Geodesic
Criteria of minimax solutions for Hamilton–Jacobi equations with coinvariant fractional-order derivatives
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 3, pp. 25-42 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider Hamilton–Jacobi equations with coinvariant fractional-order derivatives, which arise in optimal control problems for dynamical systems whose evolution is described by differential equations with Caputo fractional derivatives. For upper, lower, and minimax (generalized) solutions of such equations, a number of criteria are established, which are expressed in terms of nonlocal conditions of stability with respect to characteristic differential inclusions satisfying a certain set of requirements and in the form of inequalities for suitably introduced derivatives of functionals in multivalued directions. In particular, these criteria make it possible to establish a correspondence between the results on the existence and uniqueness of minimax solutions of boundary value problems for the considered Hamilton–Jacobi equations obtained earlier under various assumptions.
Keywords: Hamilton–Jacobi equations, coinvariant derivatives, fractional-order derivatives, minimax solutions, characteristic differential inclusions, derivatives in multivalued directions. 
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M. I. Gomoyunov. Criteria of minimax solutions for Hamilton–Jacobi equations with coinvariant fractional-order derivatives. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 3, pp. 25-42. http://geodesic.mathdoc.fr/item/TIMM_2021_27_3_a2/

[1] Subbotin A.I., Minimaksnye neravenstva i uravneniya Gamiltona - Yakobi, Nauka, M., 1991, 216 pp.

[2] Subbotin A.I., Generalized solutions of first order PDEs: The dynamical optimization perspective, Birkhauser, Basel, 1995, 314 pp. | DOI

[3] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp.

[4] Subbotin A.I., Chentsov A.G., Optimizatsiya garantii v zadachakh upravleniya, Nauka, M., 1981, 288 pp.

[5] Krasovskii N.N., Krasovskii A.N., Control under lack of information, Birkhauser, Berlin etc., 1995, 322 pp.

[6] Krasovskii N.N., “K zadache unifikatsii differentsialnykh igr”, Dokl. AN SSSR, 226:6 (1976), 1260–1263 | Zbl

[7] Ushakov V.N., Parshikov G.V., “Unifikatsiya differentsialnogo vklyucheniya s parametrom v zadache o sblizhenii”, Dokl. AN, 477:4 (2017), 406–409 | DOI | Zbl

[8] Subbotin A.I., Tarasev A.M., Ushakov V.N., “Obobschennye kharakteristiki uravnenii Gamiltona - Yakobi”, Izvestiya AN. Tekhnicheskaya kibernetika, 1993, no. 1, 190–197

[9] Subbotina N.N., “Metod kharakteristik dlya uravnenii Gamiltona - Yakobi i ego prilozheniya v dinamicheskoi optimizatsii”, Sovremennaya matematika i ee prilozheniya, 20 (2004), 1–129

[10] Subbotin A.I., Tarasyev A.M., “Stability properties of the value function of a differential game and viscosity solutions of Hamilton - Jacobi equations”, Problems Control Inform. Theory, 15:6 (1986), 451–463 | Zbl

[11] Gomoyunov M.I., “Minimaksnye resheniya odnorodnykh uravnenii Gamiltona - Yakobi s koinvariantnymi proizvodnymi drobnogo poryadka”, Tr. Instituta matematiki i mekhaniki UrO RAN, 26:4 (2020), 106–125 | DOI

[12] Gomoyunov M.I., “Minimax solutions of Hamilton - Jacobi equations with fractional coinvariant derivatives”, statya poslana v pechat, ESAIM: Control Optim. Calc. Var., 2021, arXiv: 2011.11306 | Zbl

[13] Gomoyunov M.I., “Dynamic programming principle and Hamilton - Jacobi - Bellman equations for fractional-order systems”, SIAM J. Control Optim., 58:6 (2020), 3185–3211 | DOI | Zbl

[14] Samko S.G., Kilbas A.A., Marichev O.I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987, 688 pp.

[15] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential equations, Elsevier, N Y, 2006, 540 pp. | Zbl

[16] Diethelm K., The analysis of fractional differential equations, Springer, Berlin, 2010, 247 pp. | DOI | Zbl

[17] Kim A.V., Functional differential equations. Application of i-smooth calculus, Springer, Dordrecht, 1999, 168 pp. | DOI

[18] Lukoyanov N.Yu., “O svoistvakh funktsionala tseny differentsialnoi igry s nasledstvennoi informatsiei”, Prikl. matematika i mekhanika, 65:3 (2001), 375–384 | Zbl

[19] Lukoyanov N.Yu., “Functional Hamilton - Jacobi type equations in ci-derivatives for systems with distributed delays”, Nonlinear Funct. Anal. Appl., 8:3 (2003), 365–397 | Zbl

[20] Lukoyanov N.Yu., “Minimaksnye i vyazkostnye resheniya v zadachakh optimizatsii nasledstvennykh sistem”, Tr. Instituta matematiki i mekhaniki UrO RAN, 15:4 (2009), 183–194

[21] Lukoyanov N.Yu., Funktsionalnye uravneniya Gamiltona - Yakobi i zadachi upravleniya s nasledstvennoi informatsiei, Izd-vo Ural. federal. un-ta, Ekaterinburg, 2011, 243 pp.

[22] Gomoyunov M.I., “K teorii differentsialnykh vklyuchenii s drobnymi proizvodnymi Kaputo”, Differents. uravneniya, 56:11 (2020), 1419–1432 | DOI