On a finite time interval, a differential game for the minimax–maximin of a given cost functional is considered. In this game, the motion of a conflict-controlled dynamical system is described by functional differential equations of neutral type in Hale’s form. Under assumptions more general than those considered previously, a theorem on the existence of the value and saddle point of the game in classes of players’ closed-loop control strategies with memory of the motion history is proved. The proof involves the technique of the corresponding path-dependent Hamilton–Jacobi equations with coinvariant derivatives and the theory of minimax (generalized) solutions of such equations. In order to construct optimal strategies, which constitute a saddle point of the game, a recent result on the existence and uniqueness of a suitable minimax solution and a special Lyapunov–Krasovskii functional are used.
Keywords: differential game, neutral-type equation, game value, optimal strategies, path-dependent Hamilton–Jacobi equation, coinvariant derivatives, minimax solution
Received June 26, 2024
Revised July 11, 2024
Accepted July 15, 2024
Funding Agency: This work was supported by the Ministry of Science and Higher Education of the Russian Federation within a state contract (project FEWS-2024-0009).
Mikhail Igorevich Gomoyunov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Udmurt State University, Izhevsk, 426034 Russia, e-mail: m.i.gomoyunov@gmail.com
Nikolai Yur’evich Lukoyanov, Dr. Phys.-Math. Sci., RAS Academician, Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: nyul@imm.uran.ru
REFERENCES
1. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. NY, Springer, 1988, 517 p. ISBN: 978-1-4612-8318-8. Original Russian text published in Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial’nye igry, Moscow, Nauka Publ., 1974, 456 p.
2. Krasovskii N.N. Upravlenie dinamicheskoi sistemoi [Control of a dynamical system]. Moscow, Nauka Publ., 1985, 520 p.
3. Krasovskii N.N., Krasovskii A.N. Control under lack of information. Berlin etc.: Birkhäuser, 1995. 322 p. ISBN: 0-8176-3698-6 .
4. Hale J.K., Cruz M.A. Existence, uniqueness and continuous dependence for hereditary systems. Ann. Mat. Pura Appl., 1970, vol. 85, no. 1, pp. 63–81. doi: 10.1007/BF02413530
5. Akhmerov R.R., Kamenskii M.I., Potapov A.S., Rodkina A.E., Sadovskii B.N. Theory of equations of neutral type. J. Soviet Math., 1984, vol. 24, no. 6, pp. 674–719. doi: 10.1007/BF01305757
6. Hale J.K., Lunel S.M.V. Introduction to functional differential equations. NY: Springer, 1993, 447 p. doi: 10.1007/978-1-4612-4342-7
7. Lukoyanov N.Yu., Plaksin A.R. Differential games for neutral-type systems: An approximation model. Proc. Steklov Inst. Math., 2015, vol. 291, pp. 190–202. doi: 10.1134/S0081543815080155
8. Gomoyunov M.I., Lukoyanov N.Yu., Plaksin A.R. Existence of a value and a saddle point in positional differential games for neutral-type systems. Proc. Steklov Inst. Math. (Suppl.), 2017, vol. 299, suppl. 1, pp. S37–S48. doi: 10.1134/S0081543817090061
9. Gomoyunov M.I., Plaksin A.R. On basic equation of differential games for neutral-type systems. Mech. Solids, 2019, vol. 54, no. 2, pp. 131–143. doi: 10.3103/S0025654419030099
10. Lukoyanov N.Yu., Plaksin A.R. On the theory of positional differential games for neutral-type systems. Proc. Steklov Inst. Math. (Suppl.), 2020, vol. 309, suppl. 1, pp. S83–S92. doi: 10.1134/S0081543820040100
11. Plaksin A.R. Optimal positional strategies in differential games for neutral-type systems. Dyn. Games Appl., 2024. doi: 10.1007/s13235-024-00565-8
12. Garnysheva G.G., Subbotin A.I. Strategies of minimax aiming in the direction of the quasigradient. J. Appl. Math. Mech., 1994, vol. 58, no. 4, pp. 575–581. doi: 10.1016/0021-8928(94)90134-1
13. Subbotin A.I. Generalized solutions of first order PDEs: The dynamical optimization perspective. Boston: Birkhäuser, 1995, 314 p. doi: 10.1007/978-1-4612-0847-1
14. Lukoyanov N.Yu. Funktsional’nye uravneniya Gamil’tona–Yakobi i zadachi upravleniya s nasledstvennoi informatsiei [Functional Hamilton–Jacobi equations and control problems with hereditary information]. Yekaterinburg: Ural Federal University Publ., 2011, 243 p.
15. Plaksin A.R. On Hamilton–Jacobi–Isaacs–Bellman equation for neutral type systems. Vestn. Udmurt. Univ. Mat. Mekh. Komp. Nauki, 2017, vol. 27, no. 2, pp. 222–237 (in Russian). doi: 10.20537/vm170206
16. Gomoyunov M.I., Lukoyanov N.Yu. Minimax solutions of Hamiltoni–Jacobi equations in dynamical optimization problems for hereditary systems. Uspekhi Mat. Nauk, 2024, vol. 79, no. 2, pp. 43–144 (in Russian). doi: 10.4213/rm10166
17. Kim A.V. Functional differential equations. Application of i-smooth calculus. Dordrecht: Springer, 1999, 168 p. doi: 10.1007/978-94-017-1630-7
18. Plaksin A.R. Minimax solution of functional Hamilton–Jacobi equations for neutral type systems. Diff. Equat., 2019, vol. 55, no. 11. pp. 1475–1484. doi: 10.1134/S0012266119110077
19. Plaksin A.R. On the minimax solution of the Hamilton–Jacobi equations for neutral-type systems: the case of an inhomogeneous Hamiltonian. Diff. Equat., 2021, vol. 57, no. 11, pp. 1516–1526. doi: 10.1134/S0012266121110100
20. Plaksin A.R. Viscosity solutions of Hamilton–Jacobi equations for neutral-type systems. Appl. Math. Optim., 2023, vol. 88, no. 1, art. no 6. 29 p. doi: 10.1007/s00245-023-09980-6
21. Gomoyunov M.I., Lukoyanov N.Yu. Minimax solution of hereditary Hamilton–Jacobi equations for neutral type systems. Uspekhi Mat. Nauk, 2024, vol. 79, no. 4, pp. 177–178 (in Russian). doi: 10.4213/rm10182
22. Zhou J. Viscosity solutions to first order path-dependent HJB equations. 2020. 25 p.
http://Arxiv.org/abs/2004.02095 . doi: 10.48550/arXiv.2004.02095
23. Zhou J. Viscosity solutions to first order path-dependent Hamilton–Jacobi–Bellman equations in Hilbert space. Automatica, 2022, vol. 142, art. no. 110347. 15 p. doi: 10.1016/j.automatica.2022.110347
24. Gomoyunov M.I., Lukoyanov N.Yu., Plaksin A.R. Path-dependent Hamilton–Jacobi equations: the minimax solutions revised. Appl. Math. Optim., 2021, vol. 84, suppl. 1, pp. S1087–S1117. doi: 10.1007/s00245-021-09794-4
25. Filippov A.F. Differential equations with discontinuous righthand sides. Dordrecht, Kluwer, 304 p. ISBN: 90-277-2699-X. Original Russian text published in Filippov A.F. Differentsial’nye uravneniya s razryvnoi pravoi chast’yu, Moscow, Nauka Publ., 1985, 224 p.
Cite this article as: M.I. Gomoyunov, N.Yu. Lukoyanov. The value and optimal strategies in a positional differential game for a neutral-type system. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 3, pp. 86–98.
