Pursuit Game for an Infinite System of First-Order Differential Equations with Negative Coefficients

Abstract

We consider a pursuit differential game described by an infinite system of 1st-order differential equations with negative coefficients in Hilbert space. The control functions of players are subject to integral constraints. The pursuer attempts to bring the system from a given initial state to another state for a finite time and the evader’s purpose is opposite. We obtain a condition of completion of pursuit when the control resource of the pursuer is greater than that of the evader. We study a control problem as well.

1 Introduction

Differential games in finite dimensional Euclidean spaces were studied by many researchers and developed important methods (see, for instance, [10, 25, 28, 30, 36, 37].)

There are mainly two constraints on control functions of players: geometric and integral constraints. In-views of the amount of works been done in developing the differential games, the integral constraints have been extensively discussed by many researchers with various approaches (see, for example, [4, 5, 8, 11, 12, 18,19,20,21, 26, 27, 29, 31, 34, 35, 39, 42,43,44] ).

One of the powerful tools in studying the control and differential game problems in systems with distributed parameters is the decomposition method. Using this method the control or differential game problem is reduced to ones described by infinite systems of differential equations (see, for example, [2, 6, 7, 9, 13, 32, 40, 41, 45, 46]). We demonstrate briefly the method for the following parabolic equation

$$\begin{aligned} \frac{\partial z(x,t)}{\partial t} + Az(x,t) = w(x,t), \ \ z(x,0) = z_0(x), \end{aligned}$$

(1)

where $0 \le t \le T$, T is a given positive number, $x= (x_1,\ldots , x_n) \in \varOmega \subset R^n$, $n \ge 1$, $\varOmega$ is a bounded set with piecewise smooth boundary,

$$ Az=-\sum \limits ^n_{i,j=1}\frac{\partial }{\partial {x_i} }\left( a_{ij}(x)\frac{\partial z}{\partial {x_j}}\right) . $$

$a_{ij}(x)= a_{ji}(x)$, $x \in \varOmega$, and, for some $c>0$ and for all

$$(\xi _1,\ldots , \xi _n) \in R^n, x \in \varOmega , \sum \limits _{i,j = 1}^n a_{ij}(x)\xi _i\xi _j \ge c \sum \limits _{i= 1}^n\xi _i^2. $$

The domain of the operator A is the space of twice continuously differentiable functions with compact support in $\varOmega$, denoted by ${\mathop {C^2}\limits ^{\circ }}(\varOmega )$. Define inner product

$$ (z,y)_A = (Az,y), \ \ z,y \in {\mathop {C^2}\limits ^{\circ }}(\varOmega ). $$

Then ${\mathop {C^2}\limits ^{\circ }}(\varOmega )$ becomes incomplete Euclidean space. To obtain a complete Hilbert space associated with the operator A, we complete the space ${\mathop {C^2}\limits ^{\circ }}(\varOmega )$ with respect to the norm $||z||_A = \sqrt{(Az,z)}$, $z \in {\mathop {C^2}\limits ^{\circ }}(\varOmega )$. We use the fact that the operator A has countably many eigenvalues

$$ \lambda _1, \lambda _2, \ldots , \ \ 0 < \lambda _1 \le \lambda _2 \le \ldots , \ \ \lim \limits _{k \rightarrow \infty }\lambda _k = +\infty , $$

and generalized eigenfunctions $\varphi _1, \varphi _2, \ldots$, which is a complete orthonormal system in $L_2(\varOmega )$ [33].

Next, let $C(0,T; H_r(\varOmega ))$ and $L_2(0,T; H_r(\varOmega ))$ denote the spaces of continuous and measurable functions defined on [0, T] with the values in

$$ H_r(\varOmega ) = \left\{ f \in L_2(\varOmega ) \mid \ f = \sum \limits _{i=1}^{\infty }\alpha _i\varphi _i, \ \ \sum \limits _{i=1}^{\infty }\lambda _i^r\alpha _i^2 < \infty \right\} , $$

respectively, where r is a given number. The space $H_r(\varOmega )$ is a Hilbert space with inner product and norm defined as follows: if

$$ f = \sum \limits _{i=1}^{\infty }\alpha _i\varphi _i \in H_r(\varOmega ), \ \ g = \sum \limits _{i=1}^{\infty }\beta _i\varphi _i \in H_r(\varOmega ), $$

then

$$ (f,g) = \sum \limits _{i=1}^{\infty }\lambda _i^r\alpha _i\beta _i, \ \ ||f|| = \left( \sum \limits _{i=1}^{\infty }\lambda _i^r\alpha ^2_i \right) ^{1/2}. $$

