^{1}

^{2}

^{3}

This article is devoted to studying of the problem of prosecution described by differential equations of a fractional order. It has received sufficient conditions of a possibility of completion of prosecution for such operated systems.

The dynamics of the systems described by the equations of fractional order is the subject of research experts from around the middle of the XX century. The study of dynamical systems with fractional order management is actively developing in the last 5 - 8 years [

Let driving of object in a finite-dimensional Euclidean space of

where

Besides in space

belongs to a set M at the time of

This work is dedicated to the receipt of sufficient conditions for the completion of the prosecution managed fractional order systems adjacent to the study [

Let’s pass to the formulation of the main results. Everywhere further: 1) the terminal set M has an appearance

Let

Theorem 1. If in game (1) at some

That from initial situation

Let now the

Theorem 2. If in game (1) at some

That from initial situation

Let’s designate through

integral

Theorem 3. If in game (1) at some

that from initial situation

Proof of the theorem 1. Two cases are possible: 1)

theorem condition

Considering this equality, we will consider the equation

Relatively

Really, on (9) for the decision

we have ( [

As

Proof of the theorem 2. In view of a case triviality we will begin

We have (see (5), (6))

initial set

where the

Let

Thus, for the arbitrary function

Then

Further we argue similarly. As

Let’s receive

For the arbitrary measurable function

Follows from a ratio (16) that

etc. It is clear, that there is a natural number j it that: 1)

But

Therefore ((18), (19))

Similarly on formulas (18), (19), (20) finally we receive

Thus, for any point

Proof of the theorem 3. Owing to a condition of the theorem (8) we have

Let

From here owing to a condition of measurability existence of the measurable functions

We will determine function by the found measurable function

For the decision

From here

Summarizing the results, we conclude that the differential game of pursuit of fractional order (1), starting from the position can be completed in time, respectively. Thus, to solve the game problem kind of persecution (1), we used a derivative of fractional order Caputo, which is determined by the expression (2). Many (3) analogue of the so-called first integral Pontryagin, including (4) gives the first sufficient condition for the possibility of the persecution of the task. Many (5)―an analog of the second integral Pontryagin, inclusion (6) gives the second sufficient condition for the possibility of the persecution of the task. Lots (7)―analogue N. Satimova third method, and the inclusion (8) gives a sufficient condition for the third opportunity to end the game. In Theorems 1 - 3, we obtain sufficient conditions for the solution of relevant problems in this form.

Mashrabjan Mamatov,Durdimurod Durdiev,Khakim Alimov, (2016) On the Theory of Fractional Order Differential Games of Pursuit. Journal of Applied Mathematics and Physics,04,1578-1584. doi: 10.4236/jamp.2016.48167