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

Abstract

We study a simple motion pursuit differential game of many pursuers and many evaders on a nonempty convex subset of ℝ^𝑛. 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 𝑦_𝑗, 𝑗∈{1,2,...𝑘}, coincides with the position of a pursuer 𝑥_𝑖, 𝑖∈{1,...,𝑚}, at some time 𝑡_𝑗, that is, 𝑥_𝑖(𝑡_𝑗)=𝑦_𝑗(𝑡_𝑗). 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 [𝜃_𝑖−1,𝜃_𝑖] 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.

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 ℝ^𝑛. Game is described by the following equations: 𝑧̇_𝑖=𝐶_𝑖𝑧_𝑖+𝑢_𝑖−𝑣,𝑧_𝑖𝑡₀=𝑧⁰_𝑖,𝑖=1,…,𝑚, (1.1) where 𝑢_𝑖 is the control parameter of the 𝑖th pursuer and 𝑣 is that of the evader. The eigenvalues of the matrices 𝐶_𝑖 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 𝑧̇_𝑖𝑗=𝐶_𝑖𝑗𝑧_𝑖𝑗+𝑢_𝑖−𝑣_𝑗,𝑧_𝑖𝑗𝑡₀=𝑧⁰_𝑖𝑗,𝑖=1,…,𝑚,𝑗=1,…,𝑘,_∞0𝑢_𝑖(𝑠)²𝑑𝑠≤𝜌_𝑖²,𝑖=1,…,𝑚,^∞₀𝑣_𝑗(𝑠)²𝑑𝑠≤𝜎_𝑗²,𝑗=1,…,𝑘, (1.2) where 𝑢_𝑖 is the control parameter of the 𝑖th pursuer and 𝑣_𝑗 is that of the 𝑗th evader. Different from the previous work, here eigenvalues of matrices 𝐶_𝑖𝑗 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, 𝜌²₁+𝜌²₂+⋯+𝜌²_𝑚>𝜎²₁+𝜎²₂+⋯+𝜎²_𝑘, (1.3) and real parts of all eigenvalues of the matrices 𝐶_𝑖𝑗 are nonpositive, then it was proved that pursuit can be completed from any initial position. Here, game was considered in ℝ^𝑛 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: 𝑥̇_𝑖=𝑢_𝑖,𝑥_𝑖(0)=𝑥_𝑖0,𝑢_𝑖≤1,𝑖=0,1,…,𝑚, (1.4) where 𝑥_𝑖∈ℝ^𝑛,𝑢_𝑖,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 ℝ^𝑛. 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 ℝ^𝑛.

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 ℝ^𝑛, and movements of the players are described by the following equations: 𝑥̇=𝛼(𝑡)𝑢,𝑥(0)=𝑥₀,^∞₀|𝑢(𝑠)|²𝑑𝑠≤𝜌²,𝑦̇=𝛼(𝑡)𝑣,𝑦(0)=𝑦₀,^∞₀|𝑣(𝑠)|²𝑑𝑠≤𝜎². (1.5) 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 subset 𝑁 of ℝ^𝑛,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.

2. Statement of the Problem

We consider a differential game described by the following equations: 𝑥̇_𝑖=𝜑(𝑡)𝑢_𝑖,𝑥_𝑖(0)=𝑥_𝑖0𝑦̇,𝑖=1,…,𝑚,_𝑗=𝜑(𝑡)𝑣_𝑖,𝑦_𝑗(0)=𝑦_𝑗0,𝑗=1,…,𝑘, (2.1) where 𝑥_𝑖,𝑢_𝑖,𝑦_𝑗,𝑣_𝑗∈ℝ^𝑛,𝑢_𝑖 is control parameter of the pursuer 𝑥_𝑖, 𝑖=1,…,𝑚,𝑣_𝑗 is that of the evader 𝑦_𝑗, 𝑗=1,…,𝑘, and 𝜑(𝑡) is a scalar measurable function that satisfies the following conditions: 𝑎(𝜏)≜^𝜏₀𝜑²(𝑡)𝑑𝑡^1/2<∞,𝜏>0,lim_𝜏→∞𝑎(𝜏)=∞. (2.2)

