Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions

• Main Author: Nikolay Nikandrovich Petrov
• Format: Article
• Language: English
• Published: MDPI AG 2021-05-01
• Series: Mathematics
• Subjects: differential game; pursuer; evader; group pursuit; fractional derivatives
• Online Access: https://www.mdpi.com/2227-7390/9/11/1171

The problem of conflict interaction between a group of pursuers and an evader in a finite-dimensional Euclidean space is considered. All participants have equal opportunities. The dynamics of all players are described by a system of differential equations with fractional derivatives in the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>D</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msup><msub><mi>z</mi><mi>i</mi></msub><mo>=</mo><mi>a</mi><msub><mi>z</mi><mi>i</mi></msub><mo>+</mo><msub><mi>u</mi><mi>i</mi></msub><mo>−</mo><mi>v</mi><mo>,</mo><mspace width="4pt"></mspace><msub><mi>u</mi><mi>i</mi></msub><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>D</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msup><mi>f</mi></mrow></semantics></math></inline-formula> is a Caputo derivative of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>.</mo></mrow></semantics></math></inline-formula> Additionally, it is assumed that in the process of the game the evader does not move out of a convex polyhedral cone. The set of admissible controls <i>V</i> is a strictly convex compact and <i>a</i> is a real number. The goal of the group of pursuers is to capture of the evader by no less than <i>m</i> different pursuers (the instants of capture may or may not coincide). The target sets are the origin. For such a conflict-controlled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain instant of time for any counteractions of the evader. The method of resolving functions is used to solve the problem, which is used in differential games of pursuit by a group of pursuers of one evader.