On the Theory of Fractional Order Differential Games of Pursuit

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.


Introduction
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 [1] [2].The growing interest in these areas is due to two main factors.Firstly, by the middle of the last century it has been adequately worked out the mathematical foundations of fractional integro-differential calculus and the theory of differential equations of fractional order.Around the same time, it began to develop a methodology and application of fractional calculus in applications, and we started to develop numerical methods for calculating integrals and derivatives of fractional order.Secondly, in fundamental and applied physics by this time, it had accumulated a considerable amount of results, which showed the need for fractional calculus apparatus for an adequate description of a number of real systems and processes [3].Examples of real systems will mention electrochemical cells, capacitors fractal electrodes, the viscoelastic medium.These systems have typically not trivial physical properties useful from a practical standpoint [4]- [7].For example, the irregular structure of the electrodes in capacitors allows them to reach a much larger capacity, and the use of electrical circuits with elements having a transfer characteristic of fractional-power type, provides more flexible configuration of fractional order controllers used in modern control systems.For such control systems of fractional order as of today, there are no similar results Pontryagin type [8]- [11].

Methods
Let driving of object in a finite-dimensional Euclidean space of n R be described by a differential equation of a fractional order of a look ( ) where , 1; × constant matrixes, , u υ -the operating parameters, u -the operating parameter of the pur- suing player, , υ -the operating parameter of the running-away player,

( )
f t -known measurable vector function.We will understand a fractional derivative as left-side frac- tional derivative Kaputo [1]- [6].Let's remind that fractional derivative Kaputo of the random inappropriate order 0 Besides in space n R the terminal set M is allocated.The running-away player seeks to place the aim of the pursuing player to bring z to a set M, to it.The problem of prosecution about rapprochement of a trajectory of the conflict operated system (1) with a terminal set M for terminating time from the standard initial positions 0 z is considered.Let's say that differential game (1) can be finished from initial situation 0 z during ( ) belongs to a set M at the time of t T = at any measurable functions ( ) ( ) . This work is dedicated to the receipt of sufficient conditions for the completion of the prosecution managed fractional order systems adjacent to the study [12]- [22].Some results of this paper were announced at the International Labour Conference [16] [17].In such a setting the pursuit problem was studied in [8]- [11], but it was devoted to the study of control systems of the whole order.In this sense, this paper summarizes these works.

Results and Discussion
Let's pass to the formulation of the main results.Everywhere further: 1) the terminal set M has an appearance 0 1 M -subset of a subspace of L-orthogonal addition 0 M ; 2) π -operator of orthogonal projection from n R on L; 3) operation * is understood as operation of a geometrical subtraction [8].Let ( ) ( ) That from initial situation 0 z is possible will finish prosecution during 1 T τ = .
Let now the ω -arbitraries splitting a piece [ ] { } Theorem 2. If in game (1) at some That from initial situation 0 z is possible will finish prosecution during 2 T τ = .
Let's designate through ( ) Theorem 3. If in game (1) at some that from initial situation 0 z is possible will finish prosecution during 3 T τ = .Proof of the theorem 1.Two cases are possible: 1) 1 0 , then there will be vectors ( ) . Further, according to determination of integral ( ) Considering this equality, we will consider the equation and Q υ ∈ .As ( ) ( ) ŵ r w r ∈ , the equation ( 9) has the decision.We will choose the least in lexicographic sense from all solutions of the equation ( 9) and we will designate it through ( ) ∈ , is lebegovsk measurable on t and borelevsk measurable on υ [7].Therefore for any measurable function , 0 u t t t υ τ ≤ ≤ , will be lebegovsk measurable function [7].Let's put ( ) , 0 u t u t t t υ τ = ≤ ≤ and we will show that at such way of management of the parameter of u the trajectory Really, on (9) for the decision ( ),0 . Further we have ( ) From here we will receive that ( ) The theorem is proved completely.
Proof of the theorem 2. In view of a case triviality we will begin 2 0 τ = consideration with a case 2 0 τ > .
that is the trajectory, left a point 0 z , in an instant 2 t τ = turns out M on a set.The theorem is proved completely.Proof of the theorem 3. Owing to a condition of the theorem (8) we have )