On Some Pursuit and Evasion Differential Game Problems for an Infinite Number of First-Order Differential Equations

Abstract

We study pursuit and evasion differential game problems described by infinite number of first-order differential equations with function coefficients in Hilbert space l₂. Problems involving integral, geometric, and mix constraints to the control functions of the players are considered. In each case, we give sufficient conditions for completion of pursuit and for which evasion is possible. Consequently, strategy of the pursuer and control function of the evader are constructed in an explicit form for every problem considered.

1. Introduction

The books by Friedman [1], Isaacs [2], Krasovskiĭ and Subbotin [3], Lewin [4], Petrosyan [5], and Pontryagin [6] among others are fundamental to the study of differential games.

Many works are devoted to differential game problems described by both ordinary differential equations in 𝑅^𝑛 and partial differential equations. In particular, pursuit and evasion differential game problems involving distributed parameter systems are of increasing interest (see, e.g., [7–14]).

Satimov and Tukhtasinov [10, 11] studied pursuit and evasion problems described by the parabolic equation 𝑧_𝑡−𝐴𝑧=−𝑢+𝑣,𝑧|_𝑡=0=𝑧₀(𝑥),𝑧|_𝑆𝑇=0,(1.1) where 𝑧=𝑧(𝑡,𝑥) is unknown function; 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈Ω⊂𝑅^𝑛,𝑛≥1 is parameter in a bounded domain Ω; 𝑡∈[0,𝑇],𝑇>0; 𝑢=𝑢(𝑡,𝑥),𝑣=𝑣(𝑡,𝑥) are control functions of the players; 𝑆_𝑇={(𝑥,𝑡)|𝑥∈𝜕Ω,𝑡∈(0,𝑇)}, 𝐴 is a differential operator defined in the space 𝐿₂(Ω). In this problem, the authors used the fact in [15] that under certain conditions, the problem (1.1) has a unique generalized (in the sense of distribution) solution of the form 𝑧(𝑡,𝑥)=^∞_𝑘=1𝑧_𝑘(𝑡)𝜓_𝑘(𝑥),(1.2) where the functions 𝑧_𝑘(𝑡), 0≤𝑡≤𝑇, 𝑘=1,2,…, constitute the solution of the Cauchy problem for the following infinite system of differential equations and initial conditions: 𝑧̇_𝑘+𝜆_𝑘𝑧_𝑘=−𝑢_𝑘(𝑡)+𝑣_𝑘(𝑡),𝑧_𝑘(0)=𝑧_0𝑘,𝑘=1,2,…,(1.3) and 𝜆_𝑘, 𝑘=1,2,…, satisfying the condition that 0<𝜆₁≤𝜆₂≤⋯→∞, are eigenvalues of the operator 𝐴, the functions 𝜓_𝑘(𝑥),𝑘=1,2,…, constitute an orthonormal and complete system of eigenfunctions of the operator 𝐴, 𝑢_𝑘(𝑡),𝑣_𝑘(𝑡), and 𝑧_0𝑘 are the Fourier coefficients in the expansion of 𝑢(𝑡,𝑥),𝑣(𝑡,𝑥), and 𝑧₀(𝑥), respectively, in the system {𝜓_𝑘(𝑥)}.

The work above shows the significant relationship between differential game problems described by (1.1) in one side and those described by (1.3) in the other side. Therefore, it is logical to study the latter in an independent frame work (see, e.g., [16, 17]).

In the present paper, we solve pursuit and evasion problems described by system (1.3), with function coefficients 𝜆_𝑘,𝑘=1,2,… instead of constants coefficients considered in the previous study. Different forms of constraints on the controls of the players are considered.

2. Statement of the Problem

Let 𝑙₂=𝛼=𝛼₁,𝛼₂,…∶^∞_𝑘=1𝛼^2𝑘<∞,(2.1) with inner product and norm ⟨𝛼,𝛽⟩=^∞_𝑘=1𝛼_𝑘𝛽_𝑘,𝛼,𝛽∈𝑙₂,‖𝛼‖=^∞_𝑘=1𝛼^2𝑘^1/2‖𝑤(⋅)‖_{𝐿2(0,𝑇,𝑙2)},=^∞_𝑘=1^𝑇0𝑤^2𝑘(𝑠)𝑑𝑠^1/2.(2.2)

Let 𝐿₂0,𝑇,𝑙₂=𝑤(𝑡)=𝑤₁(𝑡),𝑤₂(𝑡),…∶‖𝑤(⋅)‖_{𝐿2(0,𝑇,𝑙2)}<∞,𝑤_𝑘(⋅)∈𝐿₂(0,𝑇),(2.3) where 𝑇,𝑇>0, is a given number.

