f) x Rn if:
• E-mail [email protected] tE-mail [email protected]
207
(1) the situation u e = (u!, u~) is a Nash equilibrium in the game with two players
( {N}, E, {F;(u, zM)};=1,2)'
(2) Vu; E
jRn,
W 1(U N\i>U;,zM,V;)
(4)
0,
$
(ll)
(3) Vz E jRn, there exists io E {1,2} such that 2
L miokWk(Ue, z, Vk ) ~ 0,
(12)
k=1
(2) the uncertainty zM is Slater-minimal in problem with two criteria:
then the triplet (u!, u~, zM) is the Nash M equilibrium in the game (1) for any initial position:
2
( ]Rn,
{Lm;kFk(ue,Z)} ). k=1 ;=1,2
[6] If for every (u1(t, x), u~(t, x), z(t, x)) there exists io E {I, 2} such that Vz E ]Rn
Proposition
1.
2
11
(6)
2
1110. (13)
Stage 1. Let (to, xo) E [0, tf) x jRn an initial position of the game and x e (t), to $ t $ t f be the solution of differential system (2) with
z = zM(t,x) = pM(t)X, Ui (i = 1,2), i.e.
LmiokFk(Ue,zM) $ LmiokFk(Ue,Z), (7) k=1 k=1 then the uncertainty zM is Slater minimal in the problem with two criteria (6).
Xo
dxe(t) = [A(t) dt
= ui(t,x) = G~(t)x
+ GHt) + G~(t) + pM (t)]xe(t).
(14) Substituting x = xe(t), to $ t $ tf in (10) we obtain for all t E [to, tf]:
Ill. Sufficient condition In this section, we present sufficient conditions for existence of the Nash-M equilibrium in the game (1). For (i = 1,2) we introduce the following matrix functions.
aV;
= {it
Wi
(u, Z, Vi)
+
ax' [A(t)x + f; Uj + z1 [all.]'
Integrating this relation both to and t f ' using (14) and (9) we get (15)
(8) So
2
o
2
Z' Pi Z
ie V(to,xo) E [to,tf) x
Theorem 1. Assume that the following conditions are satisfied for the game (1): there exists a continuously differentiable functions of Lyapunov-Bellman V;(t,x) (i = 1,2), the strategies u~(t,x) = GHt)x (i = 1,2) and an uncertainty zM (t, x) = pM (t)x, where GH')' pM (.) E Cnxn[to, tf]' such that
jRn,
(i
= 1,2). (17)
Stage 2. In this section we use (11) to show that the situation u e is a Nash situation in the game (4). For (i = 1) we have: Wl(Ul,U~,ZM, VI) $ O.
(18)
Integrating this relation both to and t fusing (ll) and (17) we obtain
(9)
=
(16)
Fi(U e , zM, to, xo) = V;(to, XO).
(i = 1,2);
l1) for (i
Wt(t)dt =
+ x' Qi X .
j=1
V(t,x) E [to,tf] x
i
tl
to
L u~R;juj +
+
=
jRn:
o ~
1,2) 0,
(10)
208
tl W (t)dt = i to 1
(19)
i.e
Corollary 1. It comes from the proposition 1 that
It comes from the relation (20) that
'v'U1
E !Rn. (26) There exists io E {I, 2} such that
Using the equality (17) and the inequality (21) we obtain 'v'U1 E !Rn :
[~miOkWk(t,x'Ue(t,x),Z' Vk)]
=
min ~
zElRn
k=l
2
L
In analogous way we use the relation ref to prove that 'v'U2 E !Rn
(27)
miok Wk(t, x, ue(t, x), zM, Vk).
k=l
(23)
So the Lyapunov function are the solution of partial differential equation (26) and (27) .
So the situation (ui,u~,zM) is a Nash equilibrium in the problem 4. Stage 3 In this step we show that zM is M minimal in the problem with two criteria (6) . Let io = 1 such that (12) is satisfied and i(t) , to :S t :S t f be the solution of the differential system (2) with z(t,x) = Pi, ui(t, x) = Gii, (i = 1,2). Substituting x = i(t) into (12), we obtain after integration of this relation from to to tf :
Corollary 2. It comes from the proposition (1) that \f(to , xo) E [to, t f) x Rn , the optimal value of the game (1) is
i
t!
to
tmlkWk(t,i,Ue , Z, Vk)
~ 0,
'v'z E
Fi(ue, zM,to , xo)
jRn;
k=l
m1k Vi(to,
1, 2) .
Proposition 2. Assume that the following conditions are satisfied for the game (1):
2
e m1kFk( U , Z, to , XO) ~ L
=
In this section, we show that the existence of strategy of each player and of the uncertainty in the Nash M equilibrium is given under the solution of matrix differential Riccati equation.
