MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation65372310.1155/2009/653723653723Research ArticlePursuit-Evasion Differential Game with Many Inertial PlayersIbragimovGafurjan I.SalimiMehdiSeyranianAlexander P.Department of Mathematics & Institute for Mathematical ResearchUniversiti Putra Malaysia43400 UPM SerdangSelangorMalaysiaupm.edu.my200923082009200918042009170720092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider pursuit-evasion differential game of countable number inertial players in Hilbert space with integral constraints on the control functions of players. Duration of the game is fixed. The payoff functional is the greatest lower bound of distances between the pursuers and evader when the game is terminated. The pursuers try to minimize the functional, and the evader tries to maximize it. In this paper, we find the value of the game and construct optimal strategies of the players.

1. Introduction and Preliminaries

Many books have been devoted to differential games, such as books by Isaacs , Pontryagin , Friedman , Krasovskii and Subbotin .

Constructing the player's optimal strategies and finding the value of the game are of specific interest in studying of differential games.

The pursuit-evasion differential games involving several objects with simple motions take the attention of many authors. Ivanov and Ledyaev  studied simple motion differential game of several players with geometric constraints. They obtained sufficient conditions to find optimal pursuit time in n, by using the method of the Lyapunov function for an auxiliary problem.

Levchenkov and Pashkov  investigated differential game of optimal approach of two identical inertial pursuers to a noninertial evader on a fixed time interval. Control parameters were subject to geometric constraints. They constructed the value function of the game and used necessary and sufficient conditions which a function must satisfy to be the value function .

Chodun  examined evasion differential game with many pursuers and geometric constraints. He found a sufficient condition for avoidance.

Ibragimov  obtained the formula for optimal pursuit time in differential game described by an infinite system of differential equations. In  simple motion differential game of many pursuers with geometric constraints was investigated in the Hilbert space l2.

In the present paper, we consider a pursuit-evasion differential game of infinitely many inertial players with integral constraints on control functions. The duration of the game θ is fixed. The payoff functional of the game is the greatest lower bound of the distances between the evader and the pursuers at θ. The pursuer's goal is to minimize the payoff, and the evader's goal is to maximize it. This paper is close in spirit to . We obtain a sufficient condition to find the value of the game and constructed the optimal strategies of players.

2. Formulation of the Problem

In the space l2 consisting of elements α=(α1,α2,,αk,), with k=1αk2<, and inner product (α,β)=k=1αkβk, the motions of the countably many pursuers Pi and the evader E are defined by the equations Pi:ẍi=ui,xi(0)=xi0,ẋi(0)=xi1,E:ÿ=v,y(0)=y0,ẏ(0)=y1, where xi,xi0,xi1,ui,y,y0,y1,vl2,  ui=(ui1,ui2,,uik,) is the control parameter of the pursuer Pi, and v=(v1,v2,,vk,) is that of the evader E; here and throughout the following, i=1,2,,m,. Let θ be a given positive number, and let I={1,2,,m,}.

As a real life example, one may consider the case of a missile catching an aircraft. If the initial positions and speeds (first derivative) of both missile and aircraft are given and the constraints of both missile and aircraft are their available fuel, which could be mathematically interpreted as the mean average of their acceleration function (second derivative), then the corresponding pursuit-evasion problem is described by (2.1).

A ball (resp., sphere) of radius r and center at the point x0 is denoted by H(x0,r)={xl2:x-x0r} (resp., by S(x0,r)={xl2:x-x0=r}).

Definition 2.1.

A function ui(·),  ui:[0,θ]l2, such that uik:[0,θ]R1,  k=1,2,, are Borel measurable functions and ui(·)2=(0θui(s)2ds)1/2ρi,ui=(k=1uik2)1/2, where ρi is given positive number, is called an admissible control of the ith pursuer.

Definition 2.2.

A function v(·),  v:[0,θ]l2, such that vk:[0,θ]R1,  k=1,2,, are Borel measurable functions and v(·)2=(0θv(s)2ds)1/2σ, where σ is a given positive number, is called an admissible control of the evader.

Once the players' admissible controls ui(·) and v(·) are chosen, the corresponding motions xi(·) and y(·) of the players are defined as

xi(t)=(xi1(t),xi2(t),,xik(t),),y(t)=(y1(t),y2(t),,yk(t),),xik(t)=xik0+txik1+0t0suik(r)drds,yk(t)=yk0+tyk1+0t0svk(r)drds.