We examine a pursuit and evasion differential game problems described by the following infinite system of differential equations 𝑧̇_𝑘(𝑡)+𝜆_𝑘(𝑡)𝑧_𝑘(𝑡)=−𝑢_𝑘(𝑡)+𝑣_𝑘(𝑡),𝑧_𝑘(0)=𝑧_𝑘0,𝑘=1,2,…,(2.4) where 𝑧_𝑘,𝑢_𝑘,𝑣_𝑘∈𝑅¹,𝑘=1,2,…,𝑧₀=(𝑧₁₀,𝑧₂₀,…)∈𝑙₂,𝑢_𝑘,𝑣_𝑘,…, are control parameters of pursuer and evader respectively, 𝜆_𝑘(𝑡),𝑘=1,2,…, are bounded, non-negative continuous functions on the interval [0,𝑇] such that 𝜆_𝑘(0)=0,𝑘=1,2,….

Definition 2.1. A function 𝑤(⋅),𝑤∶[0,𝑇]→𝑙₂, with measurable coordinates 𝑤_𝑘(𝑡),0≤𝑡≤𝑇,𝑘=1,2,…, subject to ^∞_𝑘=1^𝑇0𝑤^2𝑘(𝑠)𝑑𝑠≤𝜌²^∞_𝑘=1𝑤^2𝑘(𝑡)≤𝜌²,𝑡∈[0,𝑇],(2.5) where 𝜌 is a positive number, is referred to as an admissible control subject to integral constraint (resp., geometric constraint).

We denote the set of all admissible controls with respect to integral constraint by 𝑆₁(𝜌) and with respect to geometric constraint by 𝑆₂(𝜌).

The control 𝑢(⋅)=(𝑢₁(⋅),𝑢₂(⋅),…) of the pursuer and 𝑣(⋅)=(𝑣₁(⋅),𝑣₂(⋅),…) of the evader are said to be admissible if they satisfy one of the following conditions ^∞_𝑘=1^𝑇0𝑢^2𝑘(𝑠)𝑑𝑠^1/2≤𝜌,^∞_𝑘=1^𝑇0𝑣^2𝑘(𝑠)𝑑𝑠^1/2≤𝜎,(2.6)_∞_𝑘=1𝑢^2𝑘(𝑡)^1/2≤𝜌,𝑡∈[0,𝑇],^∞_𝑘=1𝑣^2𝑘(𝑡)^1/2≤𝜎,𝑡∈[0,𝑇],(2.7)_∞_𝑘=1^𝑇0𝑢^2𝑘(𝑠)𝑑𝑠^1/2≤𝜌,^∞_𝑘=1𝑣^2𝑘(𝑡)^1/2≤𝜎,𝑡∈[0,𝑇],(2.8)_∞_𝑘=1𝑢^2𝑘(𝑡)^1/2≤𝜌,𝑡∈[0,𝑇],^∞_𝑘=1^𝑇0𝑣^2𝑘(𝑠)𝑑𝑠^1/2≤𝜎,(2.9) where 𝜌 and 𝜎 are positive constants. We will call the system (2.4) in which 𝑢(⋅) and 𝑣(⋅) satisfy inequalities (2.6) (resp., (2.7), (2.8), and (2.9)), game 𝐺₁ (resp., 𝐺₂,𝐺₃,𝐺₄).

Definition 2.2. A function 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡),…),0≤𝑡≤𝑇, is called the solution of the system (2.4) if each coordinate 𝑧_𝑘(𝑡)(i)is absolutely continuous and almost everywhere on [0,𝑇] satisfies (2.4),(ii)𝑧(⋅)∈𝐶(0,𝑇;𝑙₂).

Definition 2.3. A function 𝑈(𝑡,𝑣),𝑈∶[0,𝑇]×𝑙₂→𝑙₂(2.10) is referred to as the strategy of the pursuer with respect to integral constraint if:(1)for any admissible control of the evader 𝑣=𝑣(𝑡),𝑡∈[0,𝑇], the system (2.4) has a unique solution at 𝑢=𝑢(𝑡,𝑣₁(𝑡),𝑣₂(𝑡),…),(2)𝑈(⋅,𝑣(⋅))∈𝑆₁(𝜌).

