Linear evasion differential game of one evader and several pursuers with integral constraints

An evasion differential game of one evader and many pursuers is studied. The dynamics of state variables x1,…,xm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_1,\ldots , x_m$$\end{document} are described by linear differential equations. The control functions of players are subjected to integral constraints. If xi(t)≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i(t) \ne 0$$\end{document} for all i∈{1,…,m}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i \in \{1,\ldots ,m\}$$\end{document} and t≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \ge 0$$\end{document}, then we say that evasion is possible. It is assumed that the total energy of pursuers doesn’t exceed the energy of evader. We construct an evasion strategy and prove that for any positive integer m evasion is possible.


Introduction
The field of differential games was pioneered by Isaacs (1965) in the 1960s and since then enormous amount of work has been devoted to its study (for example, see Friedman 1971;Hajek 1975;Pontryagin 1988;Krasovskii and Subbotin 1988;Petrosyan 1993;Başar and Olsder 1999;Buckdahn et al. 2011 and references therein). A substantial part of these works concern simple motion pursuit and evasion differential games of many players. Often, either geometric or integral constraints are imposed on the control parameters of players. In Croft (1964) Croft showed that in the n-dimensional Euclidean ball n lions can catch the man, while the man can escape from n − 1 lions when the controls of the players are subject to geometric constraints. A similar game problem was studied by Ivanov (1980) on any convex compact set and an estimate from above was obtained for guaranteed pursuit time.
Chernous'ko (1976) studied an evasion game of one evader and several pursuers with a state constraint, i.e. the evader was supposed to remain in a neighborhood of a given ray for the duration of the game. He proved that if the evader is faster than the pursuers then evasion is possible. This result was extended by Chernous'ko and Zak (1985) and Zak (1978Zak ( , 1981Zak ( , 1982) to more general differential game problems. Related problems of evasion from a group of pursuers were studied in Borowko et al. (1988) and Chodun (1989).
In Pshenichnii (1976) Pshenichnii considered a simple motion differential game of many pursuers and one evader in R n , when all players have the same dynamic possibilities. He proved that if the initial state of the evader belongs to the interior of convex hull of pursuers' initial states, then pursuit can be completed, otherwise evasion is possible. Based on this work, Pshenichnii et al. (1981) developed the method of resolving functions for solving linear pursuit problems with many pursuers. Later on, the results of paper (Pshenichnii 1976) were extended by many researchers to cover various cases. For example, when control sets of players are convex compact sets, Grigorenko (1990) obtained the necessary and sufficient conditions of evasion of one evader from several pursuers.
The papers Chikrii and Prokopovich (1992) and Kuang (1986) are also extensions of Pshenichnii (1976). In Kuchkarov et al. (2012) the game problem of many pursuers and one evader was studied on a cylinder. In the recent work of Kuchkarov et al. (2016), the results of Pshenichnii (1976) were extended to differential games on manifolds with Euclidean metric. In Blagodatskikh and Petrov (2009) Blagodatskikh and Petrov obtained necessary and sufficient condition of evasion in a simple motion differential game of a group of pursuers and a group of evaders in R n where all evaders use the same control. By definition, pursuit is considered completed if the state of a pursuer coincides with the state of at least one evader. Also, the works (Bannikov and Petrov 2010;Vagin and Petrov 2001) related to such games. Recently in Scott and Leonard (2018) the authors consider a pursuit-evasion game involving one pursuer and multiple evaders motivated by the seminal "selfish herd" model of Hamilton (1971). The pursuer can freely move in any direction with bounded speed and evaders move with bounded speed and bounded turning speed. Using Isaacs' heuristic argument they constructed an optimal strategy for the pursuer and concluded that the optimal strategy for the pursuer is to focus on a single evader that can be captured in minimum time. Moreover, "nontargeted" evaders are always able to escape. We refer to Kumkov et al. (2017) for a survey of results on differential games of many players with geometric constraints. In the case of integral constraints, simple motion evasion games of many players were solved in Alias et al. (2016), Ibragimov et al. (2012 and Ibragimov et al. (2018).
In the present paper, we study a linear evasion differential game of many pursuers and one evader. The controls of players are subjected to integral constraints. To the best of our knowledge no previous study has investigated the linear evasion game problem stated in the present paper. The main difficulties in solving the problem are the construction of evasion strategy and to prove the fact that the objects go around the origin on some specified time interval [0, T ] maintaining some distances from the origin. Note that we employ non-anticipative strategies in the present game model (see, for example, Cardaliaguet et al. 2000).
Note that there is a similarity between the constructed evasion strategies and proofs of the main results of current and existing works (Ibragimov et al. 2012(Ibragimov et al. , 2018. They are (i) the definition of time intervals [τ i , τ i ), (ii) the construction of a strategy for the evader which allows the evader to use a manoeuvre on [τ i , τ i ) against the i-th pursuer, (iii) estimating the distances between the evader and pursuers, and establishing that evasion is possible. We believe that these steps will be common for the most of open evasion differential game problems of many players with integral constraints described by the linear system of equations as well. However, the main difficulties in solving the open problems will remain to overcome the steps (i)-(iii) listed above.
In the case of linear equations studied in the present paper, the strategies of existing papers do not work since, for the linear equation describing the game, we have to find its own τ i . Also, by contrast, we need bounded τ i , τ i and new techniques to estimate x p (t). Note that according to the strategy of the present paper, in contrast to previous works, each object, for any control functions of pursuers, moves with a positive speed in the direction of y-axis on the time interval [0, T ]. Moreover, all the objects will become on the upper half plane by the time T , and then evasion is established. The fact that each time interval where the evader uses a manoeuver is contained in the interval [0, T ] plays a key role in establishing a number of estimates in the proof of the main result.
This work is a milestone study to undertake a detailed analysis of linear evasion differential game problem of many pursuers and one evader with integral constraints. We are confident that the construction in the present paper will be a stepping stone to open problems and will open prospects for general multi person linear evasion differential games with integral constraints and this study makes a major contribution to research on general linear evasion differential games.