One can readily see that xi(·),y(·)C(0,θ;l2), where C(0,θ;l2) is the space of functions f(t)=(f1(t),f2(t),,fk(t),)l2,t0, such that the following conditions hold:

fk(t),  0tθ,  k=1,2,, are absolutely continuous functions;

f(t),  0tθ, is a continuous function in the norm of l2.

Definition 2.3.

A function Ui(t,xi,y,v),  Ui:[0,)×l2×l2×l2l2, such that the system ẍi=Ui(t,xi,y,v),xi(0)=xi0,ẋi(0)=xi1,ÿ=v,y(0)=y0,ẏ(0)=y1, has a unique solution (xi(·),y(·)), with xi(·),y(·)C(0,θ;l2), for an arbitrary admissible control v=v(t),  0tθ, of the evader E, is called a strategy of the pursuer Pi. A strategy Ui is said to be admissible if each control formed by this strategy is admissible.

Definition 2.4.

Strategies Ui0 of the pursuers Pi are said to be optimal if infU1,,Um,Γ1(U1,,Um,)=Γ1(U10,,Um0,), where Γ1(U1,,Um,)=supv(·)infiIxi(θ)-y(θ),  Ui are admissible strategies of the pursuers Pi, and v(·) is an admissible control of the evader E.

Definition 2.5.

A function V(t,x1,,xm,,y),V:[0,)×l2××l2××l2l2, such that the countable system of equations ẍk=uk,xk(0)=xk0,ẋk(0)=xk1,k=1,2,,m,,ÿ=V(t,x1,,xm,,y),y(0)=y0,ẏ(0)=y1, has a unique solution (x1(·),,xm(·),,y(·)), with xi(·),y(·)C(0,θ,l2), for arbitrary admissible controls ui=ui(t),  0tθ, of the pursuers Pi, is called a strategy of the evader E. If each control formed by a strategy V is admissible, then the strategy V itself is said to be admissible.

Definition 2.6.

A strategy V0 of the evader E is said to be optimal if supVΓ2(V)=Γ2(V0), where Γ2(V)=infu1(·),,um(·),infiIxi(θ)-y(θ), where ui(·) are admissible controls of the pursuers Pi, and V is an admissible strategy of the evader E.

If Γ1(U10,,Um0,)=Γ2(V0)=γ, then we say that the game has the value γ .

It is to find optimal strategies Ui0 and V0 of the players Pi and E, respectively, and the value of the game. Instead of differential game described by (2.1) we can consider an equivalent differential game with the same payoff function and described by the following system: Pi:ẋi(t)=(θ-t)ui(t),xi(0)=xi0=xi1θ+xi0,E:ẏ(t)=(θ-t)v(t),y(0)=y0=y1θ+y0. Indeed, if the pursuer Pi uses an admissible control ui(t)=(ui1(t),ui2(t),),  0tθ, then according to (2.1) we have xi(θ)=xi0+xi1θ+0θ0tui(s)dsdt=xi0+xi1θ+0θ(θ-t)ui(t)dt, and the same result can be obtained by (2.9) xi(θ)=xi0+0θ(θ-t)ui(t)dt=xi0+xi1θ+0θ(θ-t)ui(t)dt. Also, for the evader the same argument can be made, therefore in the distancexi(θ)-y(θ) we can take either the solution of (2.1) or the solution of (2.9).

The attainability domain of the pursuer Pi at time θ from the initial state xi0 at time t0=0 is the closed ball H(xi0,ρi(θ3/3)1/2). Indeed, by Cauchy-Schwartz inequality xi(θ)-xi0=0θ(θ-s)ui(s)ds0θ(θ-s)ui(s)ds(0θ(θ-s)2ds)1/2·(0θui(s)2ds)1/2ρi(θ33)1/2.

On the other hand, if x̅H(xi0,ρi(θ3/3)1/2), that is, x̅-xi0ρi(θ3/3)1/2, then for the pursuer's control ui(t)=3(θ-t)θ3(x̅-xi0),0tθ, we obtain xi(θ)=x̅.

