In this paper, we present a numerical simulation of the gametheoretic model of an environmental pollution problem. This model is formalized by a noncooperative two-person di erential game in Banach space with separated dynamics of the agents and continuous payo functions depending on a game trajectory. The numerical simulation is based on the dynamic programming approach and the nite di erence method. Some numerical results are provided for two-dimensional dynamic conict model of an environmental pollution problem.
Application of the game-theoretic approach plays a special role in mathematical modeling of environmental problems. The dynamic games that deal with similar or related problems were investigated in many works, see e.g. [3{6, 9{12, 15] and references therein.
In [14], the existence of "-Nash equilibrium in a dynamic con ict model of an environmental pollution problem which is formalized by the noncooperative n-person di erential game in Banach space was proved.
In this paper, we present a numerical simulation for the two-dimensional dynamic con ict model of an environmental pollution problem. Enterprises (agents) contaminate a water reservoir by dumping a pollutant (harmful substance) of the same type during the production process.
This model is formalized by a noncooperative two-person di erential game in Banach space with separated dynamics of the agents and continuous payo functions depending on a game trajectory. For simplicity, we consider the case of separated dynamics of agents, in which the dynamics of each agent is described by the initial boundary value problem for the parabolic equation involving Dirac measure. The existence of "-Nash equilibrium in our model follows from the results of [14].
The paper gives a formal mathematical description of the model and related numerical method and presents simulation results. 2
A closed water reservoir (e.g. lake) is considered. Two enterprises dump a pollutant (harmful substance) of the same type into this water reservoir during the process of production. Both enterprises dump the pollutant at some prescribed points of the reservoir.
Furthermore, it is assumed that the reservoir has a water intake. The level of pollution at the water intake point (the total concentration of the pollutant released by all enterprises) must not exceed the maximum permissible value. It is assumed that if this value is exceeded, all the enterprises will pay a ne (penalty) as a percentage of their income. It is also assumed that the company has certain expenses associated with the cleaning of the pollutant.
The spread of the harmful substances in the reservoir occurs by di usion. Besides, a pollutant is decomposed with the rate r > 0.
The total income of an enterprise depends on its volume of production, which is tightly linked with its total volume of dumped pollutant. Besides, the total income depends on the overall cleaning expenses and possible pollution nes. The aim of each enterprise is to maximize the total income for nite period of time. 3
We will study the di erential two-person game (c0; T ) with a prescribed duration T < 1 and an initial position of the game c0. Let I = fig = f1; 2g be a set of the agents (enterprises).
We will consider the closed water reservoir as a rectangle = [0; d1] [0; d2].
Let zi(x1; x2; t) be the pollutant concentration of the agent i at the point (x1; x2) 2 at the moment t 2 [0; T ]. Denote by (xi1; xi2) 2 the prescribed point of dumping the pollutant of the agent i 2 I and by (xw; yw) the coordinates of the water intake location inside the domain .
Let us denote by ui(t) the intensity of dumping the pollutant of the agent i 2 I at the moment t. We assume that the intensity of dumping the pollutant satis es the following conditions: 0 ui(t)
Gi(t); i 2 I;
(
The dynamics of the agent i = 1; 2 in the game (c0; T ) is described by
the initial boundary value problem for the following di erential equation on the
domain = (0; d1) (0; d2) :
(
Let the function zi(x1; x2; t) satis es the following boundary conditions of impenetrability: and the following initial condition: zi(x1; x2; 0) = ci0(x1; x2);
x1 2 [0; d1]; x2 2 [0; d2]; t = 0; where ci0(x1; x2) is some given function describing the initial distribution of the pollutant concentration of the agent i in the water reservoir at the initial moment t = 0.
De nition 1. A measurable function ui = ui(t), satisfying the condition (
Let the function fi(t) 0 for all t 2 [0; T ] determine the amount of the penalty of the agent i for exceeding the maximum permissible value of pollution at the water intake point (xw; yw) as follows: fi(t)= 8 >> 0; > > > > > < > zi(xw; yw; t) >>>>> P2 zj (xw; yw; t) > :j=1 2 P zj (xw; yw; t) j=1
Cw
Cw ;
2 if P zj (xw; yw; t) j=1
Cw;
2
if P zj (xw; yw; t) > Cw;
j=1
(
Denote by qi = qi(t) the volume of production of the agent i at the moment t. Let Pi > 0 be the price of the product of the agent i.
Assume that the intensity of dumping the pollutant ui(t) linearly depends on the production volume of the agent i: ui(t) = qi(t); > 0:
Let p > 0 be the payment for the discharge of a unit of pollutant. Then the payo of the agent i at time T is de ned by the following functional: T Z 0
T Z 0 Hi(z; ui) =
Piui( )(1 pfi( ))d
Miui( )d ;
(
Game (c0; T ) is a particular case of a noncooperative n-person di erential game in Banach space that was studied in [14]. Below, for the sake of completeness, let us recall the main results of [14].
