# A Multiplayer Pursuit Differential Game on a Closed Convex Set with Integral Constraints

Authors: Gafurjan Ibragimov; Nu'man Satimov
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study a simple motion pursuit differential game of many pursuers and many evaders on a nonempty convex subset ofℝn. In process of the game, all players must not leave the given set. Control functions of players are subjected to integral constraints. Pursuit is said to be completed if the position of each evader yj, j∈{1,2,...k}, coincides with the position of a pursuer xi, i∈{1,...,m}, at some time tj, that is, xi(tj)=yj(tj). We show that if the total resource of the pursuers is greater than that of the evaders, then pursuit can be completed. Moreover, we construct strategies for the pursuers. According to these strategies, we define a finite number of time intervals [θi−1,θi] and on each interval only one of the pursuers pursues an evader, and other pursuers do not move. We derive inequalities for the resources of these pursuer and evader and, moreover, show that the total resource of the pursuers remains greater than that of the evaders.


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## Body

## 1. Related Work

Linear differential games with integral constraints on controls were examined in many works, for example, [1–16].Satimov et al. [10] studied a linear pursuit differential game of many pursuers and one evader with integral constraints on controls of players in ℝn. Game is described by the following equations:
(1.1)z˙i=Cizi+ui-v,zi(t0)=zi0,i=1,…,m,
where ui is the control parameter of the ith pursuer and v is that of the evader. The eigenvalues of the matrices Ci are assumed to be real numbers. They proved that if the total resource of controls of the pursuers is greater than that of the evader,  then under certain conditions pursuit can be completed.Ibragimov [15] examined a pursuit differential game of m pursuers and k evaders with integral constraints described by the following systems of differential equations
(1.2)z˙ij=Cijzij+ui-vj,zij(t0)=zij0,i=1,…,m,j=1,…,k,∫0∞|ui(s)|2ds≤ρi2,i=1,…,m,∫0∞|vj(s)|2ds≤σj2,j=1,…,k,
where ui is the control parameter of the ith pursuer and vj is that of the jth evader. Different from the previous work,  here eigenvalues of matrices Cij are not necessarily real, and, moreover, the number of evaders can be any. If the total resource of controls of the pursuers is greater than that of the evaders, that is,
(1.3)ρ12+ρ22+⋯+ρm2>σ12+σ22+⋯+σk2,
and real parts of all eigenvalues of the matrices Cij are nonpositive, then it was proved that pursuit can be completed from any initial position. Here, game was considered in ℝn without any state constraint.Ivanov [17] considered generalized Lion and Man problem in the case of geometric constraints. All players have equal dynamic possibilities. Motions of the players are described by the following equations:
(1.4)x˙i=ui,xi(0)=xi0,|ui|≤1,i=0,1,…,m,
where xi∈ℝn,ui,i=1,…,m,  are control parameters of the pursuers and u0 is control parameter of the evader. During the game, all players may not leave a given compact subset N of ℝn. It was shown that if the number of pursuers m does not exceed the dimension of the space n, then evasion is possible; otherwise pursuit can be completed. In other words, Ivanov [17] derived necessary and sufficient condition of evasion for the multiple Lion and Man game in ℝn.Ibragimov [16] studied a differential game problem of one pursuer and one evader with integral constraints. Game occurs on a closed convex subset S of ℝn, and movements of the players are described by the following equations:
(1.5)x˙=α(t)u,x(0)=x0,∫0∞|u(s)|2ds≤ρ2,y˙=α(t)v,y(0)=y0,∫0∞|v(s)|2ds≤σ2.
Evasion and pursuit problems were investigated. In the latter case, a formula for optimal pursuit time was found.Leong and Ibragimov [14] studied simple motion pursuit differential game of m pursuers and one evader on a closed convex subset of the Hilbert space l2. Control functions of the players are subjected to integral constraints. The total resource of the pursuers is assumed to be greater than that of the evader. Strategies of pursuers were constructed to complete the pursuit from any initial position.In the present paper, we study a pursuit differential game of many pursuers and many evaders on a nonempty convex subsetN of ℝn,n≥2. In process of the game, all players must not leave the set N. Control functions of the players are subjected to integral constraints. We will show that if the total resource of the pursuers is greater than that of the evaders, then pursuit can be completed. In Table 1, we can compare the cases studied in the works of the previous researches and the present paper.Table 1
Brief summary of obtained results.
Author(s)
Number of pursuers
Number of evaders
The set where the pursuit occurs
Constraint on controls
Satimov et al. (1983) [10]
m
1
ℝ
n
Integral
Ibragimov (2004) [15]
m
k
ℝ
n
Integral
Ivanov (1980) [17]
m
1
Closed convex subsetN of ℝn
Geometric
Ibragimov (2002) [16]
1
1
Closed convex subsetN of ℝn
Integral
Leong and  Ibragimov (2008) [14]
m
1
Closed convex subsetN of l2
Integral
Present paper
m
k
Closed convex subsetN of ℝn
Integral


