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We consider a linear pursuit differential game of one pursuer and one evader. Controls of the pursuer and evader are subjected to integral and geometric constraints, respectively. In addition, phase constraint is imposed on the state of evader, whereas pursuer moves throughout the space. We say that pursuit is completed, if inclusion

Pursuit and evasion differential game problems under different type of constraints on controls of players were studied by many researchers, for example, [

The present paper is continuation of [

In the present paper, the following technique is proposed to control the phase vector of the pursuer: first-phase vector of the pursuer is brought to a given bounded, convex set, regardless of the behavior of evading player and then, using the evasion control and initial positions of the players, the pursuit control that ensures the completion of pursuit is constructed.

New sufficient conditions of completion of pursuit from all initial positions for a finite time have been obtained and results have been illustrated on a model example. This paper is closely related to [

Consider a linear differential game of one pursuer and one evader described by

The terminal set has the form

Measurable functions

One says that pursuit can be completed from initial points

As it is known, if

Find conditions of completion of pursuit in game (

Let

There exist numbers

There exist continuous function

for each pair of points

where

It follows from Assumption

For any

Let Assumptions

Let

Therefore,

Next, assuming that

Let us turn to case (ii). By Assumption

Next, we show admissibility of control (

There exist numbers

Assumptions

Let Assumptions

Let

Let

Thus, it has been proven that control (

Admissibility of pursuit control (

From Theorems

Let, for point

Let, for point

Thus, Corollaries

Consider a simple motion differential game:

It is not difficult to verify that if

We have proposed a technique to solve pursuit problem which consists of two steps: first-step state of the pursuer is brought to a given bounded convex set, regardless of the behavior of evading player and then, in the second step, pursuit is completed.

New sufficient conditions of completion of pursuit from all initial positions for a finite time have been obtained. Moreover, results have been illustrated on a model example.

Note that Assumptions

The authors declare that there are no competing interests regarding the publication of this paper.