# Simple Motion Evasion Differential Game of Many Pursuers and One Evader with Integral Constraints on Control Functions of Players

Authors: Gafurjan Ibragimov; Yusra Salleh
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We consider an evasion differential game of many pursuers and one evader with integral constraints in the plane. The game is described by simple equations. Each component of the control functions of players is subjected to integral constraint. Evasion is said to be possible if the state of the evader does not coincide with that of any pursuer. Strategy of the evader is constructed based on controls of the pursuers with lag. A sufficient condition of evasion from many pursuers is obtained and an illustrative example is provided.


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## Body

## 1. Introduction

### 1.1. Related Works

Among a large number of papers devoted to differential games with integral constraints (see, e.g., [1–14]), a few works are dedicated to evasion games with integral constraints (see, e.g., [1, 4, 7, 12]). Note that integral constraints arise if we consider constraints on energy, resources, and so forth. In [1], an evasion differential game was considered. In the paper [1], the dynamics of the object z(t) is given by the equation
(1.1)z˙=Az-u+v,z(0)=z0,
where z∈ℝn,A is a constant matrix, and u and v are control parameters of the pursuer and evader, respectively. The control functions u(t)and v(t) of the pursuer and evader, respectively, are defined as measurable functions satisfying conditions
(1.2)∫0∞|u(t)|2dt≤ρ2,∫0∞|v(t)|2dt≤σ2.
The terminal set M is a subspace of ℝn.  By definition evasion is possible if  z(t)∉M,t≥0. In this work Pontryagin’s method was extended to the evasion game with integral constraints and sufficient conditions of evasion from any initial position were obtained.In [4], a differential game described by (1.1) was studied.  The control functions  u(t) and v(t) satisfy conditions (1.2). Evasion is said to be possible if z(t)∉M, where M=∪i=1mMi, Mi are subspaces of ℝn. The pursuers’ purpose is to take the point z(t) onto the terminal set M; the evader's one, to guarantee evasion from the set M. By definition, l-escape is possible if  |z(t)|≥l(t)>0,t≥0 for some function l(t). Sufficient conditions for l-escape were obtained.In the book [12] (chapter 11), linear evasion differential game with integral constraints was examined. The game is described by the equation
(1.3)z˙=Az-Bu+Cv,z(0)=z0,
where z∈ℝn,A,B, and C are constant matrices and u and v are control parameters of the pursuer and evader, respectively.  The control functions u(t) and v(t) of the pursuer and evader are subjected to the conditions (1.2). The terminal set M is a subspace of ℝn.  By definition evasion is possible if  z(t)∉M,t≥0. Some sufficient conditions of evasion were obtained.In [5], a differential game of countably many pursuers and one evader is described  by the following infinite system of differential equations:
(1.4)z˙ik=-λkzik-uik+vk,zik(t0)=zik0,k=1,2,…,
which was studied in the Hilbert space lr+12, where
(1.5)zik,uik,vk,t0,zik0∈ℝ,zi0=(zi10,zi20,…)∈lr+12,zi0≠0,ui=(ui1,ui2,…) is the control parameter of ith pursuer,  i=1,2,…, and v=(v1,v2,…) is that of the evader.  The control functions of the players are subject to integral constraints
(1.6)∫t0T∥ui(t)∥2dt≤ρi2,∫t0T∥v(t)∥2dt≤σ2,∥v(t)∥=(∑k=1∞‍λkrvk2(t))1/2,T>0.
Avoidance of contact is said to be possible in the game with the initial position z0={z10,z20,…,zm0,…},zi0∈lr+12, if zi(t)=(zi1(t),zi2(t),…)≠0, for all t∈[0,T] and i=1,2,….  It was proven that if σ2>∑k=1∞ρk2, then avoidance of contact is possible in the game from any initial position z0={z10,…,zm0,…},zi0∈lr+12,zi0≠0,i=1,2,….The work [7] deals with the evasion differential game of m pursuers x1,…,xm and one evader y described by equations
(1.7)x˙i=ui,xi(0)=xi0,y˙=v,y(0)=y0,xi0≠y0,i=1,…,m.
The game occurs in the plane and the following conditions:
(1.8)∫0∞|ui(s)|2ds≤ρi2,i=1,…,m,∫0∞|v(s)|2ds≤σ2
are imposed on control functions of the players. The main result of the paper is as follow: if ρ12+ρ22+⋯+ρm2≤σ2, then evasion is possible. The important point to note here is the fact that if ρ12+ρ22+⋯+ρm2>σ2, then pursuit can be completed [13].It should be noted that in all of these works, an integral constraint was imposed on the whole control function of each player, such as(1.9)∫0∞|v(s)|2ds≤σ2.
However, in contrast to other works, in the present paper, we study a differential game with integral constraints imposed on each coordinate of control functions (see the constraints (1.12)-(1.13)).At first glance, the inequalitiesρ1i2+ρ2i2+⋯+ρmi2>σi2,i=1,2 seem to be the conditions for completing pursuit. It turns out, as shown in the example, evasion is possible from some initial positions even though these inequalities hold.