In a similar way, we define strategy of the pursuer with respect to geometric constraint.

Definition 2.4. One will say that pursuit can be completed in the game 𝐺₁ (resp., 𝐺₂,𝐺₃) from an initial position 𝑧₀, if there exists a strategy of the pursuer to ensure that 𝑧(𝑡)=0 for some 𝑡∈[0,𝑇] and for any admissible control of the evader 𝑣(⋅), where 𝑧(𝑡) is the solution to (2.4).

Definition 2.5. One will say that pursuit can be completed in the game 𝐺₄ from an initial position 𝑧₀, if for arbitrary 𝜀>0, there exists a strategy of the pursuer to ensure that ‖𝑧(𝑡)‖≤𝜀 for some 𝑡∈[0,𝑇] and for any admissible control of the evader 𝑣(⋅), where 𝑧(𝑡) is the solution to (2.4).

Definition 2.6. One will say that evasion is possible in the game 𝐺₁ (resp., 𝐺₂,𝐺₃,𝐺₄) from the initial position 𝑧₀≠0, if there exists a function 𝑣(𝑡)∈𝑆₁(𝜎) (𝑣(𝑡)∈𝑆₂(𝜎),𝑣(𝑡)∈𝑆₂(𝜎),𝑣(𝑡)∈𝑆₁(𝜎)) such that, for arbitrary function 𝑢₀(𝑡)∈𝑆₁(𝜌) (𝑢₀(𝑡)∈𝑆₂(𝜌),𝑢₀(𝑡)∈𝑆₁(𝜌),𝑢₀(𝑡)∈𝑆₂(𝜌)), the solution 𝑧(𝑡) of (2.4) does not vanish, that is, 𝑧(𝑡)≠0 for any 𝑡∈[0,𝑇].
The problem is to find (1)conditions on the initial state 𝑧₀ for which pursuit can be completed for a finite time; (2)conditions for which evasion is possible from any initial position 𝑧₀≠0 in the differential game 𝐺_𝑖, for 𝑖=1,2,3,4.In problems 1 and 2, different forms of constraints on the controls of the players are to be considered.

3. Differential Game Problem

The kth equation in (2.4) has a unique solution of the form 𝑧_𝑘(𝑡)=𝑒^{−𝛼𝑘(𝑡)}𝑧^𝑘0−^𝑡0𝑢_𝑘(𝑠)𝑒^{𝛼𝑘(𝑠)}𝑑𝑠+^𝑡0𝑣_𝑘(𝑠)𝑒^{𝛼𝑘(𝑠)}𝑑𝑠,(3.1) where 𝛼_𝑘(𝑡)=∫^𝑡0𝜆_𝑘(𝑠)𝑑𝑠.

It has been proven in [18] that the solution 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡),…) of (2.4), where 𝑧_𝑘,𝑘=1,2,… defined by (3.1), belongs to the space 𝐶(0,𝑇;𝑙₂).

Let 𝑌=𝑧₀=𝑧₁₀,𝑧₂₀,…∣∃𝑘=𝑗∶𝑧^2𝑗0≤𝜌²4,𝐴^𝑗𝑌(𝑡)≥1₁,(𝑇)=𝑧₀=𝑧₁₀,𝑧₂₀,…∣^∞_𝑘=1𝑧^2𝑘0𝑌₂𝐴_𝑘(𝑇)≤(𝜌−𝜎)²,(𝑇)=z₀=𝑧₁₀,𝑧₂₀,…∣^∞_𝑘=1𝑧^2𝑘0𝑌₃𝐵^2𝑘(𝑇)≤(𝜌−𝜎)²,(𝑇)=𝑧₀=𝑧₁₀,𝑧₂₀,…∣‖‖𝑧₀‖‖+𝜎²𝜀sup_𝑘𝐴_𝑘(𝑇)≤𝜌𝑇,𝜀>0,(3.2) where 𝐴_𝑘(𝑇)=∫^𝑇0𝑒^{2𝛼𝑘(𝑠)}𝑑𝑠 and 𝐵_𝑘(𝑇)=∫^𝑇0𝑒^{𝛼𝑘(𝑠)}𝑑𝑠.

3.1. Pursuit Differential Game

Theorem 3.1. If 𝜌≥𝜎 then from the initial position 𝑧₀∈𝑌₁(𝑇), pursuit can be completed in the game 𝐺₁.

