^{1}

^{2}

This article is devoted to obtaining sufficient conditions for the completion of pursuit for control systems of fractional order described with divided dynamics. The results are illustrated on model examples of gaming problems with a simple matrix and separated fractional-order motions.

At the present time, there is a noticeable increase in the attention of researchers to fractional calculus. The development of the theory of equations with derivatives of fractional order is stimulated by the development of the theory of differential equations of the whole order. The role of fractional calculus in the theory of equations of mixed type is well known, in the theory of problems with displacement, in the theory of degenerate equations. In addition, equations of fractional order, essentially supplementing the picture of the general theory of differential equations, can reveal a connection between phenomena that, remaining within the framework of integer differentiation, appear to be independent. The dynamics of systems described by differential equations of fractional order is an object of study of specialists from about the middle of the 20^{th} century [

At the present time, under the influence of rapid scientific and technical progress, fractional calculus has turned into a powerful scientific direction, including both fundamental and applied research. This is due to the need to more accurately describe the physical systems and processes that have become objects of interest of modern researchers. The distinguishing features of such systems and processes are their non-local character and the phenomenon of memory. For example, this applies to micro and nanostructured media, deterministic and chaotic including “fractal-chaotic” processes in nature and engineering.

In addition to research in the field of modeling fractional dynamical systems, research in management problems such as differential games has been actively developed in recent years. The present article is devoted to obtaining sufficient conditions for the completion of pursuit for differential games of fractional order, described with divided dynamics [

Let the movement of the first player, whom we call the pursuer, be described by equation

D α x = A x + u , x ∈ R m 1 (1)

where D α ―operator of fractional differentiation of order α , n 1 − 1 < α < n 1 , n 1 ∈ ℕ , t ∈ [ 0 , T ] , A ― m 1 × m 1 -constant matrix. The movement of the second player, which we will call escaping, is given by equation

D β y = B y + υ , y ∈ R m 2 (2)

where D β ―operator of fractional differentiation of order β , n 2 − 1 < β < n 2 , n 2 ∈ ℕ , t ∈ [ 0 , T ] , B ― m 2 × m 2 constant matrix, u , υ ―control parameters, u ―controlling parameter of the pursuer, u ∈ P ⊂ R m 1 , υ ―the controlling parameter of the evading player, υ ∈ Q ⊂ R m 2 , P and Q ―compacts. The fractional derivative will be understood in the sense of Caputo [

We recall that the fractional derivative of order γ , ( n − 1 < γ < n , n ∈ Ν ) from sometimes n continuously differentiable function z ( t ) , z : R + → R m in Caputo’s sense is defined by the expression

D γ z ≡ D ( γ ) z ( t ) = 1 Γ ( n − γ ) ∫ 0 t z ( n ) ( τ ) ( t − τ ) γ − n + 1 d τ . (3)

where Γ ( ⋅ ) ―gamma-function, which is defined as follows Γ ( θ ) = ∫ 0 ∞ t θ − 1 e − t d t . The main property of the gamma function is expressed by the reduction formula Γ ( θ + 1 ) = θ Γ ( θ ) . If θ ―positive integer, than Γ ( θ ) = ( θ − 1 ) ! ; Γ ( θ + 1 2 ) = 1 ⋅ 3 ⋯ ( 2 θ − 1 ) 2 θ π . When 0 < θ < 1 we have formula Γ ( θ ) Γ ( 1 − θ ) = π sin ( π θ ) .

To define a terminal set, we introduce the notation s ( s ≤ min ( m 1 , m 2 ) ) , M 1 = M 0 1 + M 1 , M 0 1 ⊂ R s 1 , L 1 × M 0 1 = R s 1 and M 2 = M 0 2 + M 1 , M 0 2 ⊂ R s 2 , L 2 × M 0 2 = R s 2 . Across Π 1 , Π 2 , denote operators orthogonal to the projections respectively from R m 1 on L 1 and from R m 2 on L 2 and пусть M = { ( x ; y ) , x ∈ R m 1 , y ∈ R m 2 : Π 1 x − Π 2 y ∈ M 1 } . The game is considered to be over, if the conditions are fulfilled. The aim of the pursuing player is to withdraw ( x ; y ) on the set M, the escaping player tries to prevent it.