### 1.2. Statement of the Problem

Inℝ2, we consider an evasion differential game of many pursuers x1,…,xm and one evader y described by the equations
(1.10)x˙i=ui,xi(0)=xi0,i=1,…,m,(1.11)y˙=v,y(0)=y0,
and assume that xi0≠y0,xi0=(xi10,xi20),i=1,…,m,y0=(y10,y20).Definition 1.1.
A measurable functionui(t)=(ui1(t),ui2(t)),t≥0 is called an admissible control of the pursuer xi if
(1.12)∫0∞|ui1(s)|2ds≤ρi12,∫0∞|ui2(s)|2ds≤ρi22,
where ρi1,ρi2,i=1,…,m  are given positive numbers.Definition 1.2.
A measurable functionv(t)=(v1(t),v2(t)), t≥0 is called an admissible control of the evader y if
(1.13)∫0∞|v1(s)|2ds≤σ12,∫0∞|v2(s)|2ds≤σ22,
where σ1 and  σ2 are given positive numbers.Definition 1.3.
A function of the form(1.14)V(t)={0,0≤t≤ε,f(u1(t-ε),…,um(t-ε)),t>ε
is called a strategy of the evader, where ε is a positive number, f:ℝ2m→ℝ2 is a continuous function, and u1(t),…,um(t),t≥0 are admissible  controls of the pursuers.Definition 1.4.
We say that evasion is possible from the initial position(x10,x20,…,xm0,y0) in the game (1.10)–(1.13) if there exists a strategy of the evader V such that xi(t)≠y(t), t≥0, i=1,…,m for any admissible controls of the pursuers.We can now state the evasion problem.Problem 1.
Find conditions for all initial positions(x10,x20,…,xm0,y0) and parameters  σ1, σ2, ρi1,ρi2,i=1,…,m  which guarantee evasion in the game (1.10)–(1.13).This is the main problem to be investigated in the present paper. It should be noted that, in the evasion game, the pursuers use arbitrary admissible controlsu1(t),…,um(t), t≥0, and the evader uses a strategy. By Definition 1.3 this strategy is constructed based on values u1(t-ε),…,um(t-ε).After the proof of the main result (Theorem2.1) we will give illustrative examples. For the initial positions, which do not satisfy hypotheses of the theorem, we will show that pursuit can be completed. Therefore, at first we have to give a definition for “pursuit can be completed”. To this end we need to define strategies of the pursuers.Definition 1.5.
A Borel measurable functionUi(v)=(Ui1(v),Ui2(v)),Ui:ℝ2→ℝ2 is called a strategy of the pursuer xi if for any control of the evader v(t),t≥0 and the inequalities
(1.15)∫0∞|Ui1(v(s))|2ds≤ρi12,∫0∞|Ui2(v(s))|2ds≤ρi22
hold.Definition 1.6.
We say that pursuit can be completed from the initial position(x10,x20,…,xm0,y0) in the game (1.10)–(1.13) if there exist strategies of the pursuers Ui, i=1,…,m such that for any admissible control of the evader the equality xi(τ)=y(τ) holds at some i∈{1,…,m} and τ≥0.
Note that in the pursuit game, the pursuers use strategies and the evader uses any admissible controlv(t) (see Definition 1.6). According to Definition 1.5 at current time t the pursuers use v(t) to construct their strategies.