Definition 2.1. A measurable function 𝑢_𝑖(𝑡)=(𝑢_𝑖1(𝑡),…,𝑢_𝑖𝑛(𝑡)), 𝑡≥0, is called admissible control of the pursuer 𝑥_𝑖 if ^∞₀𝑢_𝑖(𝑠)²𝑑𝑠≤𝜌²_𝑖, (2.3) where 𝜌_𝑖,𝑖=1,…,𝑚, are given positive numbers. We denote the set of all admissible controls of the pursuer 𝑥_𝑖 by 𝑆(𝜌_𝑖).

Definition 2.2. A measurable function 𝑣_𝑗(𝑡)=(𝑣_𝑗1(𝑡),…,𝑣_𝑗𝑘(𝑡)),𝑡≥0, is called an admissible control of the evader if ^∞₀𝑣_𝑗(𝑠)²𝑑𝑠≤𝜎²_𝑗, (2.4) where 𝜎_𝑗,𝑗=1,…,𝑘, are given positive numbers.

Definition 2.3. A Borel measurable function 𝑈_𝑖(𝑥_𝑖,𝑦₁,…,𝑦_𝑘,𝑣₁,…,𝑣_𝑘), 𝑈_𝑖∶ℝ^{(2𝑘+1)𝑛}→ℝ^𝑛, is called a strategy of the pursuer 𝑥_𝑖 if for any control of the evader 𝑣(𝑡),𝑡≥0, the initial value problem 𝑥̇_𝑖=𝜑(𝑡)𝑈_𝑖𝑥_𝑖,𝑦₁,…,𝑦_𝑘,𝑣₁(𝑡),…,𝑣_𝑘(𝑡),𝑥_𝑖(0)=𝑥_𝑖0,𝑦̇_𝑗=𝜑(𝑡)𝑣_𝑖(𝑡),𝑦_𝑗(0)=𝑦_𝑗0,𝑗=1,…,𝑘, (2.5) has a unique solution (𝑥_𝑖(𝑡),𝑦₁(𝑡),…,𝑦_𝑘(𝑡)) and the inequality ^∞₀𝑈_𝑖𝑥_𝑖(𝑠),𝑦₁(𝑠),…,𝑦_𝑘(𝑠),𝑣₁(𝑠),…,𝑣_𝑘(𝑠)²𝑑𝑠≤𝜌²_𝑖 (2.6) holds.

Definition 2.4. We say that pursuit can be completed from the initial position {𝑥₁₀,…,𝑥_𝑚0,𝑦₁₀,…,𝑦_𝑘0} for the time 𝑇 in the game (2.1)–(2.4), if there exist strategies 𝑈_𝑖,𝑖=1,…,𝑚, of the pursuers such that for any controls 𝑣₁=𝑣₁(⋅),…,𝑣_𝑘=𝑣_𝑘(⋅) of the evaders and numbers 𝑗=1,2,…𝑘, the equality 𝑥_𝑖(𝑡_𝑗)=𝑦_𝑗(𝑡_𝑗) holds for some 𝑖∈{1,…,𝑘} at some time 𝑡_𝑗∈[0,𝑇].
Given nonempty convex subset 𝑁 of ℝ^𝑛, according to the rule of the game all players must not leave the set 𝑁, that is, 𝑥_𝑖0,𝑥_𝑖(𝑡),𝑦_𝑗0,𝑦_𝑗(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 that 𝜑(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 𝜑₁(𝑡)=𝜑(𝑡),0≤𝑡≤𝑡₁,𝜑𝑡+𝑡₂−𝑡₁,𝑡>𝑡₁. (3.1)

We now formulate the main result of the paper.