The dynamics of the agent i 2 I = fig = f1; : : : ; ng in the n-person di erential game (c0; T ) is described by the initial boundary value problem for the following di erential equation: Here D(x; y; t) > 0 is the di usion coe cient; r > 0 is the coe cient characterizing the pollution decomposition; ui 2 Ui is a control parameter of the agent i, Ui Rmi is a compact set in Euclidean space. The function i(x; y) = (x xi; y yi) gives the location of the agent i inside the domain .
Let the function zi(x; y; t) satis es the following boundary and initial conditions: D(t; x; y)
= 0; zi(x; y; 0) = ci0(x; y); (x; y) 2 S; t 2 [0; T ]; (x; y) 2 ; t = 0; where m is an outward normal to the boundary surface S [0; T ], ci0(x; y) is some given function describing the initial distribution of the pollutant concentration of the agent i in the water reservoir at the initial moment t = 0.
Let us represent the problem (
A(t)ci(t) =
i(t); t 2 [0; T ];
(
The equation (
We assume that the coe tients D and r in (
Let us denote by Fi(ci0; t0; t) the set of the points ci( ) 2 Wp1( ) for which there exists an admissible control ui(t) such that the game goes from the state ci(t0) = ci0 to the state ci(t + t0) for the time interval [t0; t]. The set Fi(ci0; t0; t) is a bounded set of the space Wp1( ). It is known [7] that if the boundary of the domain is smooth then a bounded set of the space Wp1( ) is a compact set in Lp( ). This implies that Fi(ci0; t0; t) is a compact set for all ci0 2 (Wq2=p 1( ))0, t0; t 2 [0; T ] as well. Fi(ci0; t0; t0) = ci0 for all ci0 2 (Wq2=p 1( ))0, t0 2 [0; T ]. The set Fi(ci0; t0; t) has semigroup property.
The set Fi(ci0; t0; t) is called the attainability set of the player i; i = 1; n from the initial state ci0 on the time interval [t0; t].
Let us denote by Fbi(ci0; t0; t); i 2 I the set of trajectories c^i(ci0; t t0) of
(
At every moment t 2 [0; T ] of the game (c0; T ) the agents know the realized trajectory of the game, the dynamics and the duration T of the game.
Let c^i( ) 2 Fbi(ci0; 0; T ) be the trajectory of (
that, if c^i( ) = c^0i( ) for if (t + T ) T and Moreover, 2 (0; 1] there exists a map
2 [0; t], then i(c^i)( ) = i(c^i)( ) = i(c^0i)( ) for i : Fbi(ci0; 0; T ) ! Fbi( ) such i(c^0i)( ) for 2 [0; t + T ] 2 [0; T ] if (t + T ) > T . "i( ) =
sup kc^i c^i2Fbi( ) i(c^i)k
Let us x the permutation p = (i1; : : : ; ik; : : : ; in) and consider n-person multistep game p (c0; T ) at every step which the agents i1; : : : ; in choose in sequence controls ui1 ; : : : ; uik ; : : : ; uin .
'ipk : Fbik ( ) = Y Fbj ( ) ! Fbik ( );
j6=ik of the agent ik in the game p (c0; T ) is a mapping such that if c^j ( ) = c^0j ( ) for j < ik, 2 [0; l T ] and if c^j ( ) = c^0j ( ) for j > ik, 2 [0; (l 1) T ], then 'ipk (c^ ik ( )) = 'ipk (c^ 0ik ( )), 2 [0; l T ]. Here = 1=2N ; l = 1; 2; : : : ; 2N .
Let us denote by ipk the set of the strategies of the agent ik in the game p (c0; T ). In the game p (c0; T ) the players i1; :::; in choose in sequence the strategies 'ip1 ; : : : ; 'ipn . The trajectory ( 'p) is uniquely de ned for every ntuple 'p = ( 'ip1 ; : : : ; 'ipn ) stepwise on successive intervals [0; T ]; : : :, [T T; T ]. The payo function of the agent i 2 I in the game p (c0; T ) is de ned as follows:
Hi (c0; 'p) = Hi( ( 'p));
(
So, the n-person di erential game p (c0; T ) with the prescribed duration T is de ned in a normal form: p (c0; T ) = hI; f p n i g1 ; fHi g1ni:
Using the Zermelo-Neumann theorem, the existence of "-equilibrium for any " > 0 in the multistep game p (c0; T ) can be proved.
The previous Lemma 1 implies the following lemma.
Lemma 2. If ik > i1, 'ipk 2 ipk , then ik 'ipk 2 pe is a permutation of the set I n ik; moreover, for c ik 2 Fbik ( ) b
ipkik , where pik = (ik; pe), k 'ipk (bc ik ) ( ik 'ipk )(c ik )k b "( )
Lemma 3. Let the game H0 = hI; fXi0g1n; fHi0g1ni be obtained from the game
H = hI; fXig1n; fHig1ni by the epimorphic mapping i : Xi ! Xi0, i = 1; : : : ; n, with kH(x)
H0( x)k "; x = ( 1(x1); : : : ; n(xn)): Then, if x is an "-equilibrium of the game the game H0 .