Proof. Let define the pursuer's strategy as 𝑢_𝑘(𝑡)=𝑧^𝑘00,A^𝑘−1(𝑇)𝑒^{𝛼𝑘(𝑡)}+𝑣_𝑘(𝑡),0𝑡>≤𝑇𝑡.≤𝑇,(3.3) The admissibility of this strategy follows from the relations ^∞_𝑘=1^𝑇0𝑢^2𝑘(𝑠)𝑑𝑠^1/2=^∞_𝑘=1^𝑇0||𝑧_𝑘0𝐴^𝑘−1(𝑇)𝑒^{𝛼𝑘(𝑠)}+𝑣_𝑘(𝑠)||²𝑑𝑠^1/2≤^∞_𝑘=1^𝑇0||𝑧_𝑘0𝐴^𝑘−1(𝑇)𝑒^{𝛼𝑘(𝑠)}||+||𝑣_𝑘(𝑠)||²𝑑𝑠^1/2≤^∞_𝑘=1^𝑇0||𝑧_𝑘0𝐴^𝑘−1(𝑇)𝑒^{𝛼𝑘(𝑠)}||²𝑑𝑠^1/2+^∞_𝑘=1^𝑇0||𝑣_𝑘(𝑠)||²𝑑𝑠^1/2≤^∞_𝑘=1=𝜌−𝜎𝑧+^2𝑘0𝜎𝐴=^𝑘−1(𝜌𝑇,)^1/2+𝜎(3.4) here we used the Minkowski inequality and the fact that 𝑧₀∈𝑌₁(𝑇).
Suppose that the pursuer uses the strategy (3.3), one can easily see that for any admissible control of the evader 𝑧_𝑘(𝑇)=0,𝑘=1,2,…, that is, 𝑧_𝑘(𝑇)=𝑒^{−𝛼𝑘(𝑇)}𝑧^𝑘0=𝑒_{−𝛼𝑘(𝑇)}−^𝑇0𝑧_𝑘0𝐴^𝑘−1(𝑇)𝑒^{2𝛼𝑘(𝑠)}𝑑𝑠𝑧_𝑘0−𝑧_𝑘0=0.(3.5) Therefore, pursuit can be completed in the game 𝐺₁. This ends the proof of the theorem.

Theorem 3.2. If 𝜌≥𝜎 then from the initial position 𝑧₀∈𝑌₂(𝑇), pursuit can be completed in the game 𝐺₂.

Proof. We define the pursuer's strategy as 𝑢_𝑘(𝑡)=𝑧^𝑘00,𝐵^𝑘−1(𝑇)+𝑣_𝑘(𝑡),0𝑡>≤𝑇𝑡.≤𝑇,(3.6) The inclusion 𝑢(⋅)∈𝑆₂(𝜌) follows from the relations ^∞_𝑘=1𝑢^2𝑘(𝑡)^1/2=^∞_𝑘=1||𝑧_𝑘0𝐵^𝑘−1(𝑇)+𝑣_𝑘(𝑡)||²^1/2≤^∞_𝑘=1||𝑧_𝑘0𝐵^𝑘−1(𝑇)||+||𝑣_𝑘(𝑡)||²^1/2≤^∞_𝑘=1||𝑧_𝑘0𝐵^𝑘−1(𝑇)||²^1/2+^∞_𝑘=1||𝑣_𝑘(𝑡)||²^1/2=𝜌−𝜎+𝜎=𝜌,(3.7) here we used the Minkowski inequality and the fact that 𝑧₀∈𝑌₂(𝑇).
Suppose that the pursuer uses the strategy (3.6). One can easily see that 𝑧_𝑘(𝑇)=0,𝑘=1,2,…, that is, 𝑧_𝑘(𝑇)=𝑒^{−𝛼𝑘(𝑇)}𝑧^𝑘0=𝑒_{−𝛼𝑘(𝑇)}−^𝑇0𝑧_𝑘0𝐵^𝑘−1(𝑇)𝑒^{𝛼𝑘(𝑠)}𝑑𝑠𝑧_𝑘0−𝑧_𝑘0=0.(3.8) Therefore, pursuit can be completed in the game 𝐺₂. This completes the proof of the theorem.

Theorem 3.3. If 𝜌≥𝜎 and 𝑧₀∈𝑌₁(𝑇) at some 𝑇∈(0,𝑇], then pursuit can be completed in the game 𝐺₃.