## 2. Statement of the Problem

We consider a differential game described by the following equations:(2.1)x˙i=φ(t)ui,xi(0)=xi0,i=1,…,m,y˙j=φ(t)vi,yj(0)=yj0,j=1,…,k,
where xi,ui,yj,vj∈ℝn,ui is control parameter of the pursuer xi, i=1,…,m,vj is that of the evader yj, j=1,…,k, and φ(t) is a scalar measurable function that satisfies the following conditions:
(2.2)a(τ)≜(∫0τφ2(t)dt)1/2<∞,τ>0,limτ→∞a(τ)=∞.Definition 2.1.
A measurable functionui(t)=(ui1(t),…,uin(t)), t≥0, is called admissible control of the pursuer xi if
(2.3)∫0∞|ui(s)|2ds≤ρi2,
where ρi,i=1,…,m, are given positive numbers.  We denote the set of all admissible controls of the pursuer xi by  S(ρi).Definition 2.2.
A measurable functionvj(t)=(vj1(t),…,vjk(t)),t≥0, is called an admissible control of the evader if
(2.4)∫0∞|vj(s)|2ds≤σj2,
where σj,j=1,…,k, are given positive numbers.Definition 2.3.
A Borel measurable functionUi(xi,y1,…,yk,v1,…,vk), Ui:ℝ(2k+1)n→ℝn, is called a strategy of the pursuer xi if for any control of the evader v(t),t≥0, the initial value problem
(2.5)x˙i=φ(t)Ui(xi,y1,…,yk,v1(t),…,vk(t)),xi(0)=xi0,y˙j=φ(t)vi(t),yj(0)=yj0,j=1,…,k,
has a unique solution (xi(t),y1(t),…,yk(t)) and the inequality
(2.6)∫0∞|Ui(xi(s),y1(s),…,yk(s),v1(s),…,vk(s))|2ds≤ρi2
holds.Definition 2.4.
We say that pursuit can be completed from the initial position{x10,…,xm0,y10,…,yk0} for the time T in the game (2.1)–(2.4), if there exist strategies Ui,i=1,…,m, of the pursuers such that for any controls v1=v1(·),…,vk=vk(·) of the evaders and numbers j=1,2,…k, the equality xi(tj)=yj(tj) holds for some i∈{1,…,k} at some time tj∈[0,T].
Given nonempty convex subsetN of ℝn, according to the rule of the game all players must not leave the set N, that is, xi0,xi(t),yj0,yj(t)∈N, t≥0, i=1,…,m, j=1,…,k. This information describes a differential game of many players with integral constraints on control functions of players.Problem 1.
Find a sufficient condition of completing pursuit in the game (2.1)–(2.4).