It was proved [2] that if $w(\cdot ) \in L_2(0,T; H_r(\varOmega ))$, then the initial value problem (1) has a unique solution $z(\cdot ) \in C(0,T; H_{r+1}(\varOmega ))$. Next, represent the functions z(x, t) and w(x, t) as

$$ z(x,t)=\sum \limits _{k=1}^{\infty }z_k(t)\varphi _k(x), \ \ w(x,t)=\sum \limits _{k=1}^{\infty }w_k(t)\varphi _k(x), \ \ z_k(\cdot ), w_k(\cdot ) \in L_2(0,T), \ \ k=1,2,\ldots , $$

and substitute them into the Eq. (1), and then equate the coefficients at $\varphi _k(x)$ to obtain

$$ \dot{z}_k + \lambda _kz_k= w_k, \ \ z_k(0) = z_{k0}, \ k=1,2,\ldots , $$

where $w_k, z_k, z_{k0} \in R^1$, $k=1,2,\ldots$, $w_k$, are control parameters, $z_{k0} = (z_0, \varphi _k)$. Thus, we have obtained an infinite system of differential equations. Usually, the control function is subjected to geometric or integral constraint. The geometric and integral constraints for the control function $w \in H(0,T; H_r(\varOmega ))$ of the form

$$ ||w(x,t)|| \le \rho , \ \ \int \limits _0^{T}||w(x,t)||^2dt \le \rho ^2, $$

respectively, can be written as follows

$$ \left( \sum \limits _{k=1}^{\infty }\lambda _k^rw^2_k(t)\right) ^{1/2} \le \rho , \ \ \sum \limits _{k=1}^{\infty }\lambda _k^r\int \limits _0^{T}w^2_k(t)dt \le \rho ^2, $$

respectively.

Hence, there is an important connections between control problems described by PDE and those described by infinite system of differential equations. Control and differential game problems described by infinite system of differential equations are of independent interest and can be investigated within one theoretical framework independently of those described by PDE assuming that the coefficients $\lambda _k$, $k=1,2,\ldots$, are any real numbers. Of course, in the case where $\lambda _k$ are any real numbers, we must give adequate definitions of state space, solution of infinite system of differential equations. Also, we have to prove the existence-uniqueness of solution in the state space.

There are several works devoted to control or differential game problems described by infinite system of differential equations (see, for example, [1, 3, 14, 16, 17, 22,23,24, 38]).

In the paper [14] a differential game problem described by the following infinite system of differential equations

$$\begin{aligned} \dot{z}_{k} + \lambda _kz_{k}= -u_{k}+v_k, \ z_{k}(0)=z_{k0}, \ k=1,2,\ldots , \end{aligned}$$

(2)

where $z_{k}, u_{k}, v_k \in \mathbb {R}^1$, and $\lambda _k$, $k=1,2,\ldots$, are positive numbers, was studied when integral constraints are subjected to control functions of the players.

In the present paper, we study a pursuit differential game problems described by (2) in the case of negative coefficients $\lambda _k$, $k=1,2,\ldots$. Pursuer tries to bring the state of the system from an initial state $z^0$ to another given one $z^1$ for a finite time. Previous studies of differential games described by infinite system of differential equations have only dealt with the case $z^1=0$. We obtain sufficient conditions of completion of pursuit.