## 1.1. Related Works

Among a large number of papers devoted to differential games with integral constraints (see, e.g., [1–14]), a few works are dedicated to evasion games with integral constraints (see, e.g., [1, 4, 7, 12]). Note that integral constraints arise if we consider constraints on energy, resources, and so forth. In [1], an evasion differential game was considered. In the paper [1], the dynamics of the object z(t) is given by the equation
(1.1)z˙=Az-u+v,z(0)=z0,
where z∈ℝn,A is a constant matrix, and u and v are control parameters of the pursuer and evader, respectively. The control functions u(t)and v(t) of the pursuer and evader, respectively, are defined as measurable functions satisfying conditions
(1.2)∫0∞|u(t)|2dt≤ρ2,∫0∞|v(t)|2dt≤σ2.
The terminal set M is a subspace of ℝn.  By definition evasion is possible if  z(t)∉M,t≥0. In this work Pontryagin’s method was extended to the evasion game with integral constraints and sufficient conditions of evasion from any initial position were obtained.In [4], a differential game described by (1.1) was studied.  The control functions  u(t) and v(t) satisfy conditions (1.2). Evasion is said to be possible if z(t)∉M, where M=∪i=1mMi, Mi are subspaces of ℝn. The pursuers’ purpose is to take the point z(t) onto the terminal set M; the evader's one, to guarantee evasion from the set M. By definition, l-escape is possible if  |z(t)|≥l(t)>0,t≥0 for some function l(t). Sufficient conditions for l-escape were obtained.In the book [12] (chapter 11), linear evasion differential game with integral constraints was examined. The game is described by the equation
(1.3)z˙=Az-Bu+Cv,z(0)=z0,
where z∈ℝn,A,B, and C are constant matrices and u and v are control parameters of the pursuer and evader, respectively.  The control functions u(t) and v(t) of the pursuer and evader are subjected to the conditions (1.2). The terminal set M is a subspace of ℝn.  By definition evasion is possible if  z(t)∉M,t≥0. Some sufficient conditions of evasion were obtained.In [5], a differential game of countably many pursuers and one evader is described  by the following infinite system of differential equations:
(1.4)z˙ik=-λkzik-uik+vk,zik(t0)=zik0,k=1,2,…,
which was studied in the Hilbert space lr+12, where
(1.5)zik,uik,vk,t0,zik0∈ℝ,zi0=(zi10,zi20,…)∈lr+12,zi0≠0,ui=(ui1,ui2,…) is the control parameter of ith pursuer,  i=1,2,…, and v=(v1,v2,…) is that of the evader.  The control functions of the players are subject to integral constraints
(1.6)∫t0T∥ui(t)∥2dt≤ρi2,∫t0T∥v(t)∥2dt≤σ2,∥v(t)∥=(∑k=1∞‍λkrvk2(t))1/2,T>0.
Avoidance of contact is said to be possible in the game with the initial position z0={z10,z20,…,zm0,…},zi0∈lr+12, if zi(t)=(zi1(t),zi2(t),…)≠0, for all t∈[0,T] and i=1,2,….  It was proven that if σ2>∑k=1∞ρk2, then avoidance of contact is possible in the game from any initial position z0={z10,…,zm0,…},zi0∈lr+12,zi0≠0,i=1,2,….The work [7] deals with the evasion differential game of m pursuers x1,…,xm and one evader y described by equations
(1.7)x˙i=ui,xi(0)=xi0,y˙=v,y(0)=y0,xi0≠y0,i=1,…,m.
The game occurs in the plane and the following conditions:
(1.8)∫0∞|ui(s)|2ds≤ρi2,i=1,…,m,∫0∞|v(s)|2ds≤σ2
are imposed on control functions of the players. The main result of the paper is as follow: if ρ12+ρ22+⋯+ρm2≤σ2, then evasion is possible. The important point to note here is the fact that if ρ12+ρ22+⋯+ρm2>σ2, then pursuit can be completed [13].It should be noted that in all of these works, an integral constraint was imposed on the whole control function of each player, such as(1.9)∫0∞|v(s)|2ds≤σ2.
However, in contrast to other works, in the present paper, we study a differential game with integral constraints imposed on each coordinate of control functions (see the constraints (1.12)-(1.13)).At first glance, the inequalitiesρ1i2+ρ2i2+⋯+ρmi2>σi2,i=1,2 seem to be the conditions for completing pursuit. It turns out, as shown in the example, evasion is possible from some initial positions even though these inequalities hold.