Theorem 3.1. If 𝜌²₁+𝜌²₂+⋯+𝜌²_𝑚>𝜎²₁+𝜎²₂+⋯+𝜎²_𝑘, (3.2) 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: 𝑥̇=𝜑(𝑡)𝑢,𝑥(0)=𝑥₀,𝑦̇=𝜑(𝑡)𝑣,𝑦(0)=𝑦₀, (3.3) where 𝑥,𝑦∈ℝ^𝑛,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 ℝ^𝑛 without any state constraint.
Set 𝑢(𝑡)=𝜑(𝑡)𝑎²(𝜃)𝑦₀−𝑥₀+𝑣(𝑡),𝑡≥0, (3.4) where 𝜃 is an arbitrary fixed number satisfying 𝑎²(𝜃)≥𝑦₀−𝑥₀²𝜌−𝜎². (3.5) Let 𝜌²(𝑡)=𝜌²−^𝑡_𝑜|𝑢(𝑠)|²𝑑𝑠,𝜎²(𝑡)=𝜎²−^𝑡_𝑜|𝑣(𝑠)|²𝐾𝑑𝑠,𝜃,𝑥₀,𝑦₀=1𝑎(𝜃)𝑦₀−𝑥₀²+2𝜎𝑦₀−𝑥₀. (3.6)
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, 𝜌²(𝜃)≥𝜌²−𝜎²−𝐾𝜃,𝑥₀,𝑦₀𝑎(𝜃). (3.7)
(ii) If 𝜌≤𝜎, then either 𝑥(𝜃)=𝑦(𝜃) or 𝜎²(𝜃)≤𝜎²−𝜌²+𝐾𝜃,𝑥₀,𝑦₀𝑎(𝜃). (3.8)

Proof of the Lemma. Let 𝜌>𝜎. In this case, we show that the control (3.4) is admissible and ensures the equality 𝑥(𝜃)=𝑦(𝜃). Indeed, clearly 𝑥(𝜃)=𝑥₀+^𝜃₀𝜑(𝑡)𝜑(𝑡)𝑎²(𝜃)𝑦₀−𝑥₀+𝑣(𝑡)𝑑𝑡=𝑦₀+^𝜃₀𝜑(𝑡)𝑣(𝑡)𝑑𝑡=𝑦(𝜃). (3.9)
To show the admissibility of the strategy (3.4), we use the Cauchy-Schwartz inequality as follows: 𝑦₀−𝑥₀^𝜃₀𝜑(𝑡)𝑣(𝑡)𝑑𝑡≤𝑦₀−𝑥₀^𝜃₀𝜑(𝑡)|𝑣(𝑡)|𝑑𝑡≤𝑦₀−𝑥₀𝑎(𝜃)𝜎. (3.10)
Then by this inequality we have ^𝜃₀|𝑢(𝑡)|²1𝑑𝑡=𝑎²(𝜃)𝑦₀−𝑥₀²1+2𝑎²(𝜃)𝑦₀−𝑥₀^𝜃₀𝜑(𝑡)𝑣(𝑡)𝑑𝑡+^𝜃₀|𝑣(𝑡)|²≤1𝑑𝑡𝑎²(𝜃)𝑦₀−𝑥₀²+2𝑎(𝜃)𝑦₀−𝑥₀𝜎+^𝜃₀|𝑣(𝑡)|²=𝐾𝑑𝑡𝜃,𝑥₀,𝑦₀𝑎(𝜃)+^𝜃₀|𝑣(𝑡)|²𝑑𝑡. (3.11)
According to (3.5) and the condition 𝑣(⋅)∈𝑆(𝜎), the right-hand side of (3.11) is less than or equal to 𝜎−𝜌²+2𝜌−𝜎𝜎+𝜎²=𝜌². (3.12)
At this point, we have proved admissibility of the control (3.4).
In particular, using (3.11) we obtain 𝜌²(𝜃)=𝜌²−^𝜃₀|𝑢(𝑡)|²𝑑𝑡≥𝜌²−𝜎²−𝐾𝜃,𝑥₀,𝑦₀𝑎(𝜃). (3.13)
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: ^𝜃₀|𝑢(𝑡)|²𝑑𝑡=^𝜃₀𝜑(𝑡)𝑎²(𝜃)𝑦₀−𝑥₀+𝑣(𝑡)²𝑑𝑡≤𝜌², (3.14) 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 ^𝜃₀𝜑(𝑡)𝑎²(𝜃)𝑦₀−𝑥₀+𝑣(𝑡)²𝑑𝑡>𝜌². (3.15)
From this inequality by using calculations similar to (3.11), we then obtain ^𝜃₀|𝑣(𝑡)|²𝑑𝑡>𝜌²−𝐾𝜃,𝑥₀,𝑦₀𝑎(𝜃), (3.16) and hence 𝜎²(𝜃)=𝜎²−^𝜃₀|𝑣(𝑡)|²𝑑𝑡<𝜎²−𝜌²+𝐾𝜃,𝑥₀,𝑦₀𝑎(𝜃). (3.17)
This completes the proof of the lemma.