2 Statement of Problem

Consider the following Hilbert space

$$ l^2_r=\left\{ \alpha =(\alpha _1,\alpha _2,\ldots )|\ \sum \limits ^\infty _{k=1}|\lambda _k|^r\alpha ^2_k < \infty \right\} , $$

where, r is a real number and $\lambda _1,\lambda _2,\ldots$, is a bounded sequence of negative numbers, with inner product and norm defined by

$$ \langle \alpha , \beta \rangle _r = \sum \limits _{k = 1}^{\infty }|\lambda _k|^r \alpha _k\beta _k, \ \alpha , \ \beta \in l^2_r, \ \ ||\alpha || = \left( \sum \limits _{k = 1}^{\infty } |\lambda _k|^r\alpha _k^2 \right) ^{1/2}. $$

Let

$$ L_2(0, T, l^2_r)=\left\{ w(\cdot )=(w_1(\cdot ), w_2(\cdot ),\ldots )|\ \Vert w(\cdot )\Vert _{L_2(0, T, l^2_r)} < \infty ,\ w_k(\cdot ) \in L_2(0, T) \right\} , $$

where $T>0$ is a given sufficiently big number,

$$ \Vert w(\cdot )\Vert _{L_2(0, T, l^2_r)}=\left( \sum \limits ^\infty _{k=1}|\lambda _k|^r \int _0^Tw^2_k(t)dt \right) ^{1/2}, $$

We examine control and pursuit differential game problems described by the following infinite system of differential equations

$$\begin{aligned} \dot{z}_k + \lambda _kz_k= -u_k+v_k, \ z_k(0) = z^0_k, \ k = 1, 2, \dots , \end{aligned}$$

(3)

where $z_k, u_k, v_k \in \mathbb {R}^1, \ k = 1, 2, \dots$; $u = (u_1, u_2, \dots )$ is the control parameter of pursuer and $v = (v_1, v_2, \dots )$ is that of evader, $z^0 = (z^0_1, \ z^0_2, \dots ) \in l^2_{r+1}$.

Let

$$ S(\rho _0) =\left\{ w(\cdot ) \in L_2(0, T, l^2_r)|\ \Vert w(\cdot )\Vert _{L_2(0, T, l^2_r)} \le \rho _0 \right\} , $$

where $\rho _0$ is a given positive number.

Definition 1

Functions $w(\cdot ) \in S(\rho _0)$, $u(\cdot ) \in S(\rho )$, and $v(\cdot ) \in S(\sigma )$ are called admissible control, admissible control of pursuer, and admissible control of evader, respectively, where $\rho$ and $\sigma$ are given positive numbers.

It’s assumed that $\rho > \sigma$.

Definition 2

Let $w(\cdot ) \in S(\rho _0)$. A function $z(t)=(z_1(t), z_2(t), \dots )$, $0 \le t \le T$, with $z_k(0)=z^0_k$, $k = 1, 2, \dots$, is called solution of the initial value problem

$$\begin{aligned} \dot{z}_k(t) + \lambda _kz_k(t) = w_k(t), \ z_k(0)=z^0_k, \ k = 1, 2, \dots , \end{aligned}$$

(4)

if $z_k(t)$, $k=1,2,\ldots$, are absolutely continuous and almost everywhere on [0, T] satisfy the Eq. (4).

Let $C (0, T; l^2_{r+1})$ be the space of continuous functions $z(t)=(z_1(t),z_2(t),\ldots ) \in l^2_{r+1}$ defined on $[0,\ T]$. We need the following proposition [15].

Proposition 1

If $w(\cdot ) \in S(\rho )$, then infinite system of differential equations (4) has the only solution $z(t)=(z_1(t),z_2(t),\ldots )$, $0 \le t \le T$, in the space $C (0, T; l^2_{r+1})$, where

$$ z_k(t) =e^{\beta _kt} \left( z^0_k + \int _0^tw_k(s)e^{-\beta _ks}ds \right) ,\ k=1,2,\ldots , $$

with $\beta _k = -\lambda _k >0$.

Note that this existence-uniqueness theorem for the system (4) was proved for any finite interval $[0,\ T]$. Therefore, we investigate the system (3) and (4) on $[0,\ T]$.