Proof. Suppose, as contained in the hypothesis of the theorem, that 𝑧₀∈𝑌₁(𝑇), 𝑇∈(0,1] and let 𝑣₀(𝑡) be an arbitrary admissible control of the evader.
Let the pursuer use the strategy 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…) defined by 𝑢_𝑘(𝑡)=𝑧^𝑘00,𝐴^𝑘−1(𝑇)𝑒^{𝛼𝑘(𝑡)}+𝑣_0𝑘(𝑡),0𝑡>≤𝑇𝑡.≤𝑇,(3.9) Then, using (3.1), we have 𝑧(𝑇)=𝑒^{−𝛼𝑘(𝑇)}𝑧^𝑘0=𝑒_{−𝛼𝑘(𝑇)}−^𝑇0𝑧_𝑘0𝐴^𝑘−1(𝑇)𝑒^{2𝛼𝑘(𝑡)}𝑑𝑠𝑧_𝑘0−𝑧_𝑘0=0.(3.10) We now show the admissibility of the strategy used by the pursuer. From the inclusion 𝑣₀(𝑡)∈𝑆₂(𝜎) we can deduce that ^∞_𝑘=1^𝑇0𝑣^20𝑘(𝑠)𝑑𝑠^1/2≤𝜎√𝑇,(3.11)_∞_𝑘=1^𝑇0𝑢^2𝑘(𝑠)𝑑𝑠^1/2=^∞_𝑘=1^𝑇0||𝑧_𝑘0𝐴^𝑘−1(𝑇)𝑒^{𝛼𝑘(𝑠)}+𝑣_0𝑘(𝑠)||²𝑑𝑠^1/2≤^∞_𝑘=1^𝑇0||𝑧_𝑘0𝐴^𝑘−1(𝑇)𝑒^{𝛼𝑘(𝑠)}||+||𝑣_𝑘(𝑠)||²𝑑𝑠^1/2≤^∞_𝑘=1^𝑇0||𝑧_𝑘0𝐴^𝑘−1(𝑇)𝑒^{𝛼𝑘(𝑠)}||²𝑑𝑠^1/2+^∞_𝑘=1^𝑇0||𝑣_0𝑘(𝑠)||²𝑑𝑠^1/2≤^∞_𝑘=1=𝜌−𝜎𝑧+^2𝑘0𝜎𝐴√^𝑘−1(𝑇)^1/2+𝜎√𝑇𝑇≤𝜌,(3.12) recall that 𝑇∈(0,1] and (3.11). This completes the proof.

Theorem 3.4. For arbitrary 𝜌>0,𝜎>0 and initial position 𝑧₀∈𝑌₃(𝑇), pursuit can be completed in the game 𝐺₄.