The last inequality can be interpreted as follows. Though 𝑥(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^𝜏₀|𝑢(𝑡)|²𝑑𝑡=𝜌²,0<𝜏<𝜃. (3.18)

Then, of course, the pursuer cannot move anymore and automatically 𝑢(𝑡)≡0,𝑡≥𝜏.

(2⁰) Fictitious pursuers (FP): We now prove the theorem. To this purpose, introduce fictitious pursuers 𝑧₁,…,𝑧_𝑚 by equations 𝑧̇_𝑖=𝜑(𝑡)𝑤_𝑖,𝑧_𝑖(0)=𝑥_𝑖0,𝑤_𝑖∈𝑆𝜌_𝑖,𝑖=1,…,𝑚, (3.19) where 𝑤_𝑖 is control parameter of the pursuer 𝑧_𝑖. FPs may go out of the set 𝑁. They move without any state constraint in ℝ^𝑛. The aim of the FPs is to complete the pursuit as earlier as possible. Set 𝑤_𝑚(𝑡)=𝜑(𝑡)𝑎²𝜃₁𝑦_𝑘0−𝑥_𝑚0+𝑣_𝑘(𝑡),0≤𝑡≤𝜃₁,𝑤_𝑖(𝑡)≡0,𝑖=1,…,𝑚−1,0≤𝑡≤𝜃₁, (3.20)(3.21) where 𝜃₁ is any number satisfying inequalities 𝑎²𝜃₁≥𝑦_𝑘0−𝑥_𝑚0²𝜌_𝑚−𝜎_𝑘²,𝜌²₁+𝜌²₂+⋯+𝜌²_𝑚>𝜎²₁+𝜎²₂+⋯+𝜎²_𝑘+𝐾𝜃₁,𝑥_𝑚0,𝑦_𝑘0𝑎𝜃₁. (3.22)(3.23)

