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Solution to a Zero-Sum Differential Game with Fractional Dynamics via Approximations

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Abstract

The paper deals with a zero-sum differential game in which the dynamical system is described by a fractional differential equation with the Caputo derivative of an order \(\alpha \in (0, 1).\) The goal of the first (second) player is to minimize (maximize) a given quality index. The main contribution of the paper is the proof of the fact that this differential game has the value, i.e., the lower and upper game values coincide. The proof is based on the appropriate approximation of the game by a zero-sum differential game in which the dynamical system is described by a first-order functional differential equation of a retarded type. It is shown that the values of the approximating differential games have a limit, and this limit is the value of the original game. Moreover, the optimal players’ feedback control procedures are proposed that use the optimally controlled approximating system as a guide. An example is considered, and the results of computer simulations are presented.

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References

  1. Bannikov AS (2017) Evasion from pursuers in a problem of group pursuit with fractional derivatives and phase constraints. Vestn Udmurtsk Univ Mat Mekh Komp Nauki 27(3):309–314 (in Russian)

    MathSciNet  MATH  Google Scholar 

  2. Baranovskaya LV (2015) A method of resolving functions for one class of pursuit problems. East-Eur J Enterp Technol 74(4):4–8 (in Russian)

    Google Scholar 

  3. Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, Basel

    MATH  Google Scholar 

  4. Başar T, Olsder GJ (1999) Dynamic noncooperative game theory. Classics in applied mathematics, 2nd edn. SIAM, Philadelphia

    MATH  Google Scholar 

  5. Bellman R, Cooke KL (1963) Differential-difference equations. Academic Press, London

    MATH  Google Scholar 

  6. Cardaliaguet P, Quincampoix M, Saint-Pierre P (2007) Differential games through viability theory: old and recent results. In: Jørgensen S, Quincampoix M, Vincent TL (eds) Advances in dynamic game theory: numerical methods, algorithms, and applications to ecology and economics. Birkhäuser, Boston, pp 3–35

    MATH  Google Scholar 

  7. Chikrii A, Matychyn I (2011) Riemann–Liouville, Caputo, and sequential fractional derivatives in differential games. In: Breton M, Szajowski K (eds) Advances in dynamic games: theory, applications, and numerical methods for differential and stochastic games. Birkhäuser, Boston, pp 61–81

  8. Diethelm K (2010) The analysis of fractional differential equations. Springer, Berlin

    MATH  Google Scholar 

  9. Fleming WH, Soner HM (2006) Controlled Markov processes and viscosity solutions, 2nd edn. Springer, New York

    MATH  Google Scholar 

  10. Flores-Tlacuahuac A, Biegler LT (2014) Optimization of fractional order dynamic chemical processing systems. Ind Eng Chem Res 53(13):5110–5127

    Google Scholar 

  11. Friedman A (1971) Differential games. Wiley, New York

    MATH  Google Scholar 

  12. Gomoyunov MI (2018) Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems. Frac Calc Appl Anal 21(5):1238–1261

    MathSciNet  MATH  Google Scholar 

  13. Gomoyunov MI (2019) Approximation of fractional order conflict-controlled systems. Progr Fract Differ Appl 5(2):143–155

    Google Scholar 

  14. Gomoyunov MI, Lukoyanov NYu (2012) Guarantee optimization in functional-differential systems with a control aftereffect. J Appl Math Mech 76(4):369–377

    MathSciNet  MATH  Google Scholar 

  15. Gomoyunov MI, Lukoyanov NYu (2018) On the numerical solution of differential games for neutral-type linear systems. Proc Steklov Inst Math 301(suppl 1):44–56

    MathSciNet  MATH  Google Scholar 

  16. Gomoyunov MI, Lukoyanov NYu, Plaksin AR (2017) Existence of a value and a saddle point in positional differential games for neutral-type systems. Proc Steklov Inst Math 299(suppl 1):37–48

    MathSciNet  MATH  Google Scholar 

  17. Gomoyunov MI, Plaksin AR (2018) On Hamilton–Jacobi equations for neutral-type differential games. IFAC-PapersOnLine 51(14):171–176

    Google Scholar 

  18. Hajipour A, Hajipour M, Baleanu D (2018) On the adaptive sliding mode controller for a hyperchaotic fractional-order financial system. Phys A 497:139–153

    MathSciNet  Google Scholar 

  19. Hale JK, Lunel SMV (1993) Introduction to functional differential equations. Springer, New York

    MATH  Google Scholar 

  20. Idczak D, Kamocki R (2011) On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in \(\mathbb{R}^{n}\). Fract Calc Appl Anal 14(4):538–553

    MathSciNet  MATH  Google Scholar 

  21. Isaacs R (1965) Differential games. Wiley, New York

    MATH  Google Scholar 

  22. Jajarmi A, Hajipour M, Mohammadzadeh E, Baleanu D (2018) A new approach for the nonlinear fractional optimal control problems with external persistent disturbances. J Frankl Inst 355:3938–3967

    MathSciNet  MATH  Google Scholar 

  23. Kaczorek T (2016) Minimum energy control of fractional positive electrical circuits with bounded inputs. Circuits Syst Signal Process 35(6):1815–1829

    MathSciNet  MATH  Google Scholar 

  24. Kheiri H, Jafari M (2018) Optimal control of a fractional-order model for the HIV/AIDS epidemic. Int J Biomath 11(7):1850086 (23 pages)