The pursuer's control is admissible because 0θui(t)2dt=(3θ3)2θ33x̅-xi02(ρi(θ33)1/2)2·3θ3=ρi2. Likewise, the attainability domain of the evader E at time θ from the initial state y0 at time t0=0 is the closed ball H(y0,σ(θ3/3)1/2).

3. An Auxiliary Game

In this section we fix the index i and study an auxiliary differential game of two players Pi and E, also for simplicity we drop the index i and use the notion ρi=ρ,xi0=x0 and xi=x. Let X={zl2:2(y0-x0,z)θ33(ρ2-σ2)+y02-x02},ρσ, if x0y0; if x0=y0, then X={zl2:(p,z-y0)ρ(θ33)1/2}, where p is an arbitrary fixed unit vector.

Consider the one-pursuer game described by the equations P:ẋ=(θ-t)u(t),x(0)=x0,E:ẏ=(θ-t)v(t),y(0)=y0, with the state of the evader E being subject to y(θ)X. The goal of the pursuer P is to realize the equality x(τ)=y(τ) at some τ,  0τθ, and that of the evader E is opposite.

We define the pursuer's strategy as follows: if x0=y0, then we set u(t)=v(t),0tθ, and if x0y0, then we set u(t)=v(t)-(v(t),e)e+e(3θ3(θ-t)2(ρ2-σ2)+(v(t),e)2)1/2,0tτ, where e=(y0-x0)/y0-x0, and u(t)=v(t),τ<tθ, where τ,  0τθ, is the time instant at which x(τ)=y(τ) for the first time.

Lemma 3.1.

If σρ and y(θ)X, then the pursuer's strategy (3.4), (3.5), and (3.6) in the game (3.3) ensures that x(θ)=y(θ).

Proof.

If x0=y0, then from (3.4) we have x(t)=y(t),  0tθ, because x(t)=x0+0t(θ-s)u(s)ds=y0+0t(θ-s)v(s)ds=y(t). In particular, x(θ)=y(θ).

Let x0y0. By (3.5) and (3.6), we have y(t)-x(t)=ef(t), where f(t)=y0-x0+0t(θ-s)(v(s),e)ds-0t(θ-s)(3θ3(θ-s)2(ρ2-σ2)+(v(s),e)2)1/2ds. Obviously f(0)=y0-x0>0. Now we show that f(θ)0. This will imply that f(τ)=0 for some τ[0,θ].

To this end we consider the following two-dimensional vector function: g(t)=((3θ3)1/2(θ-t)2(ρ2-σ2)1/2,(θ-t)(v(t),e)),0tθ. For the last integral of (3.8) we have 0θ(θ-s)(3θ3(θ-s)2(ρ2-σ2)+(v(s),e)2)1/2ds=0θ(3θ3(θ-s)4(ρ2-σ2)+(θ-s)2(v(s),e)2)1/2ds=0θ|g(s)|ds|  0θg(s)ds|=|(0θ(3θ3)1/2(θ-s)2(ρ2-σ2)1/2ds,0θ(θ-s)(v(s),e)ds)|=|((θ33)1/2(ρ2-σ2)1/2,0θ(θ-s)(v(s),e)ds)|=(θ33(ρ2-σ2)+(0θ(θ-s)(v(s),e)ds)2)1/2. Then   f(θ)y0-x0+0θ(θ-s)(v(s),e)ds-(θ33(ρ2-σ2)+(0θ(θ-s)(v(s),e)ds)2)1/2  .

