Some Properties of the Sum and Geometric Differences of Minkowski

The sets of Minkowski algebraic sum and geometric difference are considered. The purpose of the research in this paper is to apply the properties of Minkowski sum and geometric difference to fractional differential games. This paper investigates the geometric properties of the Minkowski algebraic sum and the geometric difference of sets. Various examples are considered that calculate the geometric differences of sets. The results of the research are presented and proved as a theorem. At the end of the article, the results were applied to fractional differential games.


Introduction
Minkowski sums and geometric differences are important operations. They are used in many fields, such as: image processing, robotics, computer-aided design, mathematical morphology and spatial planning. Minkowski sums and geometric differences are used in various fields of science, such as differential games and optimal control [1] [2] [3], computer-aided design and production [2], computer animation and morphing [3], morphological image analysis [4] [5], measures for convex polyhedral [6], dynamic modeling [7], robot motion planning [8] and so on.
If our activity in the study of vector algebra in ordinary space n  begins with the addition of two vectors, then this activity extends to the addition of a vector or vectors belonging to one set to vectors belonging to another set. It is important to understand intuitively that adding a set to a vector is a combination of vectors formed by adding each element of the set to the vector [9] [10]  , you need to copy the set A onto each element of the set B and get the combination, or vice versa. This process is not difficult to imagine, even in arbitrary of n dimensional space, and it can be seen that the geometric nature of the sets A and B does not depend on their location relatively to the origin [12] [13] [14] [15].
Minkowski operators were first used in the work of L.S. Pontryagin to study differential games. L.S. Pontyagin's 1967 article "On linear differential games II" [16] provides definitions and several properties of the Minkovsky algebraic sum and the Minkovsky geometric difference.
Also, the application of Minkovsky's operator to differential games is described by N.Yu. Satimov, G.E. Ivanov, B.N. Pshenichny. In a 1973 article by N.Yu. Satimov, a linear differential game in n-dimensional Euclidean space is considered. In this work, N.Yu. Satimov finds in the linear differential game a sufficient condition that ensures that the chaser finishes the game in real time in the action of any possible line of the runner and proves it in the form of a theorem. He used Minkowski's difference and its properties to prove these theorems.
In the above work, the Minkowski sum and geometric difference are applied to whole-order differential games. In this article, we have tried to solve the fol- 3) Application of Minkowski's sum and difference to fractional differential games.

Research Methodology
be nonempty sets on the linear space E. The Minkowski sum and difference of two sets X and Y are defined to be the sets Definition 2. The multiplication of set X and number λ is defined to be the Definition 3. The Minkowski sum of any vector a E ∈ and nonempty set By the definition of the Minkowski difference of sets, the set X Y * means the intersection of movement of the set X to vector d in here S E ⊂ be any set.
In especial condition multiple-valued function G is constant ( ) 0 It is very important to know that, Minkowski sum and difference of the given sets are open or closed set. Therefore, we are writing following lemmas and theorems.

Analysis and Results
Lemma Proof. To prove this lemma we show that every element of X Y * will be an element of ( ) and on the contrary. Let a X Y * ∈ be any vector. By the definition of the Minkowski difference of sets we can write a Y X + ⊂ . For all y Y ∈ vectors a y X + ∈ . We add y Y − ∈ − vector to both sides of this relation. Hence Lemma 3. For any nonempty sets X and Y on the linear space E following relation is true:     (13). Lemma has been proved.

Lemma 6. For any nonempty sets , ,
A B C and D on the linear space E following relation is true: Therefore, is an element of set X or an element of set Y or an element of both sets. This implies that for y Y ∀ ∈ there exists vector x X ∃ ∈ or vector z Z ∃ ∈ such that y x = or y z = . By the definition of liner space we multiply both sides of these equalities by number λ and it follows that y Proof. Let x X ∈ and 1 1 y Y ∈ such that 1 2 a x y = + .
Lemma has been proved.
Lemma 9. Let 1 2 1 2 , , , X X Y Y be nonempty sets on the linear space E. If a X Y * ∈ be any vector. By the definition of the Minkowski dif- According to the condition of the theorem, 2 1 Y Y ⊂ and by the lemma 8, Lemma 10. For any nonempty sets X, Y and Z on the linear space E following relation is true: We add vector x X ∀ ∈ to both sides of these relations and we obtain x t Z x Y X Y + + ⊂ + ⊂ + . It means that a Z X Y + ⊂ + . Hence, a X Y Z * ∈ + .
Following example shows that each a X Y Z * ∈ + vector does not belong to ( )

given sets. Journal of Applied Mathematics and Physics
Then we obtain Lemma 11. For any nonempty sets X, Y and Z on the linear space E following relation is true: Following example shows that each vector a X Y Z * ∈ + does not belong to set ( ) Lemma 12. For any nonempty set X on the liner space E following equality is true.
the definition of the intersection of sets, it follows that a X Z * ∈ and a Y Z * ∈ .
By the definition of Minkowsli difference of sets, we have a Z X + ⊂ and a Z Y Theorem has been proved. Hence, We can show that every vector by using this method. Theorem has been proved. Therefore Following theorems may be proved by using of the lemmas given above.

The Discussion of the Results
In this section we give possible applications of the results of the previous paragraph.

Fractional Differential Games with Lumped Parameters
Let the motion of an object in a finite-dimensional Euclidean space n R is described by a differential equation of fractional order of the form A-n × n, B-p × n and G-q × n constant matrices, , u υ -control parameters u-chasing player control parameter, p u P R ∈ ⊂ , υ -runaway control parameter, q Q R υ ∈ ⊂ , P and Q-compacts, ( ) f t -known measurable vector function. The fractional derivative will be understood as the left-side fractional derivative of Caputo [11]. Recall that the Caputo fractional derivative of an arbitrary non-target order Also in space n R terminal set is allocated M. Chasing player goal to deduce z to many M, the fleeing player seeks to prevent this.
We consider the pursuit problem of approximating the trajectory of a con- In [12], it was proved that if in the game (27) for some

Fractional Differential Games with Distributed Parameters
A controllable distributed system is described, described by equations of fractional order [11] ( ) where z-unknown function from class , T-arbitrary positive constant; , x y C C -thermal conductivity coefficients; 0 where 1 2    where , , a b c -some constants. Therefore, the difference scheme (45) approximates Equation (32) with the order 2 α