    MathSciNet  MATH  Google Scholar 

  25. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam

    MATH  Google Scholar 

  26. Kim AV (2015) I-smooth analysis: theory and applications. Wiley, London

    MATH  Google Scholar 

  27. Kolmanovskii V, Myshkis A (1992) Applied theory of functional differential equations. Kluwer, Dordrecht

    MATH  Google Scholar 

  28. Krasovskii AN, Krasovskii NN (1995) Control under lack of information. Birkhäuser, Berlin

    MATH  Google Scholar 

  29. Krasovskii NN (1985) Control of a dynamical system: problem on the minimum of guaranteed result. Nauka, Moscow (in Russian)

    Google Scholar 

  30. Krasovskii NN, Kotelnikova AN (2012) Stochastic guide for a time-delay object in a positional differential game. Proc Steklov Inst Math 277(suppl 1):145–151

    MathSciNet  Google Scholar 

  31. Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New York

    Google Scholar 

  32. Li C, Zeng F (2015) Numerical methods for fractional calculus. Chapman and Hall, New York

    MATH  Google Scholar 

  33. Lukoyanov NYu (2000) A Hamilton–Jacobi type equation in control problems with hereditary information. J Appl Math Mech 64(2):243–253

    MathSciNet  MATH  Google Scholar 

  34. Lukoyanov NYu (2003) Functional Hamilton–Jacobi type equations with ci-derivatives in control problems with hereditary information. Nonlinear Funct Anal Appl 8(4):535–555

    MathSciNet  MATH  Google Scholar 

  35. Lukoyanov NYu (2011) Functional Hamilton–Jacobi equations and control problems with hereditary information. Ural Federal University Publ, Ekaterinburg (in Russian)

    MATH  Google Scholar 

  36. Lukoyanov NYu, Gomoyunov MI, Plaksin AR (2017) Hamilton–Jacobi functional equations and differential games for neutral-type systems. Dokl Math 96(3):654–657

    MathSciNet  MATH  Google Scholar 

  37. Lukoyanov NYu, Plaksin AR (2015) Differential games for neutral-type systems: an approximation model. Proc Steklov Inst Math 291(1):190–202

    MathSciNet  MATH  Google Scholar 

  38. Lukoyanov NYu, Plaksin AR (2015) On approximations of time-delay control systems. IFAC-PapersOnLine 28(25):178–182

    Google Scholar 

  39. Lukoyanov NYu, Reshetova TN (1998) Problems of conflict control of high dimensionality functional systems. J Appl Math Mech 62(4):545–554

    MathSciNet  Google Scholar 

  40. Maksimov VI (1981) A differential game of guidance for systems with a deviating argument of neutral type. In: Chentsov AG, Khodakovskii AN (eds) Problems of dynamic control. IMM UNTs AN SSSR, Sverdlovsk, pp 33–45 (in Russian)

    Google Scholar 

  41. Mamatov M, Alimov K (2018) Differential games of persecution of frozen order with separate dynamics. J Appl Math Phys 6:475–487

    Google Scholar 

  42. Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York

    MATH  Google Scholar 

  43. Nikol’skii MS (1972) Linear differential pursuit games in the presence of lags. Differ Uravn 8(2):260–267 (in Russian)

    MathSciNet  Google Scholar 

  44. Osipov YuS (1971) Differential games of systems with aftereffect. Dokl Akad Nauk SSSR 196(4):779–782

    MathSciNet  Google Scholar 

  45. Petrov NN (2018) A multiple capture in a group pursuit problem with fractional derivatives. Trudy Inst Mat Mekh UrO RAN 24(1):156–164 (in Russian)

    MathSciNet  Google Scholar 

  46. Podlubny I (1999) Fractional differential equations. Academic Press, New York

    MATH  Google Scholar 

  47. Pontryagin LS (1981) Linear differential games of pursuit. Math USSR-Sbornik 40(3):285–303

    MathSciNet  MATH  Google Scholar 

  48. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives. Theory and applications. Gordon & Breach Sci. Publishers, London

    MATH  Google Scholar 

  49. Shen J, Lam J (2014) State feedback \(H_\infty \) control of commensurate fractional-order systems. Int J Syst Sci 45(3):363–372

    MathSciNet  MATH  Google Scholar 

  50. Subbotin AI (1995) Generalized solutions of first-order PDEs: the dynamical optimization perspective. Birkhäuser, Boston

    Google Scholar 

  51. Subbotin AI, Chentsov AG (1981) Guarantee optimization in control problems. Nauka, Moscow (in Russian)

    MATH  Google Scholar 

  52. Toledo-Hernandez R, Rico-Ramirez V, Rico-Martinez R, Hernandez-Castro S, Diwekar UM (2014) A fractional calculus approach to the dynamic optimization of biological reactive systems. Part II: numerical solution of fractional optimal control problems. Chem Eng Sci 117:239–247

    Google Scholar 

  53. Vasil’ev FP (1972) Concerning conditions of existence of a saddle point in determinate games for integro-differential systems with a neutral type delay. Avtomat i Telemekh 2:40–50 (in Russian)

    Google Scholar 

  54. Wang J, Zhou Y (2011) A class of fractional evolution equations and optimal controls. Nonlinear Anal Real World Appl 12(1):262–272

    MathSciNet  MATH  Google Scholar 

Download references

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  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, S. Kovalevskaya Str., 16, Ekaterinburg, Russia

    Mikhail Gomoyunov

  2. Ural Federal University, Mira Str., 32, Ekaterinburg, Russia

    Mikhail Gomoyunov

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  1. Mikhail Gomoyunov
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Correspondence to Mikhail Gomoyunov.

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Gomoyunov, M. Solution to a Zero-Sum Differential Game with Fractional Dynamics via Approximations. Dyn Games Appl 10, 417–443 (2020). https://doi.org/10.1007/s13235-019-00320-4

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