In this paper, we consider for the first time the group problem of pursuit for linear fractional differential systems with several pursuers and one evader and with pure delay. We have developer an outline of the Method of Resolving Functions and the First Direct Method of Pontryagin for such conflict-controlled processes using the latest representation of the Cauchy formula. We compare the game end times guaranteed by these two methods. Sufficient conditions for ending the group pursuit game and the practical finding method of resolving functions are formulated.
functions are typically determined from specific quadratic equations, making this technique a convenient and universal means of solving certain problems.
Delay differential equations are widely used in various fields, including control theory, computer engineering, signal analysis, and damage theory. Generally, these mathematical models have a peculiarity, which is that the rate of change of these processes is determined by their history. In 2003, Khusainov, Shuklin, and Pospíšil [7-9] represented the solutions of linear differential equations, proposing to consider the so-called exponential matrix function with a delay. Based on the presented analogs of the Cauchy formula for conflict-controlled processes described by differential-difference systems, sufficient conditions for solving pursuit games have been found in research papers [10, 11]. The work [12] presents a modification of the Method of Resolving Functions for differential-difference pursuit games for a group of pursuers and one evader.
Representations of solutions of fractional differential equations with a linear delay are increasingly considered. Li and Wang considered a representation of the solution of the linear homogeneous fractional differential systems with the pure delay with order ∈ delayed matrix Mittag-Leffler functions [13]. The basic studies by Chikry–Eidelman [14-19] using contain sufficient conditions for solving the pursuit problem for systems with fractional derivatives of arbitrary order ∈
0, 1 .
In 2018, Liang et al [20] presented the solution of the linear homogeneous fractional differential system with the pure delay with a term that includes Caputo double derivatives with order ∈
0, 1 . In 2021, Liu et al [21] obtained exact solutions for a nonhomogeneous fractional oscillation equation with pure delay by constructing two functions derived from the extension of the Mittag-Leffler function.
In 2021, Elshenhab and Wang [22] introduced a new delay matrix of the Mittag–Leffler type with two Liu delay matrices. This study is based on the latest achievements in the presentation of analogs of the Cauchy formula.
In this section, we present some necessary definitions and lemmas for linear fractional systems with pure delay used in our subsequent discussions.
Consider the system where zero, where constant real nonzero matrix, : 0, ∞ → ℝ be a given function, denotes the Caputo fractional derivative of order ∈ is a solution satisfying (1) for every 0,
A is an with the initial conditions ≡ , , ℎ 0, Definition 1. [22]. The two-parameter Mittag-Leffler function is defined as , , … , : ℎ, ∞ → ℝ is an arbitrary differentiable function.
(1) Especially, if 1, then , ,
and ,
are defined as follows: Definition 2. [22]. The delayed Mittag-Leffler type matrix functions , , , ≡
Γ
, , 0, ∈ ℂ. Γ 1
, 0. Let
be a solution of the system (2) under the initial condition vector functions of its variables;
and V are nonempty compacts. where are square matrices of order ; , , : → ℝ , are jointly continuous ≡ , ≡ , ℎ 0, 1, 2, . . . , .
, ∗ ⊂ ℝ , each having the form 3. Problem statement
, ∈ ℝ , evolve in a finite-dimensional where complement
is a linear subspace in ℝ to subspace in the space ℝ .
is a convex compact belonging to the orthogonal The valid controls and are the measurable Lebesgue functions, ∈ and ∈ . Ω Ω : : ∈ , ∈
0, ∞ ∈ , ∈ 0, ∞ .
, The function ∙ )∈ Ω is chosen by the evader based on knowledge of the initial condition : ∙ ∈ Ω , ∈ 0, , is the prehistory of the control of evader 4. each moment 0, and condition 3 it assigns 4 a Lebesgue measurable functio n
The group pursuit game (2), (3) terminates when ∈ ∗ for some i. 4. Scheme of the Method of the Resolving Functions maps
Denote by the operator of orthogonal projection from ℝ onto . Consider the set-valued Condition 1. ∅
for all 1, 2, . . . , , 0.