Definition. We say that a differential game (1)-(3) can be completed from the initial position x 0 = ( x 0 0 , x 1 0 , x 2 0 , x 3 0 , ⋯ , x n 1 − 1 0 ) , y 0 = ( y 0 0 , y 1 0 , y 2 0 , ⋯ , y n 2 − 1 0 ) during T = T ( x 0 , y 0 ) , if there exists a measurable function u ( t ) = u ( z 0 , υ ( t ) ) ∈ P , t ∈ [ 0 , T ] , that the solutions of equations

D α x = A x + u ( t ) , x ∈ R m 1 , n 1 − 1 < α < n 1 , x ( 0 ) = x 0 , (4)

D β y = B y + υ ( t ) , y ∈ R m 2 , n 2 − 1 < β < n 2 , y ( 0 ) = y 0 , (5)

satisfies the condition ( x ; y ) ∈ M , those Π 1 x − Π 2 y belongs to the set M 1 in the moment t = T for any measurable functions υ ( t ) , υ ( t ) ∈ Q , 0 ≤ t ≤ T .

We now turn to the formulation of the main results. Let E η ( G ; μ ) = ∑ k = 0 ∞ G k Γ ( k η − 1 + μ )

-generalized Mittag-Lefler matrix function [

x ( k ) ( 0 ) = x k 0 , k = 0 , 1 , ⋯ , n 1 − 1 , y ( l ) ( 0 ) = y l 0 , l = 0 , 1 , ⋯ , n 2 − 1. (6)

Then the solution of Equations ((4), (5)) with initial conditions (6) has the form

x ( t ) = ∑ k = 0 n 1 − 1 t k E 1 α ( A t α ; k + 1 ) x k 0 + ∫ 0 t ( t − τ ) α − 1 E 1 α ( A ( t − τ ) α ; α ) u ( τ ) d τ . (7)

y ( t ) = ∑ l = 0 n 2 − 1 t l E 1 β ( B t β ; l + 1 ) y l 0 + ∫ 0 t ( t − τ ) β − 1 E 1 β ( B ( t − τ ) β ; β ) υ ( τ ) d τ . (8)

For r ≥ 0 , define u ^ ( r ) = Π 1 t α − 1 E 1 α ( A t α ; α ) P , υ ^ ( r ) = Π 2 t β − 1 E 1 β ( B t β ; β ) Q , w ^ ( r ) = u ^ ( r ) * υ ^ ( r ) ;

W ( τ ) = ∫ 0 τ w ^ ( r ) d r , τ > 0 , W 1 ( τ ) = − M 1 + W ( τ ) . (9)

For convenience, we introduce the notation h x ( x 0 , t ) = ∑ k = 0 n 1 − 1 t k E 1 α ( A t α ; k + 1 ) x k 0 , h y ( y 0 , t ) = ∑ l = 0 n 2 − 1 t l E 1 β ( B t β ; l + 1 ) y l 0 .

Theorem 1. If in the game (1)-(3) for some τ = τ 1 , the inclusion

− Π 1 h x ( x 0 , τ ) + Π 2 h y ( y 0 , τ ) ∈ W 1 ( τ ) (10)

then from the initial position x 0 , y 0 you can complete the pursuit of time T = τ 1 .

Now suppose that ω ―an arbitrary partition of the interval [ 0 , τ ] , ω = { 0 = t 0 < t 1 < ⋯ < t p = τ } , i = 1 , 2 , ⋯ , p , and A 0 = − M 1 ,

A i ( M 1 , τ ) = ( A i − 1 ( M 1 , τ ) + ∫ t i − 1 t i Π 1 r α − 1 E 1 α ( A r α ; α ) P d r ) * ∫ t i − 1 t i Π 2 r β − 1 E 1 β ( B r β ; β ) Q d r , i = 1 , 2 , ⋯ , p , W 2 ( τ ) = ∩ ω A i ( M 1 , τ ) . (11)

Theorem 2. If in the game (1)-(3) for some τ = τ 2 , the inclusion,

− Π 1 h x ( x 0 , τ ) + Π 2 h y ( y 0 , τ ) ∈ W 2 ( τ ) (12)

then from the initial position x 0 , y 0 you can complete the pursuit of time T = τ 2 .

We denote by w ^ ( r , τ ) a bunch of [ − 1 τ M 1 + u ^ ( r ) ] * υ ^ ( r ) defined for all r ≥ 0 , τ > 0 . Consider the integral

W 3 ( τ ) = ∫ 0 τ w ^ ( r , τ ) d r . (13)

Theorem 3. If in the game (1)-(3) for some τ = τ 3 , the inclusion

− Π 1 h x ( x 0 , τ ) + Π 2 h y ( y 0 , τ ) ∈ W 3 ( τ ) (14)

then from the initial position x 0 , y 0 you can complete the pursuit of time T = τ 3 .

