MATRIX RESOLVING FUNCTIONS IN THE LINEAR GROUP PURSUIT PROBLEM WITH FRACTIONAL DERIVATIVES

Alena I. Machtakova     (Udmurt State University, 1 Universitetskaya Str., Izhevsk, 426034, Russia; Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)
Nikolai N. Petrov     (Udmurt State University, 1 Universitetskaya Str., Izhevsk, 426034, Russia; Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)

Abstract


In finite-dimensional Euclidean space, we analyze the problem of pursuit of a single evader by a group of pursuers, which is described by a system of differential equations with Caputo fractional derivatives of order \(alpha.\) The goal of the group of pursuers is the capture of the evader by at least \(m\) different pursuers (the instants of capture may or may not coincide). As a mathematical basis, we use matrix resolving functions that are generalizations of scalar resolving functions. We obtain sufficient conditions for multiple capture of a single evader in the class of quasi-strategies. We give examples illustrating the results obtained.


Keywords


Differential game, Group pursuit, Pursuer, Evader, Fractional derivatives

Full Text:

PDF

References


  1. Blagodatskikh A.I. Simultaneous multiple capture in a conflict-controlled process. J. Appl. Math. Mech., 2013. Vol. 77, No. 3. P. 314–320. DOI: 10.1016/j.jappmathmech.2013.09.007
  2. Blagodatskikh A.I., Petrov N.N. Konfliktnoe vzaimodejstvie grupp upravlyaemyh ob”ektov [Conflict Interaction of Groups of Controlled Objects]. Izhevsk: Udmurt State University, 2009. 266 p. (in Russian)
  3. Blaquière A., Gérard F., Leitmann G. Quantitative and Qualitative Differential Games. New York, London: Academic Press, 1969. 172 p.
  4. Bopardikar S.D., Suri S. \(k\)-Capture in multiagent pursuit evasion, or the lion and the hyenas. Theoret. Comput. Sci., 2014. Vol. 522. P. 13–23. DOI: 10.1016/j.tcs.2013.12.001
  5. Caputo M. Linear model of dissipation whose Q is almost frequency independent-II. Geophys. R. Astr. Soc., 1967. Vol. 13, No. 5. P. 529–539. DOI: 10.1111/j.1365-246X.1967.tb02303.x
  6. Chikrii A.A. Conflict-Controlled Processes. Dordrecht: Springer, 1997. 404 p. DOI: 10.1007/978-94-017-1135-7
  7. Chikrii A.A. Quasilinear controlled processes under conflict. J. Math. Sci. 1996. Vol. 80. No 1. P. 1489–1518.
  8. Chikrii A.A., Chikrii G.Ts. Matrix resolving functions in game problems of dynamics. Proc. Steklov Inst. Math., 2015. Vol. 291. Suppl. 1. P. 56–65. DOI: 10.1134/S0081543815090047
  9. Chikrii A.A., Machikhin I.I. On an analogue of the Cauchy formula for linear systems of any fractional order. Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky. 2007. No. 1. P. 50–55. (in Russian)
  10. Chikrii A.A., Rappoport I.S. Method of resolving functions in the theory of conflict-controlled processes. Cybern. Syst. Anal., 2012. Vol. 48, No. 4. P. 512–531. DOI: 10.1007/s10559-012-9430-y
  11. Chikrii A.O., Chikrii G.Ts. Matrix resolving functions in dynamic games of approach. Cybern. Syst. Anal., 2014. Vol. 50, No. 2. P. 201–217. DOI: 10.1007/s10559-014-9607-7
  12. Chikriy A.A., Rappoport I.S. Measurable many-valued maps and their selectors in dynamic pursuit games. J. Autom. Inform. Sci., 2006. Vol. 38. No. 1. P. 57–67. DOI: 10.1615/J Automat Inf Scien.v38.i1.60
  13. Dzhrbashyan M.M. Integral’nye preobrazovaniya i predstavleniya funkcij v kompleksnoj oblasti [Integral Transforms and Representations of Functions in the Complex Domain]. Moscow: Nauka, 1966. 672 p. (in Russian)
  14. Friedman A. Differential Games. New York: John Wiley & Sons. 1971. 350 p.
  15. Gomoyunov M.I. Extremal shift to accompanying points in a positional differential game for a fractional order system. Proc. Steklov Inst. Math., 2020. Vol. 308. P. 83–105. DOI: 10.1134/S0081543820020078
  16. Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V. Mittag–Leffler Functions, Related Topics and Applications. Berlin Heidelberg: Springer-Verlag, 2014. 443 p. DOI: 10.1007/978-3-662-43930-2
  17. Grigorenko N.L. Simple pursuit evasion game with a group of pursuers and one evsder. Vestnik Moskov. Univ. Ser. XV Vychisl. Matematika i Kibernetika. 1983. No. 1. P. 41–47. (in Russian)