## 1.2. Statement of the Problem

Inℝ2, we consider an evasion differential game of many pursuers x1,…,xm and one evader y described by the equations
(1.10)x˙i=ui,xi(0)=xi0,i=1,…,m,(1.11)y˙=v,y(0)=y0,
and assume that xi0≠y0,xi0=(xi10,xi20),i=1,…,m,y0=(y10,y20).Definition 1.1.
A measurable functionui(t)=(ui1(t),ui2(t)),t≥0 is called an admissible control of the pursuer xi if
(1.12)∫0∞|ui1(s)|2ds≤ρi12,∫0∞|ui2(s)|2ds≤ρi22,
where ρi1,ρi2,i=1,…,m  are given positive numbers.Definition 1.2.
A measurable functionv(t)=(v1(t),v2(t)), t≥0 is called an admissible control of the evader y if
(1.13)∫0∞|v1(s)|2ds≤σ12,∫0∞|v2(s)|2ds≤σ22,
where σ1 and  σ2 are given positive numbers.Definition 1.3.
A function of the form(1.14)V(t)={0,0≤t≤ε,f(u1(t-ε),…,um(t-ε)),t>ε
is called a strategy of the evader, where ε is a positive number, f:ℝ2m→ℝ2 is a continuous function, and u1(t),…,um(t),t≥0 are admissible  controls of the pursuers.Definition 1.4.
We say that evasion is possible from the initial position(x10,x20,…,xm0,y0) in the game (1.10)–(1.13) if there exists a strategy of the evader V such that xi(t)≠y(t), t≥0, i=1,…,m for any admissible controls of the pursuers.We can now state the evasion problem.Problem 1.
Find conditions for all initial positions(x10,x20,…,xm0,y0) and parameters  σ1, σ2, ρi1,ρi2,i=1,…,m  which guarantee evasion in the game (1.10)–(1.13).This is the main problem to be investigated in the present paper. It should be noted that, in the evasion game, the pursuers use arbitrary admissible controlsu1(t),…,um(t), t≥0, and the evader uses a strategy. By Definition 1.3 this strategy is constructed based on values u1(t-ε),…,um(t-ε).After the proof of the main result (Theorem2.1) we will give illustrative examples. For the initial positions, which do not satisfy hypotheses of the theorem, we will show that pursuit can be completed. Therefore, at first we have to give a definition for “pursuit can be completed”. To this end we need to define strategies of the pursuers.Definition 1.5.
A Borel measurable functionUi(v)=(Ui1(v),Ui2(v)),Ui:ℝ2→ℝ2 is called a strategy of the pursuer xi if for any control of the evader v(t),t≥0 and the inequalities
(1.15)∫0∞|Ui1(v(s))|2ds≤ρi12,∫0∞|Ui2(v(s))|2ds≤ρi22
hold.Definition 1.6.
We say that pursuit can be completed from the initial position(x10,x20,…,xm0,y0) in the game (1.10)–(1.13) if there exist strategies of the pursuers Ui, i=1,…,m such that for any admissible control of the evader the equality xi(τ)=y(τ) holds at some i∈{1,…,m} and τ≥0.
Note that in the pursuit game, the pursuers use strategies and the evader uses any admissible controlv(t) (see Definition 1.6). According to Definition 1.5 at current time t the pursuers use v(t) to construct their strategies.