(24)
L
(i
IV. Riccati equation
k=l
2
= Vi(to,xo)
XO).
a)
k=l
Ru
Using that
< 0,
R22
< 0;
(28)
b) there exists io E {1 , 2} such that 2
so
P(mio) = L 2
miokPk
> 0;
(29)
k=l
2
Lm1kFk(ue,Z) ~ LmlkFk(ue,zM). (25) k=l
c) the solution (K 1 (t),K 2 (t)) of the system of
k=l
differential Riccati equation
So the uncertainty problem
zM
is Slater minimal in the K1
+
2
( Rn, {LmikFk(ue,z,to,xo)} k=l
).
209
-K1A(t) - A'(t)K1
+
K1Rl/ K1 K2R221 K1
+ +
K2R2l R12R221 K2 - Q1 K( mio)p- l( mi o)K1 K1P- 1(mio )K( mio )
+
K( mio )?(mio1)K (mio) ;
i=1 ,2
Remark 1. We note that we can do the same procedure if (12) is satisfied for io = 2 to show the existence of M uncertainty situation.
=
+ K1R221 K2
K2
= -K2A(t) - A'(t)K2
+ +
K2R2lK2 KlR"]} K2 + K2R"]} Kl KlR"]}R2lR"]}Kl - Q2 K (miO )p-l (miO)K2 K2P-l( mio)K( mio) K(mio)i'(mi02)K(mio)
+ +
and
which hold the realization of (26). We consider now the function
Wet, x, uHt, x), U2(t, x), zM (t, x» 2
(30)
L miokWk(t,x, ub U2, z, Vk(t,X». k=l
where 2
K(mio)
Substituting (32) and (33) in precedent equation, we obtain Vt E [to, t f], Vx E Rn.
= L miok.Kk k=l
W(t,x,Ui(t,X),U2(t,X),ZM(t,X» = O. (34) We have
and
[aa~ (t,X,U H t,x),U 2(t,X),Z(t,x»]
=
2
exists on [to, t f].
2K(mio»X + 2{L miokPdz
Then, for any initial position, (to, xo) E [0, t f) x (]Rn \ On), the Nash-M equilibrium (u e , zM) in the game (1) is given by
uHt, x) = -R;/ Ki(t)X
(i = 1,2),
(31)
zM(t,x) = {- LmiokPd-lK(mio)x.
(32)
k=l
and
[a;;!' (t,x,Ui(t,X),U2(t,X),Z(t,x»] =
2
k=l
Proof. We use the proposition 1. Let \ti(t, x) = x' (t)Ki(t)X(t), (i = 1,2) where (K 1 (t), K 2 (t» are the solutions of the differential system (30). Substituting (31) and (32) in (10) we obtain
W i (ui,u 2,zM, V;)
=0
Vx
E]Rn
(i
min W(t,x,ui(t,x),u2(t,x),z(t, x»
zERn
W(t,x,ui(t,X),U2(t,x),zM(t,x» = 0,
= 1,2). (33)
We have
[~:i(t'X,Ui,U:;"'\i'Z' Vi)] = 2RiiKi(t)x+2~iUi' And
Since
which hold the realization of (27). Finally it comes from the proposition 1 that uHt, x), (i = 1,2) and zM (t, x) given by the relation (31) and (32) define Nash-M equilibrium in the game (1).
Corollary 3. It comes from the proposition 2 and the corollary 2 that the optimal value of the game is ~i
< 0, then
Fi(ue,zM,to,XO) = X~Ki(tO»xo
max W 1 (t,X,Ul,U2,zM, Vd "I
ERn
210
(i = 1,2). (35)
V. Conclusion In this paper we use Bellman approach to construit sufficient conditions for existence of situation of the Nash-M equilibrium in differential game under uncertainty. As in the undisturbed game these conditions depend on the existence of coupled matrix differential equation. We also satisfy some other conditions which are related to eigenvalues of matrix coefficient arising in payoff function which is the main inconvenient condition since the spectrum of the sum of matrix is not well knew from the literature. This inconvenient may be surmounted by an adequate choice of matrix M. We give also an expression of an analytic solution of the game as an expression of the solution of differential matrix Riccati equation. We note that the Nash M equilibrium may be extended to any game with N > 2 players.
References [1] H Abou-Kandil, Analytic solution for a class of linear quadratic open loop N ash games, INT.J.CONTROL vol. 43 (1986), no. no 3, 997-1002. [2] G.Jank and G.Kun, Optimal control of disturbed linear quadratic differential game, European Journal of Control vol. 08 (2002), 152-162. [3] J.C.Engwerda, On the open loop Nash equilibrium in LQ games, Journal of Economic Dynamics and Control vol. 22 (1998), 729762. [4] T.Bazar and G.Olsder, Dynamic non cooperative game theory, Academic Press, 1995. [5] T.Eisele, Non existence and no uniqueness of open loop equilibria in linear quadratic game, Journal of Optimization Theory and its Applications vol. 37 (1982), no. 4, 443468. [6] V.I.Zhukovskiy and M.E.Salukvadze, The vector valued maximin, Academic Press, Inc.N.Y, 1993. [7] V.I.Zhukovskiy and M.S.Radjef, Objection and counter objection in differential game under uncertainty, Optimization, Vol 50 pp.459-475, 2001.
211