By assumption, y(θ)X, therefore 2(y0-x0,y(θ))θ33(ρ2-σ2)+y02-x02, so (e,y(θ))d, where d=(θ3/3)(ρ2-σ2)+y02-x022y0-x0. As (e,y(θ))=(e,y0+0θ(θ-s)v(s)ds)d, then we obtain 0θ(θ-s)(v(s),e)dsd-(y0,e). On the other hand, ψ(t)=y0-x0+t-((θ3/3)(ρ2-σ2)+t2)1/2 is an increasing function on (-,). Then it follows from (3.11) and (3.14) that   f(θ)y0-x0+d-(y0,e)-(θ33(ρ2-σ2)+(d-(y0,e))2)1/2. Now we show that the right-hand side of the last inequality is equal to zero. We show y0-x0+d-(y0,e)=(θ33(ρ2-σ2)+(d-(y0,e))2)1/2. The left part of this equality is positive, since y0-x0+d-(y0,e)=(θ3/3)(ρ2-σ2)+y02-x02+2(x02-2(x0,y0)+y02)-2y02+2(x0,y0)2y0-x0=(θ3/3)(ρ2-σ2)+x02-2(x0,y0)+y022y0-x0=(θ3/3)(ρ2-σ2)+x0-y022y0-x0>0. Therefore taking square we have y0-x02+(d-(y0,e))2+2y0-x0(d-(y0,e))=θ33(ρ2-σ2)+(d-(y0,e))2, then y0-x02+2y0-x0((θ3/3)(ρ2-σ2)+y02-x022y0-x0-(y0,e))=θ33(ρ2-σ2). The above equality is true since y0-x02+y02-x02-2y0-x0y0=0. So f(θ)=0, consequently f(τ)=0 for some τ,  0τθ. Therefore, x(τ)=y(τ). Further, by (3.6), u(t)=v(t) at τ<tθ. Then x(θ)=x(τ)+τθ(θ-s)u(s)ds=y(τ)+τθ(θ-s)v(s)ds=y(θ), and the proof of the lemma is complete.

4. Main Result

Now consider the game (2.9). We will solve the optimal pursuit problem under the following assumption.

Assumption 4.1.

There exists a nonzero vector p0 such that (y0-xi0,p0)0 for all iI.

Let

γ=inf{l0:H(y0,σ(θ33)1/2)i=1H(xi0,ρi(θ33)1/2+l)}.

Theorem 4.2.

If Assumption 4.1 is true and σρi+γ(3/θ3)1/2 for all iI, then the number γ given by (4.1) is the value of the game (2.9).

Proof of the above theorem relies on the following lemmas.

Consider the sphere S(y0,r) and finitely or countably many balls H(xi0,Ri) and H(y0,r), where xi0y0 and r and Ri,  iI are positive numbers.

Lemma 4.3 (see [<xref ref-type="bibr" rid="B4">10</xref>]).

Let Xi={zl2:2(y0-xi0,z)Ri2-r2+y02-xi02}, if xi0y0, and Xi={zl2:(z-y0,p0)Ri}, if xi0=y0. If Assumption 4.1 is valid and H(y0,r)iIH(xi0,Ri), then H(y0,r)iIXi.

Lemma 4.4 (see [<xref ref-type="bibr" rid="B4">10</xref>]).

Let infiIRi=R0>0. If Assumption 4.1 is true and for any 0<ε<R0 the set iIH(xi0,Ri-ε) does not contain the ball H(y0,r), then there exists a point y̅S(y0,r) such that y̅-xi0Ri for all iI.

Proof of Theorem <xref ref-type="statement" rid="thm4.1">4.2</xref>.

We prove this theorem in three parts.

(1) Construction of the Pursuers' Strategies. We introduce counterfeit pursuers zi, whose motions are described by the equations żi=(θ-t)wiε,zi(0)=xi0,(0θwiε(s)2ds)1/2ρ̅i(ε)=ρi+γ(3θ3)1/2+εki(3θ3)1/2, where ki=max{1,ρi} and ε,  0<ε<1, is an arbitrary positive number. It is obvious that the attainability domain of the counterfeit pursuer zi at time θ from an initial state xi0 is the ball H(xi0,ρ̅i(ε)(θ3/3)1/2)=H(xi0,ρi(θ3/3)1/2+γ+ε/ki).

The strategies of the counterfeit pursuers zi are defined as follows: if xi0=y0, then we set wiε(t)=v(t),0tθ, and if xi0y0, then we set wiε(t)=v(t)-(v(t),ei)ei+ei(3θ3(θ-t)2(ρ̅i2(ε)-σ2)+(v(t),ei)2)1/2,0tτi, where ei=(y0-xi0)/y0-xi0, and wiε(t)=v(t),τi<tθ, where τi,  0τiθ, is the time instant at which zi(τi)=y(τi) for the first time if it exists. Note that τi need not to exist in [0,θ].