Proof of Theorem 1. There are two possible cases:1) τ 1 = 0 ; 2) τ 1 > 0 . Case 1) is trivial, since when τ 1 = 0 from (9) and inclusion (10) we have − Π 1 h x ( x 0 , 0 ) + Π 2 h y ( y 0 , 0 ) ∈ − M 1 and Π 1 x 0 0 − Π 2 y 0 0 ∈ M 1 , which is equivalent to including ( x 0 ; y 0 ) ∈ M . Now let the case 2) τ 1 > 0 . By the conditions of the theorem (10)

− Π 1 h x ( x 0 , τ 1 ) + Π 2 h y ( y 0 , τ 1 ) ∈ W 1 ( τ 1 ) , then there are vectors d ∈ M 1 и w ∈ ∫ 0 τ 1 w ^ ( r ) d r such that (show (9), (10)) d + w = − Π 1 h x ( x 0 , τ 1 ) + Π 2 h y ( y 0 , τ 1 ) . Further, in accordance with the definition of the integral ∫ 0 τ 1 w ^ ( r ) d r there exists a summable function w ( r ) , 0 ≤ r ≤ τ 1 , w ( r ) ∈ w ^ ( r ) , when w = ∫ 0 τ 1 w ( r ) d r . Taking this equality into account, we consider the equation

Π 1 ( τ 1 − t ) α − 1 E 1 α ( A ( τ 1 − t ) α ; α ) u − Π 2 ( τ 1 − t ) β − 1 E 1 α ( B ( τ 1 − t ) β ; β ) υ = w ( τ 1 − t ) (15)

Relatively u ∈ P for fixed t ∈ [ 0 , τ 1 ] and υ ∈ Q . As w ( r ) ∈ w ^ ( r ) , then Equation (15) has a solution. From all solutions of (15) we choose the smallest in the lexicographic sense and denote it by u ( t , υ ) . Function u ( t , υ ) , 0 ≤ t ≤ τ 1 , υ ∈ Q , It is Lebesgue measurable with respect to and Borel measurable in υ [

Indeed, according to (15), for the solution of x ( t ) , y ( t ) , 0 ≤ t < ∞ , equations

D α x = A x + u ( t ) , x ( k ) ( 0 ) = x k 0 , k = 0 , 1 , ⋯ , n 1 − 1 (16)

D β y = B y + υ ( t ) , y ( l ) ( 0 ) = y l 0 , l = 0 , 1 , ⋯ , n 2 − 1 (17)

in view of (7), (8), (16), (17) we have [

− Π 1 x ( τ 1 ) + Π 2 y ( τ 1 ) = − Π 1 h x ( x 0 , τ 1 ) + Π 2 h y ( y 0 , τ 1 ) − ∫ 0 τ 1 [ Π 1 ( τ 1 − t ) α − 1 E 1 α ( A ( τ 1 − t ) α ; α ) u ( t ) − Π 2 ( τ 1 − t ) β − 1 E 1 α ( B ( τ 1 − t ) β ; β ) υ ( t ) ] d t = − Π 1 h x ( x 0 , τ 1 ) + Π 2 h y ( y 0 , τ 1 ) − ∫ 0 τ 1 w ( τ 1 − t ) d t = − d + w − ∫ 0 τ 1 w ( τ 1 − t ) d t = − d + ∫ 0 τ 1 w ( r ) d r − ∫ τ 1 0 w ( r ) d r = − d − ∫ 0 τ 1 w ( r ) d r + ∫ 0 τ 1 w ( r ) d r = − d = − M 1

Π 1 x ( τ 1 ) − Π 2 y ( τ 1 ) = d ∈ M 1 , Π 1 x ( τ 1 ) − Π 2 y ( τ 1 ) ∈ M 1 , (18)

As − d − w = Π 1 h x ( x 0 , τ 1 ) − Π 2 h y ( y 0 , τ 1 ) . Further we have Π 1 x ( τ 1 ) − Π 2 y ( τ 1 ) ∈ M 1 . From this [

