Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions
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 <...
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MDPI AG
2021-05-01
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| author | Nikolay Nikandrovich Petrov |
| author_facet | Nikolay Nikandrovich Petrov |
| author_sort | Nikolay Nikandrovich Petrov |
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| description | 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. |
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| spelling | doaj.art-23ed8d68e01f44668c396000d6a554942023-11-21T20:57:00ZengMDPI AGMathematics2227-73902021-05-01911117110.3390/math9111171Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase RestrictionsNikolay Nikandrovich Petrov0Laboratory of Mathematical Control Theory, Udmurt State University, 426034 Izhevsk, RussiaThe 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.https://www.mdpi.com/2227-7390/9/11/1171differential gamepursuerevadergroup pursuitfractional derivatives |
| spellingShingle | Nikolay Nikandrovich Petrov Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions Mathematics differential game pursuer evader group pursuit fractional derivatives |
| title | Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions |
| title_full | Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions |
| title_fullStr | Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions |
| title_full_unstemmed | Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions |
| title_short | Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions |
| title_sort | multiple capture in a group pursuit problem with fractional derivatives and phase restrictions |
| topic | differential game pursuer evader group pursuit fractional derivatives |
| url | https://www.mdpi.com/2227-7390/9/11/1171 |
| work_keys_str_mv | AT nikolaynikandrovichpetrov multiplecaptureinagrouppursuitproblemwithfractionalderivativesandphaserestrictions |