Proof. Let 𝑣₀ be an arbitrary admissible control function of the evader. When the pursuer uses the admissible control function 𝑢_𝑘(𝑡)=𝑧_𝑘0𝑇¹⁻¹𝑒^{−𝛼𝑘(𝑡)},𝑘=1,2,…,0≤𝑡≤𝑇₁,(3.13) for time 𝑇₁=‖𝑧₀‖𝜌⁻¹, the solution (3.1) of (2.4) becomes 𝑧_𝑘𝑇₁=𝑒^{−𝛼𝑘(𝑇1)}^𝑇1₀𝑣_0𝑘(𝑠)𝑒^{𝛼𝑘(𝑠)}𝑑𝑠.(3.14)
Then for arbitrary positive number 𝜀, it is obvious that either(1)‖𝑧(𝑇₁)‖≤𝜀, or(2)‖𝑧(𝑇₁)‖>𝜀.If (1) is true then the proof is complete. Obviously 𝑇₁≤𝑇.
Suppose that (1) is not true then (2) must hold. We now assume that 𝑧₀=𝑧(𝑇₁) and repeat previous argument by setting 𝑢_𝑘(𝑡)=𝑧_𝑘0𝑇₁𝑇²⁻¹𝑒^{−𝛼𝑘(𝑡)},𝑘=1,2,…,0≤𝑡≤𝑇₂,(3.15) with time 𝑇₂=‖𝑧(𝑇₁)‖𝜌⁻¹ (we will later prove that the sum of 𝑇_𝑖 is less than or equal to 𝑇). For this step the solution (3.1) becomes 𝑧_𝑘𝑇₁+𝑇₂=𝑒^{−𝛼𝑘(𝑇2)}^𝑇2₀𝑣_0𝑘𝑇₁+𝑠𝑒^{𝛼𝑘(𝑠)}𝑑𝑠.(3.16)
Yet again, we have either of the following cases holding:(1)‖𝑧(𝑇₁+𝑇₂)‖≤𝜀, or(2)‖𝑧(𝑇₁+𝑇₂)‖>𝜀.If (1) holds then the game is completed in the time 𝑇₁+𝑇₂, else we assume 𝑧₀=𝑧₀(𝑇₁+𝑇₂) and repeat the process again and so on.
We now proof a claim that the game will be completed before 𝑛th finite step, where 𝑛=𝜎²sup_𝑘𝐴_𝑘(𝑇)𝜀².(3.17) Note that the existence of the supreme of the sequence 𝐴₁(𝑇),𝐴₂(𝑇),…, follows from the fact that 𝜆₁(𝑡),𝜆₂(𝑡),… is a bounded sequence of continuous functions and 𝑡∈[0,𝑇].
Suppose that it is possible that the game can continue for 𝑛th step. In this case, we must have ^𝑛_𝑖=1𝜎^2𝑖≤𝜎².(3.18) But in the first instance, we have ||𝑧_𝑘𝑇₁||²≤𝑒^{−2𝛼𝑘(𝑇1)}^𝑇1₀𝑣^20𝑘(𝑠)𝑑𝑠^𝑇1₀𝑒^{2𝛼𝑘(𝑠)}𝑑𝑠≤sup_𝑘𝐴_𝑘(𝑇)^𝑇1₀𝑣^20𝑘(𝑠)𝑑𝑠,(3.19) here we used (3.14) and Cauchy-Schwarz inequality.
Therefore, ‖‖𝑧(𝑇₁)‖‖²≤sup_𝑘𝐴_𝑘(𝑇)^∞_𝑘=1^𝑇1₀𝑣^20𝑘(𝑠)𝑑𝑠=sup_𝑘𝐴_𝑘(𝑇)𝜎²¹,(3.20) and by using the assumption that ‖𝑧(𝑇₁)‖>𝜀, we have 𝜎²¹>𝜀²sup_𝑘𝐴_𝑘(𝑇).(3.21) Since the right hand side of this inequality is independent of 𝑛, we can conclude that 𝜎^2𝑛>𝜀²sup_𝑘𝐴_𝑘(𝑇).(3.22) Using this inequality and definition of 𝑛, we have ^𝑛_𝑖=1𝜎^2𝑖>𝑛𝜀²sup_𝑘𝐴_𝑘(𝑇)>𝜎²,(3.23) contradicting (3.18). Hence, pursuit must be completed for the initial position 𝑧₀∈𝑌₃(𝑇) before the 𝑛th step. Furthermore, the pursuit time is given by 𝑇(𝑧₀)=𝑇₁+𝑇₂+⋯+𝑇_𝑛−1, and the inclusion 𝑇(𝑧₀)∈[0,𝑇] is satisfied. Indeed (see (3.20), definition of 𝑛 and that 𝑧₀∈𝑌₃(𝑇)), 𝑇𝑧₀=‖‖𝑧⁰‖‖𝜌+‖‖z𝑇¹‖‖𝜌+⋯+‖‖𝑧𝑇^𝑛−2‖‖≤1𝜌𝜌‖‖𝑧⁰‖‖+sup_𝑘𝐴_𝑘(𝑇)^𝑛−2_𝑖=1𝜎_𝑖≤1𝜌‖‖𝑧⁰≤1‖‖+𝜎(𝑛−2)sup_𝑘𝐴_𝑘(𝑇)𝜌‖‖𝑧⁰‖‖+𝜎²𝜀sup_𝑘𝐴_𝑘(𝑇)≤𝑇.(3.24)
This proves the theorem.

3.2. Evasion Differential Game

Theorem 3.5. If 𝜎−𝜌≥0 then evasion is possible in the game 𝐺₁ from the initial position 𝑧₀≠0.

