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A Scheme for Calculating Solvability Sets “Up to Moment” in Linear Differential Games

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Abstract

A general scheme for calculating a solvability set (backward reachability set) is proposed for a linear conflict controlled system. The backward procedure constructs each subsequent set only based on the set from the previous step. The algorithm is specified for a target set with a convex complement (concave target set). The concavity of solvability sets is justified for a concave target set. The conservation of the concavity singles out a separate class of time-optimal differential games. Note that the convexity is not conserved in the general case of linear time-optimal differential games. On the whole, the article deals with a theoretical approximation of solvability sets in linear time-optimal conflict control problems, the construction of which with structural accuracy (allowing one to distinguish barriers) is usually nontrivial even on a plane. The algorithm is illustrated by numerical calculations for two disturbed dynamics with concave target sets in the plane: the double integrator and the oscillating system.

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Data Availability

The datasets generated during the current study are available from the author on reasonable request.

References

  1. Chen M, Tomlin CJ. Hamilton–Jacobi reachability: some recent theoretical advances and applications in unmanned airspace management. Annual Review of Control, Robotics, and Autonomous Systems 2018;1:333–358. https://doi.org/10.1146/annurev-control-060117-104941.

    Article  Google Scholar 

  2. Subbotin AI. Generalized solutions of first-order PDEs. The dynamical optimization perspective. Boston: Birkhäuser; 1995. https://doi.org/10.1007/978-1-4612-0847-1.

    Book  Google Scholar 

  3. Bardi M, Capuzzo-Dolcetta I. Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Boston: Birkhäuser; 1997. https://doi.org/10.1007/978-0-8176-4755-1.

    Book  MATH  Google Scholar 

  4. Krasovskii NN, Subbotin AI. Game-theoretical control problems. New York: Springer; 1988.

    Book  Google Scholar 

  5. Mitchell IM, Bayen AM, Tomlin CJ. A time-dependent Hamilton–Jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans Autom Control 2005;50(7):947–957. https://doi.org/10.1109/TAC.2005.851439.

    Article  MathSciNet  MATH  Google Scholar 

  6. Mitchell IM. 2002. Application of level set methods to control and reachability problems in continuous and hybrid systems. PhD thesis, Stanford University.

  7. Taras’ev AM, Tokmantsev TB, Uspenskii AA, Ushakov VN. On procedures for constructing solutions in differential games on a finite interval of time. J Math Sci 2006;139(5):6954–6975. https://doi.org/10.1007/s10958-006-0400-7.

    Article  MathSciNet  MATH  Google Scholar 

  8. Botkin ND, Hoffmann K-H, Turova VL. Stable numerical schemes for solving Hamilton–Jacobi–Bellman–Isaacs equations. SIAM J Sci Comput 2011;33 (2):992–1007. https://doi.org/10.1137/100801068.

    Article  MathSciNet  MATH  Google Scholar 

  9. Cardaliaguet P. A differential game with two players and one target. SIAM J Control Optim 1996;34(4):1441–1460. https://doi.org/10.1137/S036301299427223X.

    Article  MathSciNet  MATH  Google Scholar 

  10. Cardaliaguet P, Quincampoix M, Saint-Pierre P. Set-valued numerical analysis for optimal control and differential games. Stochastic and Differential Games: Theory and Numerical Methods. Annals of International Society of Dynamic Games. In: Bardi M, Raghavan TES, and Parthasarathy T, editors. Norwell, MA: Birkhäuser; 1999. p. 177–247. https://doi.org/10.1007/978-1-4612-1592-9_4.

  11. Isaacs R. Differential games. New York: Wiley; 1965.

    MATH  Google Scholar 

  12. Pshenichnyj BN. The structure of differential games. Soviet Mathematics Doklady 1969;10:70–72.

    MATH  Google Scholar 

  13. Krasovskii NN, Subbotin AI. 1974. Positional differential games. Nauka, Moscow. (in Russian).

  14. Bardi M, Falcone M, Soravia P. Numerical methods for pursuit-evasion games via viscosity solutions. Stochastic and Differential Games. Annals of the International Society of Dynamic Games. In: Bardi M, Raghavan TES, and Parthasarathy T, editors. Boston, MA: Birkhäuser; 1999. p. 105–175. https://doi.org/10.1007/978-1-4612-1592-9_3.

  15. Ushakov VN, Khripunov AP. On the approximate construction of solutions in game-theoretic control problems. J Appl Math Mech 1997;61(3):401–408.

    Article  MathSciNet  MATH  Google Scholar 

  16. Shagalova LG. Approximation of a stable bridge in the problem of approaching a non-cylindrical target. Proceedings of the International Seminar “Control Theory and Theory of Generalized Solutions of Hamilton–Jacobi Equations” Dedicated to the 60th Birthday of Academician A.I. Subbotin, June 22–26, 2005, Ekaterinburg, Russia. In: Subbotina NN and Ushakov VN, editors. Ekaterinburg: Ural State University Publ.; 2006. p. 182–189. (in Russian).

  17. Mikhalev DK, Ushakov VN. Two algorithms for approximate construction of the set of positional absorption in the game problem of pursuit. Autom Remote Control 2007;68(11):2056–2070. https://doi.org/10.1134/S0005117907110136.