Statement of problem
Let x 1 , . . . , x m , m ≥ 1, be the points moving in R n whose dynamics are described by the equationsẋ where u 1 , . . . , u m are the control parameters of pursuers and v is that of evader, λ i > 0, Definition 2.1 Measurable functions u i (t) and v(t), t ≥ 0, that satisfy the following integral constraints are called controls of the ith pursuer and evader, respectively.
are some positive numbers (unspecified) and k is a positive integer, is called strategy of evader if the following system of equationṡ has a unique solution (x 1 (t), . . . , x m (t)), t ≥ 0, for any controls (u 1 (t), . . . , u m (t)), of pursuers and along this solution The strategy V (t, t 1 , . . . , t k , x 1 , . . . , x m , u 1 , . . . , u m ) is nonanticipatively defined with respect to the strictly increasing finite sequence of numbers t 1 , . . . , t k as follows.
i.e. the solution of (3) generated by this strategy and arbitrary controls of pursuers u 1 (t), . . . , u m (t) are then defined as the solution of the initial value probleṁ until the time t i+1 , i = 0, 1, . . . , k, occurs. The number t i+1 is defined as the first time when the points x 1 (t), . . . , x m (t) satisfy a certain condition. In this way, we define the solution of (3) on the intervals [t i , t i+1 ), j = 0, 1, . . . , k. It should be noted that the evader can predict neither the values of t 1 , . . . , t k nor the length of the interval

Definition 2.3
If there exists a strategy V of evader such that for any controls of pursuers x i (t) = 0, i = 1, . . . , m, t ≥ 0, then we say that evasion is possible.
The problem is to find a condition for evasion to be possible. Thus, the evader knows the values . . , u m at the current time t. Pursuers apply arbitrary controls u 1 (t), . . . , u m (t), t ≥ 0, and try to realize the equation x i (t) = 0 at least for one i ∈ {1, 2, . . . , m}, whereas the evader tries to maintain the inequalities x i (t) = 0 for all i = 1, . . . , m and t ≥ 0.