Now let us show that the strategies (4.6), (4.7), and (4.8) are admissible. If xi0=y0 and 0tθ, then 0θwiε(s)2ds=0θv(s)2dsσ2(ρi+γ(3θ3)1/2)2(ρi+γ(3θ3)1/2+εki(3θ3)1/2)2=ρ̅i2(ε). If xi0y0 we have 0θwiε(s)2ds=0τiwiε(s)2ds+τiθwiε(s)2ds=0τiv(s)2ds+3θ3(ρ̅i2(ε)-σ2)0τi(θ-s)2ds+τiθv(s)2ds0θv(s)2ds+3θ3(ρ̅i2(ε)-σ2)0θ(θ-s)2dsσ2+ρ̅i2(ε)-σ2=ρ̅i2(ε).

The strategies of the pursuers xi are defined as follows: ui(t)=ρiρ̅iwi(t),0tθ, where ρ̅i=ρ̅i(0)=ρi+γ(3/θ3)1/2 and wi(t)=wi0(t); that is, wi(t) is given by (4.6), (4.7), and (4.8) with ε=0 and the same τi.

(2) The value   γ  is guaranteed for the pursuers. Let us show that the above-constructed strategies of the pursuers satisfy the inequalities supv(·)infiIy(θ)-xi(θ)γ.

By the definition of γ, we have   H(y0,σ(θ33)1/2)i=1H(xi0,ρi(θ33)1/2+γ+εki). By Assumption 4.1 the inequality (y0-xi0,p0)0 holds for all iI. Then it follows from Lemma 4.3 that H(y0,σ(θ33)1/2)i=1Xiε, where Xiε={z:2(y0-xi0,z)(ρi(θ33)1/2+γ+εki)2-  σ2θ33+y02-xi02}, if xi0y0, and Xiε={z:(z-y0,p0)ρi(θ33)1/2+γ+εki}, if xi0=y0. Consequently, the point y(θ)H(y0,σ(θ3/3)1/2) belongs to some half-space Xsε,  s=s(ε)I.

By the assumption of the theorem, ρ̅i(ε)>σ; then it follows from Lemma 3.1 that if zi uses the strategy (4.6), (4.7), and (4.8), then zs(θ)=y(θ). By taking account of (4.11) we obtain y(θ)-xs(θ)=zs(θ)-xs(θ)=  0θ(θ-t)(wsε(t)-ρsρ̅sws(t))dt0θ(θ-t)(wsε(t)-ws(t))dt+0θ(θ-t)(ws(t)-ρsρ̅sws(t))dt.   Now we put aside the right-hand side of the last inequality. Let us show that limε0supiI0θ(θ-t)(wiε(t)-wi(t))dt=0.

Indeed, if xi0=y0, then by (4.6), wiε(t)=wi(t)=v(t), and the validity of (4.18) is obvious. Now let xi0y0. If there exists τi[0,θ], mentioned in (4.7) and (4.8), then 0τiwiε(t)-wi(t)2dt=0τi((3θ3(θ-t)2(ρ̅i2(ε)-σ2)+(v(t),ei)2)1/2-(3θ3(θ-t)2(ρ̅i2-σ2)+(v(t),ei)2)1/2)2dt0τi((3θ3(θ-t)2(ρ̅i2(ε)-σ2))1/2-(3θ3(θ-t)2(ρ̅i2-σ2))1/2)2dt0θ((3θ3(θ-t)2)1/2((ρ̅i2(ε)-σ2)1/2-(ρ̅i2-σ2)1/2))2dt=((ρ̅i2(ε)-σ2)1/2-(ρ̅i2-σ2)1/2)2=((2ρ̅iεki(3θ3)1/2+3θ3(εki)2+ρ̅i2-σ2)1/2-(ρ̅i2-σ2)1/2)22ρ̅iεki(3θ3)1/2+3θ3(εki)2(2(3θ3)1/2+6γθ3+3θ3)ε. In the last inequality, we have used the facts that 0<ε<1, ki1 and the inequality ρ̅iki=ρiki+γ(3/θ3)1/2ki1+γ(3θ3)1/2. So 0θ(θ-t)(wiε(t)-wi(t))dt=0τi(θ-t)(wiε(t)-wi(t))dt(0τi(θ-t)2  dt)1/2(0τiwiε(t)-wi(t)2dt)1/2Kε, where K is some positive number.