Since by the assumption 𝑎(𝑡)→∞ as 𝑡→∞, it follows from (3.2) that such 𝜃₁ exists. Equations (3.20), (3.21) mean that all FPs 𝑧₁,…,𝑧_𝑚−1 do not move on the time interval [0,𝜃₁], and only one FP 𝑧_𝑚 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,…,𝑧_𝑚 and evaders 𝑦₁,…,𝑦_𝑘 is reduced to a pursuit problem with the pursuers 𝑧₁,…,𝑧_𝑝 and evaders 𝑦₁,…,𝑦_𝑞, for which 𝜌²₁(𝜃₁)+⋯+𝜌²_𝑝(𝜃₁)>𝜎²₁(𝜃₁)+⋯+𝜎²_𝑞(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) 𝜌_𝑚≤𝜎_𝑘; (ii) 𝜌_𝑚>𝜎_𝑘. In the former case, that is, 𝜌_𝑚≤𝜎_𝑘, if the equality 𝑧_𝑚(𝜃₁)=𝑦_𝑘(𝜃₁) holds, then we eliminate the pursuer 𝑧_𝑚 and the evader 𝑦_𝑘 and then consider the pursuit problem with pursuers 𝑧₁,…,𝑧_𝑚−1 and evaders 𝑦₁,…,𝑦_𝑘−1 under the condition 𝜌²₁+⋯+𝜌²_𝑚−1>𝜎²₁+⋯+𝜎²_𝑘−1𝑝=𝑚−1,𝑞=𝑘−1. (3.24) In case of 𝑧_𝑚(𝜃₁)≠𝑦_𝑘(𝜃₁), according to (3.8), we obtain that 𝜎²_𝑘𝜃₁≤𝜎²_𝑘−𝜌²_𝑚+𝐾𝜃₁,𝑥_𝑚0,𝑦_𝑘0𝑎𝜃₁. (3.25)

Then in view of (3.23) we obtain 𝜌²₁+⋯+𝜌²_𝑚−1>𝜎²₁+⋯+𝜎²_𝑘𝜃₁𝑝=𝑚−1,𝑞=𝑘, (3.26) and at the time 𝜃₁ we consider the pursuit problem with the pursuers 𝑧₁,…,𝑧_𝑚−1 and evaders 𝑦₁,…,𝑦_𝑘.

We now turn to the case (ii), that is, 𝜌_𝑚>𝜎_𝑘. Then the pursuer 𝑧_𝑚 certainly ensures the equality 𝑧_𝑚(𝜃₁)=𝑦_𝑘(𝜃₁) and according to Lemma 3.2 (see, (3.7)) 𝜌²_𝑚𝜃₁≥𝜌²_𝑚−𝜎²_𝑘−𝐾𝜃₁,𝑥_𝑚0,𝑦_𝑘0𝑎𝜃₁. (3.27)

Then with the aid of (3.23) we obtain 𝜌²₁+⋯+𝜌²_𝑚−1+𝜌²_𝑚𝜃₁>𝜎²₁+⋯+𝜎²_𝑘−1𝑝=𝑚,𝑞=𝑘−1, (3.28) and therefore at the time 𝜃₁ we arrive at the pursuit problem with the pursuers 𝑧₁,…,𝑧_𝑚 and evaders 𝑦₁,…,𝑦_𝑘−1.

Let now 𝜃₂ be an arbitrary fixed number satisfying inequalities 𝜃₂>𝜃₁,𝑎²𝜃₂−𝜃₁≥𝑦_𝑞𝜃₁−𝑧_𝑝𝜃₁²𝜌_𝑝𝜃₁−𝜎_𝑞𝜃₁²,𝜌²₁𝜃₁+⋯+𝜌²_𝑝𝜃₁>𝜎²₁𝜃₁+⋯+𝜎²_𝑞𝜃₁+𝐾𝜃₂−𝜃₁,𝑦_𝑞𝜃₁,𝑧_𝑞𝜃₁𝑎𝜃₂−𝜃₁, (3.29) where 𝑝 and 𝑞 are the numbers of pursuers and evaders, respectively, which take part in the pursuit problem at time 𝜃₁. Set 𝑤_𝑝(𝑡)=𝜑(𝑡)𝑎²𝜃₂−𝜃₁𝑦_𝑞𝜃₁−𝑧_𝑝𝜃₁+𝑣_𝑞(𝑡),𝜃₁<𝑡≤𝜃₂,𝑤_𝑖(𝑡)≡0,𝜃₁<𝑡≤𝜃₂,𝑖∈{1,…,𝑚}⧵𝑝. (3.30)