Proof of Theorem 2. In view of the triviality of the case τ 2 = 0 we start with the case τ 2 > 0 . We have (show (11), (12)) − Π 1 h x ( x 0 , τ 2 ) + Π 2 h y ( y 0 , τ 2 ) ∈ W 2 ( τ 2 ) . W 2 ( τ 2 ) is an alternating integral with initial set A 0 = − M 1 [

W 2 ( τ 2 ) ⊂ ( W 2 ( τ 2 − ε ) + ∫ τ 2 − ε τ 2 Π 1 r α − 1 E 1 α ( A r α ; α ) P d r ) * ∫ τ 2 − ε τ 2 Π 2 r β − 1 E 1 β ( B r β ; β ) Q d r , (19)

where, ε ―arbitrary positive fixed number 0 < ε ≤ τ 2 ; υ 0 ( r ) , τ 2 − ε ≤ r ≤ τ 2 ―an arbitrary measurable function with values in Q.

Let υ = υ ( t ) , 0 ≤ t < ∞ ―arbitrary measurable function υ ( t ) ∈ Q . In accordance with the conditions of the theorem at time t = 0 the narrowing becomes known υ ( t ) , 0 ≤ t ≤ ε , function υ ( t ) , 0 ≤ t < ∞ , on the line [ 0 , ε ] . It follows from the inclusion (19) that for an arbitrary function υ ˜ ( τ 2 − r ) , τ 2 − ε ≤ r ≤ τ 2 , υ ˜ ( τ 2 − r ) ∈ Q , we have

− Π 1 h x ( x 0 , τ 2 ) + Π 2 h y ( y 0 , τ 2 ) ∈ W 2 ( τ 2 − ε ) + ∫ τ 2 − ε τ 2 Π 1 r α − 1 E 1 α ( A r α ; α ) P d r * ∫ τ 2 − ε τ 2 Π 2 r β − 1 E 1 β ( B r β ; β ) υ ˜ ( τ 2 − r ) d r , (20)

Thus, for an arbitrary function υ ˜ ( s ) , 0 ≤ s ≤ ε , there is an inclusion (20). Consequently, when υ ˜ ( s ) ≡ υ ( s ) , 0 ≤ s ≤ ε , the inclusion (17). This implies the existence of a measurable function u ( s ) , 0 ≤ s ≤ ε , such that

than

We argue further in the same way as (21), (22). As

we get

for an (23), (24) arbitrary measurable function

It follows from (25) that

etc. It is clear that there exists a natural number j such that: 1)

and

therefore (26)-(28).

Similarly, by formulas (27)-(29) we eventually obtain

Thus (30), for a point

Proof of Theorem 3. By the hypothesis of Theorem (14), we have

Let

From this (32), in view of the measurability condition, there follows the existence of measurable functions

A measurable function

From (34) here

Example 1. Let the pursuer’s motion be described by equation

where

where

the current position in

Because the A and B represent

and

Respectively (37), (38). We denote by

Now calculate the set

Thus (40), the set

The set defined by formula (9), (41)

It is (42) easy to prove that in order for the quantities

It is (43) clear that under these conditions all the conditions of the theorem are satisfied for this example 1. Thus, the quantity

Example 2. Let in the Euclidean space

and

where

and

The game is (48) considered to be over if conditions

Summarizing the results obtained, we come to the conclusion that the differential game of pursuit of fractional order (1)-(3) starting at the moment t = 0 from the initial position

The research carried out to solve fractional differential games clearly demonstrates that fractional calculus is, in general, a more general and complex field of research than the classical differential games. Similarly, the theory of fractional dynamical systems and fractional calculus of variations include systems of integer order as special cases. The development of fractional differential games is just beginning, and therefore in this area there remains an extensive field for research. In particular, there is still no single clear interpretation of the geometric and physical meaning of fractional operators. There is also no single definition of the fractional derivative: in more abstract mathematical studies, as a rule, the Riemann-Lowville definition is used, and in more applied studies related to physics or control theory, in most cases the definition of Caputo is used or the definition of Grunwald-Letnikova. At the same time, the question of constructing standardizing functions for initial, boundary and initial boundary value problems that allow one to change the form of the in homogeneity in equations and thereby reduce the corresponding problems to problems with zero boundary or initial conditions becomes urgent.

Mamatov, M. and Alimov, K. (2018) Differential Games of Persecution of Frozen Order with Separate Dynamics. Journal of Applied Mathematics and Physics, 6, 475-487. https://doi.org/10.4236/jamp.2018.63044