## 3. The Main Result

Since the control parameters of the players can take the values of opposite signs, without loss of generality, we may assume thatφ(t)≥0. Moreover, without loss of generality, we may assume that φ(t) is not identically zero on every open interval. Otherwise, if for instance φ(t)=0,t∈(t1,t2), then instead of φ(t) we consider
(3.1)φ1(t)={φ(t),0≤t≤t1,φ(t+t2-t1),t>t1.We now formulate the main result of the paper.Theorem 3.1.
If(3.2)ρ12+ρ22+⋯+ρm2>σ12+σ22+⋯+σk2,
then pursuit can be completed for a finite time T in the game (2.1)–(2.4) from any initial position.Proof.
(10) An auxiliary differential game: To prove this theorem, we first study an auxiliary differential game of one pursuer x and one evader y, which is described by the following equations:
(3.3)x˙=φ(t)u,x(0)=x0,y˙=φ(t)v,y(0)=y0,
where x,y∈ℝn,u is control parameter of the pursuer x, and v is that of the evader y. Assume that u=u(·)∈S(ρ),v=v(·)∈S(σ). Pursuit is completed if x(t′)=y(t′) at some time t′≥0. Here the players move in ℝn without any state constraint.
Set(3.4)u(t)=φ(t)a2(θ)(y0-x0)+v(t),t≥0,
where θ is an arbitrary fixed number satisfying
(3.5)a2(θ)≥|y0-x0|2(ρ-σ)2.
Let
(3.6)ρ2(t)=ρ2-∫ot|u(s)|2ds,σ2(t)=σ2-∫ot|v(s)|2ds,K(θ,x0,y0)=1a(θ)|y0-x0|2+2σ|y0-x0|.
We will now prove the following statement.Lemma 3.2.
Let the pursuer use the strategy (3.4). (i) If ρ>σ then pursuit can be completed in the game (3.3) for the time θ, and, moreover,
(3.7)ρ2(θ)≥ρ2-σ2-K(θ,x0,y0)a(θ).
(ii) Ifρ≤σ, then either x(θ)=y(θ) or
(3.8)σ2(θ)≤σ2-ρ2+K(θ,x0,y0)a(θ).Proof of the Lemma.
Letρ>σ. In this case, we show that the control (3.4) is admissible and ensures the equality x(θ)=y(θ). Indeed, clearly
(3.9)x(θ)=x0+∫0θφ(t)(φ(t)a2(θ)(y0-x0)+v(t))dt=y0+∫0θφ(t)v(t)dt=y(θ).
To show the admissibility of the strategy (3.4), we use the Cauchy-Schwartz inequality as follows:
(3.10)(y0-x0)∫0θφ(t)v(t)dt≤|y0-x0|∫0θφ(t)|v(t)|dt≤|y0-x0|a(θ)σ.
Then by this inequality we have(3.11)∫0θ|u(t)|2dt=1a2(θ)|y0-x0|2+21a2(θ)(y0-x0)∫0θφ(t)v(t)dt+∫0θ|v(t)|2dt≤1a2(θ)|y0-x0|2+2a(θ)|y0-x0|σ+∫0θ|v(t)|2dt=K(θ,x0,y0)a(θ)+∫0θ|v(t)|2dt.
According to (3.5) and the condition v(·)∈S(σ), the right-hand side of (3.11) is less than or equal to
(3.12)(σ-ρ)2+2(ρ-σ)σ+σ2=ρ2.
At this point, we have proved admissibility of the control (3.4).
In particular, using (3.11) we obtain
(3.13)ρ2(θ)=ρ2-∫0θ|u(t)|2dt≥ρ2-σ2-K(θ,x0,y0)a(θ).
Thus, (3.7) holds. We now turn to the part (ii) of the Lemma 3.2. Let ρ≤σ and the pursuer use the strategy (3.4) on the interval [0,θ]. If for a control of the evader v(t),t∈[0,θ], the following inequality is satisfied:
(3.14)∫0θ|u(t)|2dt=∫0θ|φ(t)a2(θ)(y0-x0)+v(t)|2dt≤ρ2,
then, clearly, the control (3.4) is admissible, and similar to (3.9) we obtain that x(θ)=y(θ) and the proof of the lemma follows. Hence, if x(θ)≠y(θ), then for the control (3.4) we must have
(3.15)∫0θ|φ(t)a2(θ)(y0-x0)+v(t)|2dt>ρ2.
From this inequality by using calculations similar to (3.11), we then obtain
(3.16)∫0θ|v(t)|2dt>ρ2-K(θ,x0,y0)a(θ),
and hence
(3.17)σ2(θ)=σ2-∫0θ|v(t)|2dt<σ2-ρ2+K(θ,x0,y0)a(θ).
This completes the proof of the lemma.The last inequality can be interpreted as follows. Thoughx(θ)≠y(θ), the pursuer can force to expend the evader’s energy more than ρ2-(K(θ,x0,y0)/a(θ)). At the same time, using the strategy (3.4), the pursuer will spend all his energy by a time τ(3.18)∫0τ|u(t)|2dt=ρ2,0<τ<θ.Then, of course, the pursuer cannot move anymore and automaticallyu(t)≡0,t≥τ.(20) Fictitious pursuers (FP): We now prove the theorem. To this purpose, introduce fictitious pursuers z1,…,zm by equations