Proof. Suppose that 𝜎−𝜌≥0,(3.25) and let 𝑢₀(𝑡) be an arbitrary control of the pursuer subjected to integral constraint. We construct the control function of the evader as follows: 𝑣_𝑘(𝑡)=𝐴^𝑗−1/20,(𝑇)𝜌𝑒^{𝛼𝑗(𝑡)},𝑘𝑘≠=𝑗𝑗.,(3.26) This control function belongs to 𝑆₁(𝜎). Indeed, ^∞_𝑘=1^𝑇0𝑣^2𝑘(𝑠)𝑑𝑠=𝐴^𝑗−1(𝑇)𝜌²𝐴_𝑗(𝑇)≤𝜎²(3.27) we have used (3.26) and (3.25).
Our goal now is to show that 𝑧_𝑗(𝑡)≠0 for any 𝑡∈[0,𝑇] as defined by (3.1). Substituting (3.26) into (3.1) and using the Cauchy-Schwartz inequality, we have 𝑧_𝑗(𝑡)≥=𝑧𝑒^{−𝛼𝑗(𝑡)}_𝑗0𝑧^𝑗0+𝜌𝐴_𝑗(𝑡)−𝜌𝐴_𝑗(𝑡)𝑒^{−𝛼𝑗(𝑡)}>0,(3.28) for any 𝑡∈[0,𝑇]. It follows that 𝑧(𝑡)≠0 on the interval [0,𝑇]. Hence, evasion is possible in the game 𝐺₁ from the given initial position 𝑧₀≠0. The proof of the theorem is complete.

Theorem 3.6. Suppose that 𝜎≥𝜌 or there exists a number 𝑘=𝑗 such that 𝑧_0𝑗>0 and 𝜎−𝜌√𝐴_𝑗(𝑇)≥0. Then from the initial position 𝑧₀≠0, evasion is possible in the game 𝐺₂.

Proof. Suppose that 𝜎≥𝜌 and that 𝑧₀≠0. The later condition means that 𝑧_𝑘0≠0 for some 𝑘=𝑗. We construct the control function of the evader as follows: 𝑣_𝑘(𝑡)=𝜌0,,𝑘𝑘≠=𝑗𝑗.,(3.29) It is obvious that this control belongs to the set 𝑆₂(𝜎).
To be definite, let 𝑧_𝑗0>0. Using (3.29) and (3.1), we have 𝑧_𝑗(𝑡)>𝑒^{−𝛼𝑗(𝑡)}𝑧^𝑗0=𝑒_{−𝛼𝑗(𝑡)}+𝜌^𝑡0𝑒^{𝛼𝑗(𝑠)}𝑑𝑠−𝜌^𝑡0𝑒^{𝛼𝑗(𝑠)}𝑑𝑠𝑧_𝑗0>0.(3.30) This means that evasion is possible from the initial position 𝑧₀≠0 in the game 𝐺₂.
We now prove the theorem with the alternative condition. Suppose that there exists a number 𝑘=𝑗 such that 𝑧_0𝑗>0 and 𝜎−𝜌√𝐴_𝑗(𝑇)≥0. Let the control of the evader be as follows: 𝑣_𝑘(𝑡)=⎧⎪⎨⎪⎩√𝑇𝐴_𝑗(𝑇)𝑡+𝑒+𝑇𝜌𝑒^{−𝛼𝑗(𝑡)}0,,𝑘𝑘≠=𝑗𝑗.,(3.31) We show that this control satisfies the geometric constraint: ^∞_𝑘=1𝑣^2𝑘(𝑡)=𝑇𝐴^𝑗(𝑇)(𝑡+𝑒+𝑇)²𝜌²𝑒^{−2𝛼𝑗(𝑡)}≤𝜌²𝐴_𝑗(𝑇)≤𝜎².(3.32)
When the evader uses the control (3.31), the non-vanishing of 𝑧_𝑗(𝑡) in the interval [0,𝑇] for any admissible control of the pursuer 𝑢_𝑖0, can be seen from the following (see (3.1)) 𝑧_𝑗(𝑡)≥𝑒^{−𝛼𝑗(𝑡)}𝑧^𝑗0+𝜌𝑇𝐴_𝑗(𝑇)ln(𝑡+𝑒+𝑇)−𝜌𝑇𝐴_𝑗(𝑇)>0,(3.33) we use the fact that ln(𝑡+𝑒+𝑇)>1 for any 𝑡∈[0,𝑇].
Therefore, 𝑧(𝑡)≠0,𝑡∈[0,𝑇]. This completes the proof of the theorem.

Theorem 3.7. If 𝜎−𝜌√𝑇≥0 then evasion is possible from the initial position 𝑧₀≠0 in the game 𝐺₄.