Observe that according to (3.30) all pursuers except for 𝑧_𝑝 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)If 𝜌_𝑝(𝜃₁)≤𝜎_𝑞(𝜃₁) and 𝑧_𝑝(𝜃₂)=𝑦_𝑞(𝜃₂), then starting from the time 𝜃₂ we consider a pursuit problem with the pursuers 𝑧₁,…,𝑧_𝑝−1 and evaders 𝑦₁,…,𝑦_𝑞−1 under the condition 𝜌²₁𝜃₂+⋯+𝜌²_𝑝−1𝜃₂>𝜎²₁𝜃₂+⋯+𝜎²_𝑞−1𝜃₂. (3.31)(2)If 𝜌_𝑝(𝜃₁)≤𝜎_𝑞(𝜃₁) and 𝑧_𝑝(𝜃₂)≠𝑦_𝑞(𝜃₂), then starting from the time 𝜃₂ we consider the pursuit problem with the pursuers 𝑧₁,…,𝑧_𝑝−1 and evaders 𝑦₁,…,𝑦_𝑞 under the condition 𝜌²₁𝜃₂+⋯+𝜌²_𝑝−1𝜃₂>𝜎²₁𝜃₂+⋯+𝜎²_𝑞𝜃₂. (3.32)(3)If 𝜌_𝑝(𝜃₁)>𝜎_𝑞(𝜃₁), then by Lemma 3.2 the equality 𝑧_𝑝(𝜃₂)=𝑦_𝑞(𝜃₂) holds. In this case, starting from the time 𝜃₂ the pursuit problem with the pursuers 𝑧₁,…,𝑧_𝑝 and evaders 𝑦₁,…,𝑦_𝑞−1 is considered under the condition 𝜌²₁𝜃₂+⋯+𝜌²_𝑝𝜃₂>𝜎²₁𝜃₂+⋯+𝜌²_𝑞−1𝜃₂. (3.33)

Repeated application of this procedure enables us to complete the pursuit for some finite time 𝑇 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,…,𝑢_𝑚 of the pursuers 𝑥₁,…,𝑥_𝑚 by the controls of the FPs 𝑤₁,…,𝑤_𝑚. We denote by 𝐹_𝑁(𝑥) the projection of the point 𝑥∈ℝ^𝑛 on the set 𝑁, that is, min_𝑦∈𝑁𝑥−𝑦=𝑥−𝐹_𝑁(𝑥). (3.34)

Note that 𝐹_𝑁(𝑥)=𝑥 if 𝑥∈𝑁. It is familiar that for any point 𝑥∈ℝ^𝑛 there exists a unique point 𝐹_𝑁(𝑥). Moreover, for any 𝑥,𝑦∈ℝ^𝑛, 𝐹_𝑁(𝑥)−𝐹_𝑁𝑦≤𝑥−𝑦, (3.35) and hence the operator 𝐹_𝑁(𝑥) relates any absolute continuous function 𝑧(𝑡),0≤𝑡≤𝑇, to an absolute continuous function 𝑥(𝑡)=𝐹_𝑁(𝑧(𝑡)),0≤𝑡≤𝑇, (3.36) where 𝑇 is the time in which pursuit can be completed by FPs. Define the control 𝑢_𝑖(𝑡) to satisfy 𝑥_𝑖(𝑡)=𝐹_𝑁𝑧_𝑖(𝑡),0≤𝑡≤𝑇,𝑖=1,…,𝑚. (3.37)

We first show admissibility of such defined control 𝑢_𝑖(𝑡). Indeed, from (3.35), 𝜑(𝑡)𝑢_𝑖(𝑡)=𝑥̇_𝑖(𝑡)=lim_ℎ→0𝑥_𝑖(𝑡+ℎ)−𝑥(𝑡)|ℎ|=lim_ℎ→0𝐹_𝑁(𝑧(𝑡+ℎ))−𝐹_𝑁(𝑧(𝑡))|ℎ|≤lim_ℎ→0𝑧_𝑖(𝑡+ℎ)−𝑧_𝑖(𝑡)|ℎ|=𝑧̇_𝑖(𝑡)=𝜑(𝑡)𝑤_𝑖(𝑡). (3.38)