Definition 3

A function

$$ U(t, v)=(U_1(t, v), U_2(t, v),\ldots ), \ \ U: [0,\ T] \times l_r^2 \rightarrow l_r^2, $$

with the components of the form

$$ U_k(t, v)=w_k(t)+v_k(t), \ k=1,2,\ldots , $$

is referred to as the strategy of pursuer, if, for any admissible control of evader $v(\cdot )=(v_1(\cdot ),v_2(\cdot ),\ldots )$, the system (3) has the only solution at $u(t)=U(t, v)$, where $w(\cdot )=(w_1(\cdot ),w_2(\cdot ),\ldots ) \in S(\rho -\sigma )$.

We are given another state $z^1 = (z^1_1, \ z^1_2, \dots ) \in l^2_{r+1}$.

Definition 4

We say that the game (3) can be completed for the time $\theta$ ($\theta \le T$), if there exists a strategy U of pursuer such that, for any admissible control of evader, $z(\tau ) = z^1$ at some time $\tau$, $0 \le \tau \le \theta$.

Pursuer tries to bring the state of the system (3) from $z^0$ to $z^1$, and the purpose of evader is opposite. Formulate the problems.

Problem 1

Find a condition on the states $z^0, z^1 \in l^2_{r+1}$ such that the state z(t) of the system (4) can be transferred from the initial position $z^0$ to the final position $z^1$ for a finite time.

Problem 2

Find a condition on the states $z^0, z^1 \in l^2_{r+1}$, for which pursuit can be completed in the game (3) for a finite time.

3 Control Problem

In this section, we study a control problem for transferring the system z(t) from the initial position $z^0$ to the final position $z^1$.

For the system (4), we study the control problem: find a time $\theta$ such that

$$\begin{aligned} z(0)=z^0,\ z(\theta )=z^1. \end{aligned}$$

(5)

First, we analysis the following series

$$\begin{aligned} E(t) = E_1(t) + E_2(t), \ t > 0, \end{aligned}$$

(6)

where

$$\begin{aligned} E_1(t) = 2\sum \limits _{k = 1}^{\infty }\beta _k^r |z^0_k|^2\phi _k(t), \ \ E_2(t) = 2\sum \limits _{k = 1}^{\infty }\beta _k^r|z^1_k|^2\psi _k(t), \end{aligned}$$

(7)

$$ \phi _k(t) = \frac{2\beta _k}{1 - e^{-2\beta _kt}},\ \psi _k(t)= \frac{2\beta _k}{e^{2\beta _kt} - 1}, \ k=1,2,\ldots . $$

Lemma 1

Let $z^0, z^1 \in l^2_{r+1}$. If, in addition, $z^0, z^1 \in l^2_r$, then the series E(t) converges at any $t > 0$.

Proof

Let $z^0, z^1 \in l_r^2$. To show that the series (6) converges, we show that the series $E_1(t)$ and $E_2(t)$ converge. Since $\beta _k$ is a bounded sequence of positive numbers, therefore $\beta = \sup \limits _k\beta _k < \infty$. Since $\beta _k \le \beta$, then it is not difficult to show that

$$ \phi _k(t) = \frac{2\beta _k}{1 - e^{-2\beta _kt}} \le \frac{2\beta }{1 - e^{-2\beta t}}, $$

which implies that

$$\begin{aligned} E_1(t) \le \frac{4\beta }{1 - e^{-2\beta t}}\sum \limits _{k = 1}^{\infty }\beta _k^r|z^0_k|^2. \end{aligned}$$

The series on the right hand side of this inequality is convergent since $z^0 \in l_r^2$. Thus, the series $E_1(t)$ is convergent.

We can see that $\psi _k(t) \le \frac{1}{t}$, $t>0$, $k=1,2,\ldots$. Then

$$\begin{aligned} E_2(t) \le \frac{2}{t}\sum \limits _{k = 1}^{\infty }\beta _k^r|z^1_k|^2. \end{aligned}$$

The series on the right hand side of this inequality is convergent since $z^1 \in l_r^2$. Thus, the series $E_2(t)$ is convergent. This completes the proof of Lemma 1.

We’ll need some properties of E(t).