The main result
In this section we prove a theorem about evasion. To this end, we specify the conditions to define the numbers t i and construct an explicit nonanticipative strategy for the evader.The following is the main result of the current paper.
We prove the theorem in several subsections. The proof strategy is as follows. The solution of the initial value problem (1) is given by Since x i (t) = 0 if and only if y i (t) = 0, below we study the evolution of y i (t).
We construct an evasion strategy such that the second coordinate of each point y i (t) strictly increases all the time. It remains to look at the situation with initial state y i (0) with y i2 (0) < 0 for some i.
Define an a i -approach time t = τ i as the first time for which |y i (t)| = a i and More essential for the proof of the theorem is that the sequence {τ i } is bounded such that the set for some T . This is essentially implied by the fact that y i2 (t) is strictly increasing for all i and at any time t > 0, and the definition for the a i -approach time τ i requires y i2 (t) < 0.
The "manoeuvre" is defined (see (16)) such that once some object y i (τ p ) is close to the origin, i.e. |y i (τ p )| = a p while y i2 (τ p ) < 0 for some τ p , some energy is allocated on the first coordinate in order to increase |y i1 (t)| (such that |y i (t)| ≥ a p+1 on [τ p , τ p ], (see (55)), avoiding the origin). Also, (56))). The objective is that after the distance of point y i (t) from the origin is a p at t = τ p , it is not possible for it to be again at an a q distance at some τ q with q ≥ p + 1 (see (53)-(55)) The auxiliary trajectory z p (t) is the trajectory of y p (t) if the evader applies the "manoeuvre" against the p-pursuer on the whole interval [τ p , τ p ). In the end, estimations of |y p (t) − z p (t)| and z p (t) are needed to estimate |y p (t)|.

Notations
It is sufficient to consider the case when n = 2 and ρ 2 := ρ 2 1 + · · · + ρ 2 m < σ 2 (6) (see, for example, (Ibragimov et al. 2012(Ibragimov et al. , 2018. Let α be any number satisfying the condition We choose a number a 1 such that i . It is not difficult to see that this sequence has the following for some j ∈ {1, . . . , m}, where m 0 is a positive integer. In Sect. 3.2 we'll show that τ i are defined for some points y i , i = 1, . . . , m 0 , m 0 ≤ m. Note that a i -approach times τ i may not be defined as well (m 0 = 0). First we define τ 1 if relations (10) are satisfied at i = 1 for some j. Then, we define τ 2 and so on. Therefore, τ 1 < τ 2 < · · · < τ m 0 . Note that times τ i are unspecified and depend on the evader's strategy and the controls of the pursuers. It is important to note the fact that all the numbers τ i will be in the interval [0, T ], which will be established in Sect. 3.2.
Without loss of generality we relabel y j for which |y j (τ i )| = a i , y j2 (τ i ) < 0 by y i . Note that the condition (10) can occur at the time τ i for several j. If so, we label any one of them by y i . Let Proof To prove item (1), we have Since for any a ≥ 1 and b ≥ 0 we have ln(a + b) ≤ ln a + b, therefore (11) implies that The proof of item (2) follows from (12) as follows using Property 3.2.

Strategy for the evader
Now we are ready to construct a strategy for the evader. Let u j (t), j = 1, . . . , m, be arbitrary controls of pursuers. Set Equation (15) shows that the function r = r (t) assigns the control V r (t) for v(t).
For example, if there are 5 pursuers and the numbers τ i , τ i , i = 1, . . . , 5, are arranged as in Example 3.5, then using the values of r (t) in Fig. 1 we obtain from (15) that On the intervals [0, τ 1 ), [τ 2 , τ 4 ), and [τ 5 , T ], where r (t) = 0 on [0, T ], the evader's strategy is defined by (14). If r (t) = i > 0 on some time interval, then we say that evader is applying a manoeuvre V i (t) against the ith pursuer, or the evader is under the attack of the i-th pursuer on that interval. In Example 3.5, r (t) = 0 on intervals [0, τ 1 ) and [τ 2 , τ 4 ) and so the evader is not under the attack of any pursuer on these intervals. Since r (t) = 1 on [τ 1 , τ 2 ), therefore the evader is under the attack of the first pursuer on this interval, and so evader applies the manoeuvre V 1 (t) against the first pursuer on this interval. Also, we can see other manoeuvres of evader in formula (18).
Next, by item 1) (ii) of Property 3.3 we have and the proof of item 1) of Lemma 3.6 follows. In particular, (22) implies that I 1 ⊂ [0, T ].