Hence the inequality |𝑢_𝑖(𝑡)|≤|𝑤_𝑖(𝑡)| holds almost everywhere and therefore ^𝑇₀𝑢_𝑖(𝑡)²𝑑𝑡≤^𝑇₀𝑤_𝑖(𝑡)²𝑑𝑡≤𝜌²_𝑖. (3.39)

Since for any evader 𝑦_𝑖,𝑖∈{1,…,𝑘}, the equality 𝑧_𝑛𝑖(𝑡_𝑖)=𝑦_𝑖(𝑡_𝑖) holds for some 𝑡_𝑖≤𝑇 and 𝑛_𝑖∈{1,…,𝑚}, and the evader 𝑦_𝑖(𝑡) is in 𝑁 for any 𝑡≥0; in particular, 𝑦_𝑖(𝑡_𝑖)∈𝑁, then FP 𝑧_𝑛𝑖(𝑡_𝑖) is also in 𝑁. Consequently, 𝑥_𝑛𝑖𝑡_𝑖=𝐹_𝑁𝑧_𝑛𝑖𝑡_𝑖=𝑧_𝑛𝑖𝑡_𝑖=𝑦_𝑖𝑡_𝑖. (3.40)

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: 𝑥̇_𝑖=𝜆𝑥_𝑖+𝑢_𝑖,𝑥_𝑖(0)=𝑥_𝑖0𝑦̇,𝑖=1,…,𝑚,𝜆≤0,_𝑗=𝜆𝑦_𝑗+𝑣_𝑗,𝑦_𝑗(0)=𝑦_𝑗0,𝑗=1,…,𝑘, (3.41)
It is assumed that 𝑥_𝑖0≠𝑦_𝑗0 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 𝑥_𝑖(𝑡_𝑗)=𝑦_𝑗(𝑡_𝑗) holds for some 𝑖∈{1,…,𝑘} at some time 𝑡_𝑗≥0. Players may not leave the half space 𝑁={𝑥∣(𝑥,𝑙)≥0,𝑥∈ℝ^𝑛}, where 𝑙∈ℝ^𝑛 is a nonzero vector. Since 𝑥_𝑖(𝑡)=𝑒^𝜆𝑡𝑥_𝑖(𝑡),𝑦_𝑖(𝑡)=𝑒^𝜆𝑡𝑦_𝑖(𝑡), (3.42) where 𝑥_𝑖(𝑡)=𝑥_𝑖0+^𝑡₀𝑒^−𝜆𝑠𝑢_𝑖(𝑠)𝑑𝑠,𝑦_𝑖(𝑡)=𝑦_𝑗0+^𝑡₀𝑒^−𝜆𝑠𝑣_𝑗(𝑠)𝑑𝑠, (3.43) then the equality 𝑥_𝑖(𝑡_𝑗)=𝑦_𝑗(𝑡_𝑗) is equivalent to the one 𝑥_𝑖(𝑡_𝑗)=𝑦_𝑗(𝑡_𝑗) and inclusions 𝑥_𝑖(𝑡)∈𝑁 and 𝑦_𝑖(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: 𝑥_𝑖𝑡_𝑗=𝑒^−𝜆𝑡𝑢_𝑖,𝑥_𝑖(0)=𝑥_𝑖0,𝑖=1,…,𝑚,𝜆≤0,𝑦_𝑗𝑡_𝑗=𝑒^−𝜆𝑡𝑣_𝑗,𝑦_𝑗(0)=𝑦_𝑗0,𝑗=1,…,𝑘. (3.44)
Denoting 𝜑(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=ℝ^𝑛 and 𝑘=1 (one evader), the condition (3.2) of the theorem takes the form 𝜌²₁+𝜌²₂+⋯+𝜌²_𝑚>𝜎²₁. This condition is sharp since if 𝜌²₁+𝜌²₂+⋯+𝜌²_𝑚≤𝜎²₁, then evasion is possible [13].