(c0; T ).
x is the 3"-equilibrium of De nition 3. The pair ( i; f 'pi
i g =1=2N ) is called the strategy of the agent i.
Here N 2 Z, i is a range of dyadic partition of the time interval [0; T ] and 'ipi is the strategy of the agent i in the game pi (c0; T ) for the permutation pi = (i; pe) and pe is the permutation of the set I n i.
For n-tuple ' = ('1; : : : ; 'n) the game (c0; T ) is played as follows. The smallest i = is chosen and the trajectory ( ) is constructed for n-tuple ' = ( '1p1 ; : : : ; 'pnn ). This trajectory is unique.
The game (c0; T ) is obtained from the game p (c0; T ) by the epimorphic mapping which is de ned in Lemma 2. Since in the game p (c0; T ) there exists "-equilibrium, then the existence of the 3"-equilibrium in the game (c0; T ) follows from Lemma 2 and Lemma 3.
Thus, the following theorem is valid.
Theorem 1. There exists "-equilibrium in the noncooperative n-person di erential game (c0; T ) for all " > 0. 4
The numerical method based on the dynamic programming method [2] and the nite di erence scheme [13] is proposed for the numerical solving of the auxuliary multistep game p (c0; T ).
To construct a di erence scheme for our problem, we use an approach that was developed in [13] for constructing a homogeneous di erence scheme for the nonstationary heat conduction problem with a one-point heat source that is de ned by a Dirac delta function expression.
On the rectangle = [0; d1] [0; d2], we construct the uniform grid with the
step h1 on x1 and the step h2 on x2
!h = fx1l = lh1; l = 0; : : : ; N1; x10 = 0; x1N1 = d1;
x2k = kh2; k = 0; : : : ; N2; x20 = 0; x2N2 = d2g ;
(
On every interval [ts; ts+1]; s = 0; N 1, we construct the uniform net with step
! ;s = ftj = j ; j = 0; N3; t0 = ts; tN3 = ts+1g: Here ts 2 , where is the time interval partition
= ft0 = 0 < t1 < : : : < tN = T g:
For numerical simulations, we will consider the admissible control parameters set Ui = [U i1; U i ], where U i1 = const, U i2 = const, i = 1; 2, and construct the 2 following partition on the set Ui:
1 2 i = fui;0 = U i < ui;1 < : : : < ui;N4 = U i g; i = 1; 2:
On the time interval [ts; ts+1] we will consider grid functions iys(x1l; x2k; tj) instead of functions zi(x1; x2; t) of continuous arguments (x1; x2; t) 2 [ts; ts+1], where i 2 I { the agent's number, s = 0; N 1. Here the argument of the grid function (x1l; x2k; tj) is a node of the grid !h = !h ! ;s.
Let us denote by iylj;;ks = iys(x1l; x2k; tj ) the grid function de ned on the net !h = !h
! ;s.
We construct for the problem (
! ;s for any pair of admissible controls the elimination method [13]. (16) (17) (18) (19) (20) (21) (22) (23)
The payo function of the agent i; i = 1; 2 in the game imated as follows: p (c0; T ) is approxHi(u10; : : : ; u1N 1; u02; : : : ; u2N 1) =
N 1 N2 1 X X Piuis(1
pfij )
s=0 j=0
N 1 N2 1
X X
The numerical experiments were carried out for the following input data: D =
4:4, d1 = d2 = 30, h = 0:5, T = 320, = 1, N = 8, r = 0:005, p = 3:1,
M1 = 4:5, M2 = 5:5, P1(
2 2 The results of numerical experiments are presented in Figures 1{5.
Figure 1 presents the computed distribution of the pollutant concentration of the rst and second agents. The realizations of their optimal strategies are shown in Fig. 2. The agents reduce the intensity of dumping the pollutant (stop production) to minimize nes. Then the intensity of production is resumed to the maximum values for obtaining the maximum income by the time T . The time development of the payo functions for both agents is given in Fig. 3. The dynamics of changing the pollutant concentration at the intake point is presented in the Fig. 4. The agent pays the penalty (Fig. 5) in the case of exceeding the maximum permissible value Cw. We investigated the noncooperative two-person di erential game in Banach space which models a con ict-controlled process of the contaminating a closed water reservoir. The dynamics of the agents is described by the initial boundary value problem for the two-dimensional di usion equation with a point source.
The proposed numerical algorithm for solving the considered di erential game is based on the dynamic programming method and the nite di erence scheme. It has been applied to compute the auxiliary multistep game.
The authors are grateful to the anonymous reviewers for their valuable comments and suggestions which helped to improve the quality of the paper.