Fig. 2 Initial states have negative y-coordinates
Next, we prove item 2) (i). Since by (21) y j2 (t), 0 ≤ t ≤ T 0 , increases strictly and as shown above y j2 (T 0 ) > 0, then we necessarily have that y j2 (θ ) = 0 at some unique θ , 0 < θ < T 0 . In view of (20) we then obtain for t > θ that which is the desired result. To show item 2) (ii), using x 0 j2 ≥ 0 we observe that for t ≥ 0, This completes the proof of Lemma 3.6. Item 1) (i) of Lemma 3.6 implies, in particular, that for the point y j with initial state x 0 j for which x 0 j2 < 0, the inequality y j2 (T 0 ) > 0 is satisfied. Thus, we necessarily have either |y j (τ j )| = a j and y j2 (τ j ) < 0 at some 0 < τ j < T 0 (see the point y i in Fig. 2) or y j2 (θ ) = 0 and |y j1 (θ )| ≥ a j at some 0 < θ < T 0 (see the point y j in Fig. 2). The former case will be studied in the following subsections in detail. In the latter case, we ignore the point y j (t) starting from the time θ since by Lemma 3.6 2) (i) we have y j2 (t) > 0 and so y j (t) = 0 for all t > θ. That is why in definition of  Fig. 3. We ignore the corresponding points y k (t) and y l (t) as well for t ≥ 0 since by Lemma 3.6 2) (ii) y k2 (t) > 0 and y l2 (t) > 0 for all t ≥ 0.

Auxiliary point z p
Take any p ∈ {1, . . . , m 0 } and estimate |y p (t)| on [τ p , τ p ] assuming that a p -approach was occurred at time τ p with the point y p . To this end we introduce an auxiliary point z p whose dynamics is described by the following equation Note that the point z p (t) is defined only on the interval [τ p , τ p ]. Since by (15) Next, we show that Indeed, denoting Therefore, using the Minkowskii inequality and then item (1) of Property 3.3 we obtain since by (8) a p ≤ a 1 < (σ −ρ) 2 4α , and hence (27) is true.

Estimation of |z p (t)|
Let τ p ≤ t < τ p and for definiteness assume that y p1 (τ p ) ≥ 0. Then by (16) we have V p1 (t) = α + |u p1 (t)|. Therefore, On the other hand, The integral in (29) can be estimated by using the Cauchy-Schwartz inequality as follows Since by (27) and the admissibility of control u p (s) Since by (22) τ p ≤ t ≤ τ p ≤ T , therefore t τ p e 2λ p s ds = 1 2λ p e λ p t + e λ p τ p e λ p t − e λ p τ p ≤ e T λ p e λ p t − e λ p τ p .
Also, it follows from the definition of κ and the inequality a p < 1 that a p+1 ≤ α 16σ 2 e T a 4 p ≤ α 12σ 2 e T a 2 p .
The proof of Theorem 3.1 is completed.

Conclusion
We have studied a linear evasion differential game of many pursuers and one evader. We have constructed a strategy for the evader and proved the possibility of evasion. The evader uses a manoeuvre on the set I 1 and on the set [0, T ]\I 1 evader uses the control v(t) = 0, α + m j=1 |u j (t)| 2 1/2 . The measure of the set I 1 can be made by choosing parameters a 1 and α as small as we wish. We have also shown that all the approach times τ i can occur only before a specified time T 0 , moreover τ i ≤ T . The total number of approach times τ i of all pursuers doesn't exceed the number of pursuers m. For t ≥ T , the evader uses the control v(t) = 0, m j=1 |u j (t)| 2 1/2 and there is no longer an approach time occurs. The main contributions of the paper are (i) the construction of evasion strategy, (ii) estimating the distances of objects from the origin, (iii) the possibility of evasion from many pursuers.