Proof. Suppose that 𝑧₀≠0 and that 𝜎−𝜌√𝑇≥0. We construct the control function of the evader as follows: 𝑣_𝑘(𝑡)=⎧⎪⎨⎪⎩𝜌𝑇𝐴_𝑗(𝑇)𝑒^{𝛼𝑗(𝑡)}0,,𝑘𝑘≠=𝑗𝑗.,(3.34) We now show that this control satisfies the integral constraint ^∞_𝑘=1^𝑇0𝑣^2𝑘(𝑠)𝑑𝑠=𝜌²𝑇𝐴_𝑗(𝑇)^𝑇0𝑒^{2𝛼𝑗(𝑠)}𝑑𝑠≤𝜌²𝑇≤𝜎².(3.35)
When the evader uses the control (3.34), our task is to show that 𝑧_𝑗(𝑡) does not vanish in the interval [0,𝑇] for any admissible control of the pursuer 𝑢_𝑖0.
For definiteness let 𝑧_𝑗0>0. Substituting (3.34) into (3.1), we have 𝑧_𝑗(𝑡)≥𝑒^{−𝛼𝑗(𝑡)}𝑧^𝑗0+𝜌𝑇𝐴_𝑗(𝑇)𝐴^𝑗(𝑇)−𝜌𝑇𝐴_𝑗(𝑇)=𝑒^{−𝛼𝑗(𝑡)}𝑧_𝑗0>0.(3.36) Therefore, 𝑧(𝑡)≠0,𝑡∈[0,𝑇]. This means that evasion is possible from initial position 𝑧₀≠0 in game 𝐺₄. This ends the proof of the theorem.

Theorem 3.8. If 𝑧₀∈𝑌 and 𝜎≥2𝜌𝑒^{𝛼𝑗(𝑇)} for some 𝑘=𝑗, then from the initial position 𝑧₀≠0 evasion is possible in the game 𝐺₃.

Proof. Suppose that 𝑧₀∈𝑌 and that there exists 𝑘=𝑗 such that 𝜎≥2𝜌𝑒^{𝛼𝑗(𝑇)}.(3.37) We construct the control function of the evader as follows: 𝑣_𝑘(𝑡)=2𝑧^𝑗00,+𝜌𝑒^{𝛼𝑗(𝑡)},𝑘𝑘≠=𝑗𝑗.,(3.38) The inclusion 𝑣(⋅)∈𝑆₂(𝜎) follows from the following ^∞_𝑘=1𝑣^2𝑘(𝑡)=2𝑧_𝑗0+𝜌𝑒^{𝛼𝑗(𝑡)}²≤8𝑧^2𝑗0𝑒^{2𝛼𝑗(𝑡)}+2𝜌²𝑒^{2𝛼𝑗(𝑡)}≤2𝜌²𝑒^{2𝛼𝑗(𝑡)}+2𝜌²𝑒^{2𝛼𝑗(𝑡)}≤4𝜌²𝑒^{2𝛼𝑗(𝑇)}≤𝜎².(3.39) we used (3.34); (3.37) and the inequality (𝑎+𝑏)²≤2𝑎²+2𝑏².
Let the evader use the control (3.34) and for definiteness let 𝑧_𝑗0>0. Using (3.1) and the Cauchy-Schwartz inequality, we have 𝑧_𝑗(𝑡)≥=𝑒𝑒^{−𝛼𝑗(𝑡)}_{−𝛼𝑗(𝑡)}𝑧^𝑗0−𝜌𝐴_𝑗(𝑡)+2𝑧_𝑗0𝐴_𝑗(𝑡)+𝜌𝐴_𝑗(𝑡)𝑧_𝑗0+2𝑧_𝑗0A_𝑗(𝑡)>0.(3.40) Therefore, we have 𝑧_𝑗(𝑡)>0,0≤𝑡≤𝑇, that is, evasion is possible in the game 𝐺₃. This ends the proof of the theorem.

4. Conclusion

This paper is closely related to [10, 11]. However, the game model considered in this paper is a better generalization to the one in the last cited papers. The constant coefficients of the game model considered in the cited papers are specific to function coefficients considered in this papers. Sufficient conditions for which pursuit can be completed and for which evasion is possible with various form of constraints on the control of the players have been established.

For future works, optimal pursuit and multiplayers game problems described by the model considered in this paper can be investigated. As there are four different possible combinations of geometric and integral constraints on the control functions of the two players of the game, there would be four different problems to be studied.

Acknowledgment

The authors wish to express the deepest appreciation to the reviewers for their valuable comments and observations. This research was partially supported by the Research Grant (RUGS) of the Universiti Putra Malaysia, no. 05-04-10-1005RU.