Property 1

E(t) has the following properties:

• (i) E(t) is decreasing on $(0,\ +\infty )$;

• (ii) $E(t) \rightarrow +\infty$ as $t \rightarrow 0^+$;

• (iii) $E(t) \rightarrow 4\sum \limits _{k = 1}^{\infty }\beta _k^{r +1} |z^0_k|^2$ as $t \rightarrow +\infty$.

Proof

The first property follows from the fact that $\psi _k(t)$ and $\phi _k(t)$, $k=1,2,\ldots$, are decreasing on $(0,\ +\infty )$.

The proof of the property (ii) follows from the observations that $\psi _k(t) \rightarrow +\infty$ and $\phi _k(t) \rightarrow +\infty$, as $t \rightarrow 0^+$ for each k.

Finally, we prove the property (iii). According to Lemma 1, E(t) is convergent for any $t > 0$. We fix $t_0 > 0$. Since $E(t_0)$ is convergent, then for any $\varepsilon > 0$, there exists a positive integer N such that

$$\begin{aligned} F(t_0)= \sum \limits _{k = N+1}^{\infty }\beta _{k}^{r} \left( 2|z^0_k|^2\phi _{k}(t_0) + 2|z^1_k|^2\psi _{k}(t_0) \right) < \frac{\varepsilon }{3}, \end{aligned}$$

(8)

and also

$$\begin{aligned} \sum \limits _{k =N+1}^{\infty }4\beta _k^{r +1} |z^0_k|^2< \frac{\varepsilon }{3} \end{aligned}$$

(9)

since $z^0 \in l^2_{r+1}$. Then, $F(t)< \frac{\varepsilon }{3}$ for all $t \ge t_0$ since the functions $\psi _{k}(t)$ and $\phi _{k}(t)$ are decreasing on $(0, +\infty )$ for each k.

On the other hand, there exists number $T_1 > 0$ such that, for all $t > T_1$,

$$\begin{aligned} \left| 2\sum \limits _{k = 1}^{N}\beta _{k}^{r} \left( |z^0_k|^2\phi _{k}(t) + |z^1_k|^2\psi _{k}(t)\right) -4\sum \limits _{k = 1}^{N}\beta _{k}^{r +1} |z^0_k|^2\right| < \frac{\varepsilon }{3}, \end{aligned}$$

(10)

since the sum consists of a finite number of summands and

$$ \lim _{t \rightarrow +\infty }\phi _{k}(t)=2\beta _k, \ \ \lim _{t \rightarrow +\infty }\psi _{k}(t)=0, \ \ k =1,2,\ldots $$

Thus, by (8)–(10)

$$\begin{aligned} \left| E(t) -4\sum \limits _{k = 1}^{\infty }\beta _{k}^{r +1}|z^0_k|^2\right|&\le \left| 2\sum \limits _{k = 1}^{N}\beta _{k}^{r}\left( |z^0_k|^2\phi _{k}(t) + |z^1_k|^2\psi _{k}(t) \right) -4\sum \limits _{k = 1}^{N}\beta _{k}^{r +1}|z^0_k|^2\right| \\&+ 2\sum \limits _{k =N+1}^{\infty }\beta _{k}^{r} \left( |z^0_k|^2\phi _{k}(t) + |z^1_k|^2\psi _{k}(t) \right) + 4\sum \limits _{k =N+1}^{\infty }\beta _{k}^{r +1}|z^0_k|^2\\&< \frac{\varepsilon }{3} + \frac{\varepsilon }{3} + \frac{\varepsilon }{3} = \varepsilon . \end{aligned}$$

This proves property (iii).

Next since $\dfrac{4}{1 - e^{-2\beta _kt}} >4$, $t > 0$, therefore we obtain from (i) and (iii) that

$$\begin{aligned} E(t)> 4\sum \limits _{k = 1}^{\infty }\beta _k^{r+1}|z^0_k|^2, \ t > 0. \end{aligned}$$

(11)

Property 1 and (11) imply that the equation

$$\begin{aligned} E(t) =\rho ^2_0 \end{aligned}$$

(12)

has a root $t=\theta$ if and only if

$$\begin{aligned} \rho ^2_0 > 4\sum \limits _{k = 1}^{\infty }\beta _k^{r+1}|z^0_k|^2, \end{aligned}$$

(13)

and this root is unique. Without loss of generality, we can assume that $\theta < T$ since T is sufficiently big number.