(3.19)z˙i=φ(t)wi,zi(0)=xi0,wi∈S(ρi),i=1,…,m,
where wi is control parameter of the pursuer zi. FPs may go out of the set N. They move without any state constraint in ℝn. The aim of the FPs is to complete the pursuit as earlier as possible. Set
(3.20)wm(t)=φ(t)a2(θ1)(yk0-xm0)+vk(t),0≤t≤θ1,(3.21)wi(t)≡0,i=1,…,m-1,0≤t≤θ1,
where θ1 is any number satisfying inequalities
(3.22)a2(θ1)≥|yk0-xm0|2(ρm-σk)2,(3.23)ρ12+ρ22+⋯+ρm2>σ12+σ22+⋯+σk2+K(θ1,xm0,yk0)a(θ1).Since by the assumptiona(t)→∞ as t→∞, it follows from (3.2) that such θ1 exists. Equations (3.20), (3.21) mean that all FPs z1,…,zm-1 do not move on the time interval [0,θ1], and only one FP zm moves according to (3.20).We will now show that if (3.2) holds and the FPs use the strategies (3.20), (3.21), then the pursuit problem with the pursuers z1,…,zm and evaders y1,…,yk is reduced to a pursuit problem with the pursuers z1,…,zp and evaders y1,…,yq, for which ρ12(θ1)+⋯+ρp2(θ1)>σ12(θ1)+⋯+σq2(θ1) and p+q<m+k. Hence, by the time θ1, the number of pursuers is reduced to p+q.Indeed, we consider two possible cases: (i)ρm≤σk; (ii) ρm>σk. In the former case, that is, ρm≤σk, if the equality zm(θ1)=yk(θ1) holds, then we eliminate the pursuer zm and the evader yk and then consider the pursuit problem with pursuers z1,…,zm-1 and evaders y1,…,yk-1 under the condition
(3.24)ρ12+⋯+ρm-12>σ12+⋯+σk-12(p=m-1,q=k-1).
In case of zm(θ1)≠yk(θ1), according to (3.8), we obtain that
(3.25)σk2(θ1)≤σk2-ρm2+K(θ1,xm0,yk0)a(θ1).Then in view of (3.23) we obtain
(3.26)ρ12+⋯+ρm-12>σ12+⋯+σk2(θ1)(p=m-1,q=k),
and at the time θ1 we consider the pursuit problem with the pursuers z1,…,zm-1 and evaders y1,…,yk.We now turn to the case (ii), that is,ρm>σk. Then the pursuer zm certainly ensures the equality zm(θ1)=yk(θ1) and according to Lemma 3.2 (see, (3.7))
(3.27)ρm2(θ1)≥ρm2-σk2-K(θ1,xm0,yk0)a(θ1).Then with the aid of (3.23) we obtain
(3.28)ρ12+⋯+ρm-12+ρm2(θ1)>σ12+⋯+σk-12(p=m,q=k-1),
and therefore at the time θ1 we arrive at the pursuit problem with the pursuers z1,…,zm and evaders y1,…,yk-1.Let nowθ2 be an arbitrary fixed number satisfying inequalities
(3.29)θ2>θ1,a2(θ2-θ1)≥|yq(θ1)-zp(θ1)|2(ρp(θ1)-σq(θ1))2,ρ12(θ1)+⋯+ρp2(θ1)>σ12(θ1)+⋯+σq2(θ1)+K(θ2-θ1,yq(θ1),zq(θ1))a(θ2-θ1),
where p and q are the numbers of pursuers and evaders, respectively, which take part in the pursuit problem at time θ1. Set
(3.30)wp(t)=φ(t)a2(θ2-θ1)(yq(θ1)-zp(θ1))+vq(t),θ1<t≤θ2,wi(t)≡0,θ1<t≤θ2,i∈{1,…,m}∖{p}.Observe that according to (3.30) all pursuers except for zp will not move on the time interval (θ1,θ2].Let the pursuers use the strategies (3.30). Applying the same arguments above, we arrive at the following conclusion.(1)
Ifρp(θ1)≤σq(θ1) and zp(θ2)=yq(θ2), then starting from the time θ2 we consider a pursuit problem with the pursuers z1,…,zp-1 and evaders y1,…,yq-1 under the condition
(3.31)ρ12(θ2)+⋯+ρp-12(θ2)>σ12(θ2)+⋯+σq-12(θ2).(2)
Ifρp(θ1)≤σq(θ1) and zp(θ2)≠yq(θ2), then starting from the time θ2 we consider the pursuit problem with the pursuers z1,…,zp-1 and evaders y1,…,yq under the condition
(3.32)ρ12(θ2)+⋯+ρp-12(θ2)>σ12(θ2)+⋯+σq2(θ2).(3)
Ifρp(θ1)>σq(θ1), then by Lemma 3.2 the equality zp(θ2)=yq(θ2) holds. In this case, starting from the time θ2 the pursuit problem with the pursuers z1,…,zp and evaders y1,…,yq-1 is considered under the condition