For the second integral in (4.17) we have 0θ(θ-t)(1-ρsρ̅s)ws(t)dt=(1-ρsρ̅s)0θ(θ-t)ws(t)dt(1-ρsρ̅s)(0θ(θ-t)2dt)1/2(0θws(t)2dt)1/2=(1-ρsρ̅s)(θ33)1/2ρ̅s=γ. Then it follows from (4.17) that y(θ)-xs(θ)γ+Kε.

Thus if the pursuers use the strategies (4.11), the inequality (4.12) is true.

(3) The value   γ   is guaranteed for the evader. Let us construct the evader's strategy ensuring that infu1(·),,um(·),  infiIy(θ)-xi(θ)γ, where u1(·),,um(·), are arbitrary admissible controls of the pursuers. If γ=0, then inequality (4.23) is obviously valid for any admissible control of the evader. Let γ>0. By the definition of γ, for any ε>0, the set i=1H(xi0,ρi(θ33)1/2+γ-ε), does not contain the ball H(y0,σ(θ3/3)1/2). Then, by Lemma 4.4 there exists a point y̅S(y0,σ(θ3/3)1/2), that is, y̅-y0=σ(θ3/3)1/2 such that y̅-xi0ρi(θ3/3)1/2+γ. On the other hand xi(θ)-xi0(θ33)1/2(0θui(t)2dt)1/2=ρi(θ33)1/2. Consequently y̅-xi(θ)y̅-xi0-xi(θ)-xi0ρi(θ33)1/2+γ-ρi(θ33)1/2=γ. Now by using the control v(t)=σ(3θ3)1/2(θ-t)e,0tθ,e=y̅-y0y̅-y0, we obtain y(θ)=y0+0θ(θ-s)v(s)ds=y0+0θ(θ-s)2σ(3θ3)1/2eds=y0+σ(θ33)1/2e=y̅. Then the value of the game is not less than γ, and inequality (4.23) holds. The proof of the theorem is complete.

5. Conclusion

A pursuit-evasion differential game of fixed duration with countably many pursuers has been studied. Control functions satisfy integral constraints. Under certain conditions, the value of the game has been found, and the optimal strategies of players have been constructed.

The proof of the main result relies on the solution of an auxiliary differential game problem in the half-space. Such method was used by many authors (see, e.g., [5, 6]), but the method used here for this auxiliary problem is different from those of others and requires only basic knowledge of calculus.

It should be noted that the condition given by Assumption 4.1 is relevant. If this condition does not hold, then, in general, we do not have a solution of the pursuit-evasion problem even in a finite dimensional space with a finite number of pursuers.

The present work can be extended by considering higher-order differential equations instead of (2.1). Then differential game can be reduced to an equivalent game, described by (2.1), with θ-t replaced by another function.

Acknowledgments

The authors would like to thank the referee for giving useful comments and suggestions for the improvement of this paper. The present research was supported by the National Fundamental Research Grant Scheme (FRGS) of Malaysia, no. 05-10-07-376FR.

IsaacsR.Differential Games1965New York, NY, USAJohn Wiley & SonsMR0210469PontryaginL. S.Izbrannye Trudy (Selected Works)1988Moscow, RussiaFriedmanA.Differential Games1971New York, NY, USAJohn Wiley & SonsMR0421700KrasovskiiN. N.SubbotinA. I.Game-Theoretical Control Problems1988New York, NY, USASpringerMR918771IvanovR. P.LedyaevYu. S.Optimality of pursuit time in a simple motion differential game of many objectsTrudy Matematicheskogo Instituta imeni V. A. Steklova19811588797MR662836ZBL0509.90098LevchenkovA. Y.PashkovA. G.Differential game of optimal approach of two inertial pursuers to a noninertial evaderJournal of Optimization Theory and Applications1990653501517MR105283010.1007/BF00939563ZBL0676.90108SubbotinA. I.ChentsovA. G.Optimization of Guaranteed Result in Control Problems1981Moscow, RussiaChodunW.Differential games of evasion with many pursuersJournal of Mathematical Analysis and Applications19891422370389MR101458210.1016/0022-247X(89)90007-3ZBL0679.90110IbragimovG. I.A problem of optimal pursuit in systems with distributed parametersJournal of Applied Mathematics and Mechanics2002665719724MR196461110.1016/S0021-8928(02)90002-XIbragimovG. I.Optimal pursuit with countably many pursuers and one evaderDifferential Equations2005415627635MR220067210.1007/s10625-005-0198-y