The following statement is a solution for the control problem (5).

Theorem 1

Let inequality (13) be satisfied and $z^0, z^1 \in l^2_r$. Then the system (4) can be transferred from the initial position $z^0$ to the position $z^1$ for the time $\theta$.

Proof

Define a control

$$\begin{aligned} w_k(t) = \left\{ \begin{array}{ll}-\left[ z^0_k-z^1_ke^{-\beta _k{\theta }} \right] \phi _k(\theta ){e^{-\beta _kt}}, &{} 0\le t \le \theta \\ 0, &{} t > \theta \end{array}\right. , \ \ k=1,2,\ldots . \end{aligned}$$

(14)

Show that this control is admissible. Using Eq. (12), control (14), and the obvious inequality $|x-y|^2 \le 2|x|^2 + 2|y|^2$, we proceed as follows:

$$\begin{aligned} \sum \limits _{k = 1}^{\infty }\beta _k^r\int _0^{\theta }|w_k(s)|^2ds&= \sum \limits _{k = 1}^{\infty }\beta _k^r\int _0^{\theta } \left| -\left[ z^0_k-z^1_ke^{-\beta _k{\theta }}\right] \phi _k(\theta ){e^{-\beta _ks}}\right| ^2ds \\&\le \sum \limits _{k = 1}^{\infty }\beta _k^r \left( 2|z^0_k|^2+2|z^1_k|^{2}e^{-2\beta _k{\theta }} \right) \phi _k^2(\theta ) \int _0^{\theta }{e^{-2\beta _ks}}ds \\&= 2\sum \limits _{k = 1}^{\infty }\beta _k^r \left( |z^0_k|^2\phi _k(\theta ) + |z^1_k|^2\psi _k(\theta ) \right) \\&=E(\theta ) = \rho ^2_0. \end{aligned}$$

Show that the system can be transferred from $z^0$ to $z^1$ for the time $\theta$. Indeed,

$$\begin{aligned} z_k(\theta )&= e^{\beta _k\theta }\left( z^0_k-\left[ z^0_k-z^1_ke^{-\beta _k{\theta }} \right] \phi _k(\theta ) \int _0^{\theta } e^{-2\beta _ks}ds\right) \\&=e^{\beta _k\theta }(z^1_ke^{-\beta _k\theta }) = z^1_k. \end{aligned}$$

This completes the proof of Theorem 1.

4 Pursuit Differential Game Problem

In this section, we study pursuit differential game described by the Eq. (3). It is assumed that control resources of pursuer is greater than that of evader, that is $\rho >\sigma$.

We obtain from (3) that

$$\begin{aligned} z_k(t) =e^{\beta _kt} \left( z^0_k - \int _0^tu_k(s)e^{-\beta _ks}ds + \int _0^t v_k(s)e^{-\beta _ks}ds \right) . \end{aligned}$$

(15)

In view of the previous section we can state that the equation

$$\begin{aligned} E(t) = 2\sum \limits _{k = 1}^{\infty }\beta _k^r \left( |z^0_k|^2\phi _k(t) + |z^1_k|^2\psi _k(t) \right) = (\rho -\sigma )^2 \end{aligned}$$

(16)

has a root $t=\theta _1$ if and only if

$$\begin{aligned} (\rho -\sigma )^2 > 4\sum \limits _{k = 1}^{\infty }\beta _k^{r+1}|z^0_k|^2, \end{aligned}$$

(17)

and this root is unique. We can assume, by selecting T if needed that $\theta _1 < T$.

Theorem 2

Let (17) be satisfied and $z^0, z^1 \in l^2_r$. Then pursuit can be completed in the game (3) for the time $\theta _1$.