    Article  MathSciNet  MATH  Google Scholar 

  18. Turova VL. Construction of the positional absorption set in a linear second-order differential game with non-fixed terminal time. Control with guaranteed result: Collection of scientific works. In: Subbotin AI and Ushakov VN, editors. Sverdlovsk: Urals Scientific Center of the USSR Academy of Sciences; 1987. p. 92–112. (in Russian).

  19. Pshenichnyy BN. A game with simple motion and a convex terminal set. Proceedings of the Seminar “Theory of Optimal Decisions”, vol 3. Kiev, pp 3–16; 1969. (in Russian).

  20. Averboukh Y. 2009. A transformation of the control under uncertainty problems. arXiv:0902.2556.

  21. Kolmogorov AN, Fomin SV. Elements of the theory of functions and functional analysis, Vol. 1: Metric and Normed Spaces. Rochester, NY: Graylock Press; 1957.

    Google Scholar 

  22. Ushakov VN, Guseinov KG, Latushkin YA, Lebedev PD. On the coincidence of maximal stable bridges in two approach game problems for stationary control systems. Proc Steklov Inst Math (Suppl) 2010;268(suppl. 1):240–263. https://doi.org/10.1134/S0081543810050172.

    Article  MathSciNet  MATH  Google Scholar 

  23. Bellman R. Introduction to matrix analysis. New York: McGraw-Hill; 1970.

    MATH  Google Scholar 

  24. Schneider R, Vol. 44. Convex bodies: the Brunn–Minkowski Theory. Encyclopedia of Math and its Appl. Cambridge: Cambridge University Press; 1993. https://doi.org/10.1017/CBO9780511526282.

    Book  Google Scholar 

  25. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF. The mathematical theory of optimal processes. New York: John Wiley & Sons; 1962.

    MATH  Google Scholar 

  26. Coddington EA, Levinson N. Theory of ordinary differential equations. New York: McGraw-Hill; 1955.

    MATH  Google Scholar 

  27. Cardaliaguet P, Quincampoix M, Saint-Pierre P. Some algorithms for differential games with two players and one target. Math Model Numer Anal 1994;28(4):441–461. https://doi.org/10.1051/m2an/1994280404411.

    Article  MathSciNet  MATH  Google Scholar 

  28. Pontryagin LS. Linear differential games, II. Soviet Mathematics. Doklady 1967;8:910–912.

    MATH  Google Scholar 

  29. Kamneva LV, Patsko VS. Maximal stable bridge in game with simple motions in the plane. Advances in Dynamic and Evolutionary Games. Annals of the International Society of Dynamic Games. In: Thuijsman F and Wagener F, editors. Cham: Birkhäuser; 2016. p. 139–163. https://doi.org/10.1007/978-3-319-28014-1_7.

  30. Rockafellar RT. Convex analysis. Princeton, New Jersey: Princeton University Press; 1970.

    Book  MATH  Google Scholar 

  31. Bryson AE, Ho Y-C. 1969. Applied optimal control: optimization, estimation, and control. Blaisdell Publishing Company, Waltham Mass.

  32. Pshenichnyy BN, Sagaydak MI. Differential games with fixed time. J Cybernet 1971;1(1):117–135. https://doi.org/10.1080/01969727108545833.

    Article  MathSciNet  Google Scholar 

  33. Kamneva L. Computation of solvability set for differential games in the plane with simple motion and non-convex terminal set. Dyn Games Appl 2019;9(3):724–750. https://doi.org/10.1007/s13235-018-00292-x.

    Article  MathSciNet  MATH  Google Scholar 

  34. Aubin J-P, Cellina A. Differential inclusions. Set-valued maps and viability theory. Berlin, Heidelberg: Springer; 1984. https://doi.org/10.1007/978-3-642-69512-4.

    Book  MATH  Google Scholar 

  35. Patsko VS, Turova VL. 1995. Numerical solution of two-dimensional differential games: Preprint. IMM UrO RAN, Ekaterinburg. http://sector3.imm.uran.ru/preprint1995_01/preprint_1995_eng.pdf. Accessed 30 Nov 2022.

  36. Patsko VS, Turova VL. Minimum-time problem for linear second-order conflict-controlled systems. UKACC International Conference on Control’96. Conference Publication Number 427, vol 2, pp 947–952. University of Exeter, UK; 1996. http://sector3.imm.uran.ru/contr96c/contr96c_new2.pdf. Accessed 30 Nov 2022.

  37. Patsko VS, Turova VL. Numerical solutions to the minimum-time problem for linear second-order conflict-controlled systems. Proceedings of the 7th International Colloquium on Differential Equations, Plovdiv, Bulgaria, August 18–23, 1996, pp 329–338. VSP, Utrecht, the Netherlands; Tokyo, Japan. In: Bainov D, editors; 1997. http://sector3.imm.uran.ru/stat/PtskTrvBlg.pdf. Accessed 30 Nov 2022.

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Acknowledgements

The author thanks the reviewers for careful reading of the manuscript and their insightful comments.

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  1. Krasovskii Institute of Mathematics and Mechanics, 16 S. Kovalevskaya str., Yekaterinburg, 620108, Russia

    Liudmila Kamneva

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Kamneva, L. A Scheme for Calculating Solvability Sets “Up to Moment” in Linear Differential Games. J Dyn Control Syst 29, 989–1018 (2023). https://doi.org/10.1007/s10883-022-09627-9

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