Since 0, ∞
the set-valued map them has a Borelian selection [24]. Let us denote ∙ ∙ : ∈ , 0, 1, … ,
. For fixed g∙ ∈ ⋯ ℎ subject to the equations , , 1, 2, . . . , , for 0, ℎ
0, is upper semicontinuous and each of
and we put ∙
ℎ Consider the resolving function (4) (5) (6) and
, 0 assumes finite values. for all ∈ 0, , ∈ , if and . If for some 0 T ( (), g()) inf t 0 : inf max t inf i (t, s,v)ds 1, i 1,..., ,
vV i 0 vV ∙
Theorem. Let the conflict-controlled process (2) and (4) with the initial condition (3) satisfy
Borelian selection ∙ ∈ , 1, … , , and let for given ∙
Then for at least one i a trajectory of the corresponding process (2) can be brought from ∙ to the set ∗ at the moment introduce the controlling function ℎ ℎ, , ∙ , ∙ ,
∙ 1 min ,…, , , ∙ , , ∙ , 0. controlling function becomes zero:
∙ , 0 ∗ , such that the At the same time, we can assign arbitrary control to all other pursuers.
If , ∗ ∗ on the interval 0, ∗ equals to and it is a Borelian function [25]. 0. appear as a jointly Borelian function in their variables [25]. The control of each of the pursuers lexmin , 1, … , ,
1 ∗ pursuer equal to ∈ : , ℎ ∗ ∙ , , ∗ ∙ ≡ 0 on the interval ∗, , and the control of the ∗-th , ,
the ∗ pursuer. Since ∗ , , that
From the Cauchy formula (Lemma 1) for the ∗ pursuer we have If we add and subtract from the right-hand side of (14) the value ∗ , we obtain Then ∗ ∗
∗ ∗ ∗, ∗ ∗ ∗ ∗
, . ∗ 0 ∗
∗, ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗, ∗, ∗,
∗ ℎ ∗ 2ℎ ∗ ℎ ∗ ∗
where is a geometric subtraction of the sets (Minkowski’s difference) [26-29]. ∗ ℎ ∗ ℎ ∗ ℎ ∗ , ∗ .
∗ , ∗ Taking the pursuer's control choice laws (8), (9), (11), (12) into account we deduce the inclusion ∗ ∗ ∈ ∗ , ∗ ∙ , ∗ ∙ ∗ , , ∗ ∗ ∙ , ∗ ∙ complete.
Remark. For the linear process (2) ∗ ∗ ∈ ∗ , ∗ ∙ , ∗ ∙ 1 ∗ , , ∗ , , ∗ .
Since (13) and the set ∗ is convex then ∗ ∗ ∈ ∗ .
If for some i ,
, then, because of the control choice law, in this case, formula (5), and Lemma 1, we infer the inclusion ∈ . The proof is therefore 1 is fulfilled, and let there exist continuous positive functions , and nonnegative numbers where is a unit ball centered at zero in the subspace . to be the largest roots of the quadratic equations for , 0, Then when , ∙ , ∙ ∉ , the resolving functions , , ∙ are the maximal numbers such that ∩ , ℎ ,
ℎ , ℎ 0 , ℎ 0
ℎ
∈ ℎ (15) (16) (17)
, ℎ , 2ℎ 0 ∈ ℎ
, 1, … , . 5. Scheme of the First Direct Method of Pontryagin
Since the left part of inclusion (16) is linear in , the vector lies on the boundary of the sphere ℎ for the maximal value of for each 1, … , . each 1, … , ,
In other words, the length of this vector is equal to the radius of this ball for which is demonstrated by (15). The proof is complete. terminal set ∗ at the moment .
Proof. We shall follow the i-th pursuer. The following inclusion holds Theorem 2. Let the process (2), (4) with the initial condition (3) satisfy Condition 1, and for the given initial state ∙
∙ is determined by the equality (17).
Then for at least one i a trajectory of the process (2)-(4) can be brought from ∙ to the ℎ .
Hence, there exists a point and a selector ∙ ∈ 0
, where ∈
0; P , ∈ . are Borel measurable functions in s, v.
where , ∈ , is a measurable function. Under (18) and (17), we obtain , ℎ 0 , ℎ 0
, ℎ holds if and only if there exists a selector g ∙ ∈ G such that , .
Proof. Let ,
∈ ∗. The proof is complete.
Theorem 3. Let the conflict-controlled process (2), and (4) with the initial condition (3) satisfy
Then there exists a point and a selection g ∙ ∈ G 0 reverse order, we will get the desired result.
Then for at least one i and initial state (3), there exists a selection g ∙ ∈ G , then reasoning in the
presented.
This paper elaborates the results obtained in previous research [30] and focuses on group pursuit games, which are described by fractional differential systems with pure delay. We construct outlines of the Method of Resolving Functions and the First Direct Method of Pontryagin using an analog of the Cauchy formula for these systems, and formulate sufficient conditions for the ending of the game. The game end times guaranteed by these two methods are comparable. The method of practical implementation of the resolving functions is
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