(3.33)ρ12(θ2)+⋯+ρp2(θ2)>σ12(θ2)+⋯+ρq-12(θ2).Repeated application of this procedure enables us to complete the pursuit for some finite timeT since the number of players is decreasing. Thus, we have proved that FPs can complete the pursuit.(20) Completion of the proof of the Theorem: We will now show that the actual pursuers also can complete the pursuit. Define the controls u1,…,um of the pursuers x1,…,xm by the controls of the FPs w1,…,wm. We denote by FN(x) the projection of the point x∈ℝn on the set N, that is,
(3.34)miny∈N|x-y|=|x-FN(x)|.Note thatFN(x)=x if x∈N. It is familiar that for any point x∈ℝn there exists a unique point FN(x). Moreover, for any x,y∈ℝn,
(3.35)|FN(x)-FN(y)|≤|x-y|,
and hence the operator FN(x) relates any absolute continuous function z(t),0≤t≤T, to an absolute continuous function
(3.36)x(t)=FN(z(t)),0≤t≤T,
where T is the time in which pursuit can be completed by FPs. Define the control ui(t) to satisfy
(3.37)xi(t)=FN(zi(t)),0≤t≤T,i=1,…,m.We first show admissibility of such defined controlui(t). Indeed, from (3.35),
(3.38)φ(t)|ui(t)|=|x˙i(t)|=limh→0|xi(t+h)-x(t)||h|=limh→0|FN(z(t+h))-FN(z(t))||h|≤limh→0|zi(t+h)-zi(t)||h|=|z˙i(t)|=φ(t)|wi(t)|.Hence the inequality|ui(t)|≤|wi(t)| holds almost everywhere and therefore
(3.39)∫0T|ui(t)|2dt≤∫0T|wi(t)|2dt≤ρi2.Since for any evaderyi,i∈{1,…,k}, the equality zni(ti)=yi(ti) holds for some ti≤T and ni∈{1,…,m}, and the evader yi(t) is in N for any t≥0; in particular, yi(ti)∈N, then FP zni(ti) is also in N. Consequently,
(3.40)xni(ti)=FN(zni(ti))=zni(ti)=yi(ti).This means differential game (2.1)–(2.4) can be completed for the time T. The proof of the theorem is complete.We give an illustrative example.Example 3.3.
We consider a differential game described by the following equations:(3.41)x˙i=λxi+ui,xi(0)=xi0,i=1,…,m,λ≤0,y˙j=λyj+vj,yj(0)=yj0,j=1,…,k,
It is assumed thatxi0≠yj0 for all i=1,…,m,j=1,…,k. The control functions satisfy the integral constraints (2.3) and (2.4). Pursuit is said to be completed if for any numbers j=1,2,…k the equality xi(tj)=yj(tj) holds for some i∈{1,…,k} at some time tj≥0. Players may not leave the half space N={x∣(x,l)≥0,x∈ℝn}, where l∈ℝn is a nonzero vector. Since
(3.42)xi(t)=eλtx-i(t),yi(t)=eλty-i(t),
where
(3.43)x-i(t)=xi0+∫0te-λsui(s)ds,y-i(t)=yj0+∫0te-λsvj(s)ds,
then the equality xi(tj)=yj(tj) is equivalent to the one x-i(tj)=y-j(tj) and inclusions xi(t)∈N and yi(t)∈N are equivalent to ones x-i(t)∈N and y-i(t)∈N, respectively. Therefore, differential game described by (3.41) is equivalent to the game described by the following equations:
(3.44)x-i(tj)=e-λtui,x-i(0)=xi0,i=1,…,m,λ≤0,y-j(tj)=e-λtvj,y-j(0)=yj0,j=1,…,k.
Denotingφ(t)=e-λt, we obtain a differential game of the form (2.1).


## 4.  Conclusion

We have obtained a sufficient condition (3.2) to complete the pursuit in the differential game of m pursuers and k evaders with integral and state constraints. We have constructed strategies of the pursuers and showed that pursuit can be completed from any initial position in N. In the case of N=ℝn and k=1 (one evader), the condition (3.2) of the theorem takes the form ρ12+ρ22+⋯+ρm2>σ12. This condition is sharp since if ρ12+ρ22+⋯+ρm2≤σ12, then evasion is possible [13].



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