## 2. Main Result

The main result of the paper is formulated as follows.Theorem 2.1.
Assume the following conditions hold.(1)
There exists a subsetI of the set {1,…,m} such that
(2.1)∑i∈Iρi12≤σ12,∑i∈Jρi22≤σ22,J={1,…,m}∖I.(2)
y10∉[x1′,x1′′],y20∉[x2′,x2′′], where
(2.2)x1′=mini∈Ixi10,x1′′=maxi∈Ixi10,x2′=mini∈Jxi20,x2′′=maxi∈Jxi20.
Then evasion is possible in the game (1.10)–(1.13).Proof.
Letρ=(∑i=1m∑j=12ρij2)1/2, and let ε,ε<1/(4ρ2)min{(y10-x1′′)2,(y20-x2′′)2}, be a positive number. We construct a strategy for the evader as follows:
(2.3)v1(t)=y10-xk10|y10-xk10|{0,0≤t≤ε,(∑i∈Iui12(t-ε))1/2,t>ε,(2.4)v2(t)=y20-xl20|y20-xl20|{0,0≤t≤ε,(∑i∈Jui22(t-ε))1/2,t>ε,
where u1(t),…,um(t),t≥0 are any admissible controls of the pursuers, and k∈I and  l∈J are arbitrary numbers. The pursuers use any admissible controls and the evader uses the strategy (2.3) and (2.4). We’ll prove that evasion is possible. We examine the case y10>x1′′,y20>x2′′. Other cases such as
(2.5)y10>x1′′,y20<x2′;y10<x1′,y20>x2′′;y10<x1′,y20<x2′
can also be considered similarly.  It follows from the inequalities y10>x1′′,y20>x2′′ that y10>xk10, y20>xl20, and so
(2.6)y10-xk10|y10-xk10|=1,y20-xl20|y20-xl20|=1.
Then (2.3) and (2.4) take the following form:
(2.7)v1(t)={0,0≤t≤ε,(∑i∈Iui12(t-ε))1/2,t>ε,(2.8)v2(t)={0,0≤t≤ε,(∑i∈Jui22(t-ε))1/2,t>ε.
First, we show that evasion is possible on the time interval [0,ε]. Let t∈[0,ε].  Then for any i∈I we have:
(2.9)y1(t)-xi1(t)=y10+∫0tv1(s)ds-xi10-∫0tui1(s)ds≥y10-x1′′-∫0t|ui1(s)|ds≥y10-x1′′-ρt≥y10-x1′′-ρε≥12(y10-x1′′)>0,
since by choice of ε, ρε<(1/2)(y10-x1′′).
We now show that evasion is possible on(ε,∞). If t∈(ε,∞), then according to (2.7) for any i∈I we have:
(2.10)y1(t)-xi1(t)=y10+∫εtv1(s)ds-xi10-∫0tui1(s)ds≥y10-x1′′+∫εt∑i∈Iui12(s-ε)ds-∫0tui1(s)ds≥y10-x1′′+∫0t-ε∑i∈Iui12(s)ds-(∫0t-ε+∫t-εt)|ui1(s)|ds≥y10-x1′′-∫t-εt|ui1(s)|ds≥y10-x1′′-ερ≥12(y10-x1′′)>0.
Thus, evasion from the pursuers xi,i∈I, is possible. Similarly, according to (2.8) for any j∈J  we obtain that
(2.11)y2(t)-xi2(t)≥y20-x2′′-ερ≥12(y20-x2′′)>0,t≥0.
This completes the proof of the theorem.Example 2.2.
We consider two differential games of two pursuers and one evader described by the equations(2.12)x˙i=ui,xi(0)=xi0,∫0∞uij2(s)ds≤ρij2,(2.13)y˙=v,y(0)=y0,∫0∞vj2(s)ds≤σj2,i,j=1,2.(A)
In the first differential game,ρ112=4,ρ122=1,ρ212=1,ρ222=4,σ12=σ22=1, and hence the sets I={2},J={1} satisfy the first hypothesis of the theorem. Initial positions of the players x10, x20, and y0, where x10≠y0, x20≠y0, are assumed  to be any points in the plane. It is not difficult to verify that the second hypothesis of the theorem holds, if x210≠y10, and x120≠y20 since x1′=x1′′=x210, x2′=x2′′=x120. Therefore, for this case conclusion of the theorem is true and hence from such initial positions evasion is possible. Note that the meaning of the condition x210≠y10, x120≠y20 is that the initial position of the evader y0 doesn’t lie  on the horizontal and vertical lines passing through the points  x10 and x20, respectively.We now construct the strategy for the evader, which guarantees the evasion. Letu1(t),u2(t),t≥0 be any admissible controls of the pursuers. According to (2.3) and (2.4) the strategy of the evader takes the following form:
(2.14)v1(t)={0,0≤t≤ε,ξ|u21(t-ε)|,t>ε,v2(t)={0,0≤t≤ε,η|u12(t-ε)|,t>ε,
where
(2.15)ξ=y10-x210|y10-x210|,η=y20-x120|y20-x120|.