  18. Grigorenko N.L. Matematicheskie metody upravleniya neskol’kimi dinamicheskimi processami [Mathematical methods of control a few dynamic processes]. Moscow: Moscow State University, 1990. 197 p. (in Russian)
  19. Hájek O. Pursuit Games. New York: Academic Press, 1975. 265 p.
  20. Isaacs R. Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. New York: John Wiley and Sons, 1965. 384 p.
  21. Krasovskii N.N., Subbotin A.I. Pozicionnye differencial’nye igry [Positional Differential Games]. Moscow: Nauka, 1974. 456 p. (in Russian)
  22. Leitmann G. Cooperative and Non-Cooperative Many Players Differential Games. Wien: Springer-Verlag, 1974. 77 p. DOI: 10.1007/978-3-7091-2914-2
  23. Machtakova A.I., Petrov N.N. Pursuit of Rigidly Coordinated Evaders in a Linear Problem with Fractional Derivatives, a Simple Matrix, and Phase Restrictions. In: Lect. Notes Control Inf. Sci.: Proc. Stability and Control Processes. SCP 2020 (Smirnov N., Golovkina A. (eds.)), 2022. P. 391–398. DOI: 10.1007/978-3-030-87966-2_43
  24. Petrosyan L.A. Differencial’nye igry presledovaniya [Differential Games of Pursuit]. Leningrad: Leningrad University Press, 1977. 224 p. (in Russian)
  25. Petrov N.N. Group Pursuit Problem in a Differential Game with Fractional Derivatives, State Constraints, and Simple Matrix. Diff. Equat., 2019. Vol. 55. No. 6. P. 841–848. DOI: 10.1134/S0012266119060119
  26. Petrov N.N. Matrix resolving functions in a linear problem of group pursuit with multiple capture. Tr. Inst. Mat. Mekh. UrO RAN, 2021. Vol. 27. No. 2. P. 185–196. DOI: 10.21538/0134-4889-2021-27-2-185-196
  27. Petrov N.N. Multiple capture in a group pursuit problem with fractional derivatives and phase restrictions. Mathematics, 2021. Vol. 9. No. 11. Art. no. 1171. 12 p. DOI: 10.3390/math9111171
  28. Petrov N.N. The problem of simple group pursuit with phase constraints in time scale. Vestn. Udmurt. Univ. Mat. Mekh. Komp. Nauki, 2020. Vol. 30. No. 2. P. 249–258. DOI: 10.35634/vm200208
  29. Petrov N.N., Machtakova A.I. Capture of two coordinated evaders in a problem with fractional derivatives, phase restrictions and a simple matrix. Izv. IMI UdGU, 2020. Vol. 56. P. 50–62. DOI: 10.35634/2226-3594-2020-56-05
  30. Petrov N.N., Solov’eva N.A. Multiple capture of given number of evaders in linear recurrent differential games. J. Optim. Theory Appl., 2019. Vol. 182. No. 1. P. 417–429. DOI: 10.1007/s10957-019-01526-7
  31. Petrov N.N., Solov’eva N.A. Multiple capture in Pontryagin’s recurrent example with phase constraints. Proc. Steklov Inst. Math., 2016. Vol. 293, Suppl. 1 P. 174–182. DOI: 10.1134/S0081543816050163
  32. Petrov N.N., Solov’eva N.A. Multiple capture in Pontryagin’s recurrent example. Autom. Remote Control, 2016. Vol. 77. No. 5. P. 855–861. DOI: 10.1134/S0005117916050088
  33. Petrov N.N., Narmanov A.Ya. Multiple capture of a given number of evaders in a problem with fractional derivatives and a simple matrix. Proc. Steklov Inst. Math., 2020. Vol. 309, Suppl. 1. P. S105–S115. DOI: 10.1134/S0081543820040136
  34. Pollard H. The completely monotonic character of the Mittag--Leffler function \(E_a(-x)\). Bull. Amer. Math. Soc., 1948. Vol. 54, No. 12. P. 1115–1116. DOI: 10.1090/S0002-9904-1948-09132-7
  35. Polovinkin E.S. Mnogoznachnyj analiz i differencial’nye vklyucheniya [Multivalued Analysis and Differential Inclusions]. Moscow: Fizmathlit, 2015. 524 p. (in Russian)
  36. Pontryagin L.S. Izbrannye nauchnye trudy [Selected Scientific Works]. Vol. 2. Moscow: Nauka, 1988. 576 p. (in Russian)
  37. Popov A.Yu., Sedletskii A.M. Distribution of roots of Mittag–Leffler functions. J. Math. Sci., 2013. Vol. 190. No. 2. P. 209–409. DOI: 10.1007/s10958-013-1255-3
  38. Pshenichnyi B.N. Simple pursuit by several objects. Kibernetika, 1976. No. 3. P. 145–146. (in Russian)
  39. Rappoport J.S. Strategies of group approach in the method of resolving functions for quasilinear conflict-controlled processes. Cybern. Syst. Anal., 2019. Vol. 55. No. 1. P. 128–140. DOI: 10.1007/s10559-019-00118-7
  40. Subbotin A.I., Chentsov A.G. Optimizaciya garantii v zadachah upravleniya [Optimization of a Guarantee in Problems of Control]. Moscow: Nauka, 1981. 288 p. (in Russian)




DOI: http://dx.doi.org/10.15826/umj.2022.1.008

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.