Proof

Construct a strategy for the pursuer. Set

$$\begin{aligned} u_k(t, v) = \left\{ \begin{array}{ll} \left[ z^0_k-z^1_ke^{-\beta _k{\theta _1}}\right] \phi _k(\theta _1) e^{-\beta _ks}+ v_k(t), &{} 0\le t \le \theta _1\\ 0, &{} t >\theta _1 \end{array}\right. , \ \ \textit{k}=1,2,\ldots \end{aligned}$$

(18)

Show that strategy (18) is admissible. Applying the Minkowskii inequality, we have

$$\begin{aligned} \left( \sum \limits _{k = 1}^{\infty }\beta _k^r\int _0^{\theta _1}\left| u_k(s) \right| ^{2}ds\right) ^{1/2}&= \left( \sum \limits _{k = 1}^{\infty }\beta _k^r\int _0^{\theta _1} \left| \left( z^0_k-z^1_ke^{-\beta _k{\theta _1}}\right) \phi _k(\theta _1)e^{-\beta _ks} + v_k(s) \right| ^2 ds \right) ^{1/2}\nonumber \\&\le \left( \sum \limits _{k = 1}^{\infty }\beta _k^r\int _0^{\theta _1} \left| \left( z^0_k-z^1_ke^{-\beta _k{\theta _1}}\right) \phi _k(\theta _1)e^{-\beta _ks}\right| ^2ds \right) ^{1/2}\nonumber \\&+ \left( \sum \limits _{k = 1}^{\infty }\beta _k^r\int _0^{\theta _1}\left| v_k(s) \right| ^{2}ds\right) ^{1/2}\nonumber \\&\le \left( \sum \limits _{k = 1}^{\infty }\beta _k^r |z^0_k-z^1_ke^{-\beta _k{\theta _1}} |^2 \phi _k^2(\theta _1) \int _0^{\theta _1}{e^{-2\beta _ks}}ds \right) ^{1/2} + \sigma . \end{aligned}$$

(19)

Using the obvious inequality $|x-y|^2 \le 2|x|^2+2|y|^2$ and Eq. (16), we obtain form (19) that

$$\begin{aligned} \left( \sum \limits _{k = 1}^{\infty }\beta _k^r\int _0^{\theta _1}\left| u_k(s) \right| ^{2}ds\right) ^{1/2}&\le \left( 2\sum \limits _{k = 1}^{\infty }\beta _k^r \left( |z^0_k|^2\phi _k(\theta _1) + |z^1_k|^2\psi _k(\theta _1) \right) \right) ^{1/2} + \sigma \\&=E^{1/2}(\theta _1)+ \sigma \\&=\rho - \sigma + \sigma = \rho . \end{aligned}$$

Thus the strategy (18) is admissible.

Next, we show that pursuit is completed at the time $\theta _1$. Indeed, using (15) and strategy (18), we have

$$\begin{aligned} z_k(\theta _1)&=e^{\beta _k\theta _1} \left( z^0_k - \int _0^{\theta _1} \left( \left( z^0_k-z^1_ke^{-\beta _k{\theta _1}}\right) \phi _k(\theta _1){e^{-\beta _ks}} + v_k(s) \right) e^{-\beta _ks}ds + \int _0^{\theta _1} v_k(s)e^{-\beta _ks}ds \right) \\&=e^{\beta _k\theta _1} \left( z^0_k- \int _0^{\theta _1} \left( z^0_k-z^1_ke^{-\beta _k{\theta _1}} \right) \phi _k(\theta _1) e^{-2\beta _ks}ds \right) \\&=e^{\beta _k\theta _1} \left( z^0_k- z^0_k+z^1_ke^{-\beta _k{\theta _1}} \right) =z^1_k. \end{aligned}$$

The proof of the theorem is completed.

5 Conclusion

We have studied a pursuit differential game problem described by infinite system of 1st-order differential equations with negative coefficients in the space $l^2_{r+1}$. The control functions of players are subjected to integral constraints.

We have obtained a condition for which a control problem is solvable, also we have constructed a control that transfers the system from an initial state $z^0$ to the final state $z^1$ for a finite time.

We have obtained a condition of completion of pursuit in the differential game. Moreover, a pursuit strategy has been constructed.