For this example,  ρ=(ρ112+ρ122+ρ212+ρ222)1/2 and ε satisfies the following conditions:
(2.16)ε<14ρ2min{(y10-x210)2,(y20-x120)2}.
According to the theorem if x210≠y10,x120≠y20, then the strategy of the evader (2.14) guarantees the evasion.However, the theorem gives no information if eitherx210=y10 or x120=y20 since the second hypothesis of the theorem is not satisfied for these cases. We’ll show that in each of these cases pursuit can be completed in the game (2.12) and (2.13) (see Definitions 1.5 and 1.6). In the case x210=y10, define the strategies of the pursuers as follows:
(2.17)u11(t)=0,u12(t)=0,t≥0,u21(t)=v1(t),(2.18)u22(t)={1θ(y20-x220)+v2(t),0≤t≤θ,0,t>θ,
where θ=|y20-x220|2.In the casex120=y20, define the strategies of the pursuers as follows:
(2.19)u21(t)=0,u22(t)=0,t≥0,u11(t)={1θ1(y10-x110)+v1(t),0≤t≤θ1,0,t>θ1,u12(t)=v2(t),
where θ1=|y10-x110|2.We consider the casex210=y10.  The case x120=y20 can be analyzed   in a similar fashion. Admissibility of the strategy  and (2.18) follows from the following relations:
(2.20)∫0∞|u22(t)|2dt=∫0θ|u22(t)|2dt+∫θ∞|u22(t)|2dt=∫0θ(1θ(y20-x220)+v2(t))2dt=∫0θ(1θ2|y20-x220|2+2θ(y20-x220)v2(t)+|v2(t)|2)dt=1θ|y20-x220|2+2θ∫0θ(y20-x220)v2(t)dt+∫0θ|v2(t)|2dt≤1+2θ|y20-x220|·∫0θ|v2(t)|dt+σ22≤2+2θ|y20-x220|·∫0θ12dt∫0θ|v2(t)|2dt≤2+2σ22θ|y20-x220|=4.
Here, we used the Cauchy-Schwartz inequality. Next, we show that x2(θ)=y(θ), that is, pursuit will be completed by the pursuer x2 at the time θ. We have
(2.21)x21(t)=x210+∫0tu21(s)ds=y10+∫0tv1(s)ds=y1(t)
for all t≥0. In particular, x21(θ)=y1(θ).  In addition,
(2.22)x22(θ)=x220+∫0θu22(s)ds=x220+∫0θ(1θ(y20-x220)+v2(s))ds=y20+∫0θv2(s)ds=y2(θ).
Hence,  x22(θ)=y2(θ). Thus x2(θ)=y(θ).(B)
Consider an example of a differential game described by (2.12) and (2.13), for which the second hypothesis of the theorem is not satisfied.  Let ρ112=2,ρ122=1,ρ212=2,ρ222=1,σ12=σ22=2, x10=(0,-1),   x20=(0,1), y0=(0,a), where -1<a<1.Let the strategies of the pursuers be defined by formulas(2.23)u11(t)=u21(t)=v1(t),u12(t)=1,u22(t)=-1,t≥0.
We show that pursuit is completed by the time t=1. Indeed, for all t≥0 we have
(2.24)xi1(t)=xi10+∫0tui1(s)ds=∫0tv1(s)ds=y1(t),i=1,2.
What is left is to show that  xi2(τ)=y2(τ) for some i∈{1,2} and τ,0<τ≤1.It follows from the relations(2.25)x12(0)<y2(0)<x22(0),x12(1)=x120+∫01u12(s)ds=-1+∫011ds=0,x22(1)=x220+∫01u22(s)ds=1+∫01(-1)ds=0
that one of the equalities x12(τ)=y(τ) or x22(τ)=y(τ) holds at some  τ,0≤τ≤1.  According to (2.24), xi1(τ)=y1(τ) and therefore one of the equalities x1(τ)=y(τ) or x2(τ)=y(τ) holds and hence pursuit is completed at τ. Thus, if the initial position of the evader is in the line segment (x10,x20) then pursuit can be completed by the time t=1.


## 3. Conclusion

We have studied a simple motion evasion differential game with integral constraints on control functions of the players. Unlike the traditional integral constraints, in the present paper the integral constraint is imposed on each component of control functions of the players.  Under certain conditions, we have proven that the evasion is possible, that is, we have obtained a sufficient condition of evasion. The problem is open for further investigation if the hypotheses of the theorem are not satisfied. Further studies can be done to obtain complete solution for the evasion problem (1.10)–(1.13).



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*Source: 748096-2012-10-14.xml*