Geodesic
Continuous dependence of sets in a space of measures and a program minimax problem
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 2, pp. 277-299 Cet article a éte moissonné depuis la source Math-Net.Ru

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For conflict-controlled dynamical systems satisfying the conditions of generalized uniqueness and uniform boundedness, the solvability of the minimax problem in the class of generalized controls is studied. The issues of consistency of such an extension are considered; i. e., the possibility of approximating generalized controls in the space of strategic measures by embeddings of ordinary controls is analyzed. For this purpose, the dependence of the set of measures on the general marginal distribution specified on one of the factors of the base space is studied. The continuity of this dependence in the Hausdorff metric defined by the metric corresponding to the $*$-weak topology in the space of measures is established. The density of embeddings of ordinary controls and control-noise pairs in sets of corresponding generalized controls in the $*$-weak topologies is also shown.
Keywords: generalized controls, strategic measures, minimax problem, $*$-weak convergence, Hausdorff metric. 
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A. G. Chentsov; D. A. Serkov. Continuous dependence of sets in a space of measures and a program minimax problem. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 2, pp. 277-299. http://geodesic.mathdoc.fr/item/TIMM_2024_30_2_a18/

[1] Krasovskii N. N., Subbotin A. I., “Alternativa dlya igrovoi zadachi sblizheniya”, Prikladnaya matematika i mekhanika, 34:6 (1970), 1005–1022 | DOI | Zbl

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

[3] Kryazhimskii A. V., “K teorii pozitsionnykh differentsialnykh igr sblizheniya-ukloneniya”, Dokl. AN SSSR, 239:4 (1978), 779–782 | MR | Zbl

[4] Hopenhayn H., “Entry, exit, and firm dynamics in long run equilibrium”, Econometrica, 60 (1992), 1127–1150 | DOI | MR | Zbl

[5] Bergin J., Bernhardt D., “Anonymous sequential games: Existence and characterization of equilibria”, Economic Theory, 5 (1995), 461–489 | DOI | MR | Zbl

[6] Bergin J., “On the continuity of correspondence on sets of measures with restricted marginals”, Economic Theory, 13 (1999), 471–481 | DOI | MR | Zbl

[7] Bogachev V. I., Popova S. N., “Rasstoyaniya Khausdorfa mezhdu kaplingami i optimalnaya transportirovka s parametrom”, Mat. sb., 215:1 (2024), 33–58 | DOI | MR

[8] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977, 624 pp.

[9] Warga J., “Nonsmooth problems with conflicting controls”, SIAM J. Control Optim., 29 (1991), 678–701 | DOI | MR | Zbl

[10] Warga J., Zhu Q.J., “A proper relaxation of Shiftedand Delayed Controls”, J. Math. Anal. Appl., 169 (1992), 546–561 | DOI | MR | Zbl

[11] Rosenblueth J. F., “Proper relaxation of optimal control problems”, J. Optim. Theory Appl., 74:3 (1992), 509–526 | DOI | MR | Zbl

[12] Serkov D. and Chentsov A., “On a property of continuous dependence of sets in the space of measures”, Nonlinear Analysis and Extremal Problems (NLA-2022), 7 Intern. conf.: Proceedings (Irkutsk, Russia, July 15–22, 2022), ISDCT SB RAS, Irkutsk, 2022, 106–107

[13] Chentsov A.G., “Ob odnoi igrovoi zadache upravleniya na minimaks”, Izv. AN SSSR. Ser. Tekhn. kibernetika, 1975, no. 1, 39–46 | Zbl

[14] Kuratovskii K., Mostovskii A., Teoriya mnozhestv, Mir, M., 1970, 416 pp.

[15] Bogachev V.I., Osnovy teorii mery, v. 1, NITs “Regulyarnaya i khaoticheskaya dinamika”, Moskva; Izhevsk, 2003, 545 pp.

[16] Danford N., Shvarts Dzh., Lineinye operatory, IL, M., 1962, 895 pp.

[17] Chentsov A.G., Elementy konechno-additivnoi teorii mery, I, UGTU-UPI, Ekaterinburg, 2008, 389 pp.

[18] Billingsli P., Skhodimost veroyatnostnykh mer, Nauka, M., 1977

[19] Bogachev V.I., Osnovy teorii mery, v. 2, NITs “Regulyarnaya i khaoticheskaya dinamika”, Moskva; Izhevsk, 2003, 578 pp.

[20] Engelking R., Obschaya topologiya, Mir, M., 1986, 752 pp.

[21] Neve Zh., Matematicheskie osnovy teorii veroyatnostei, Mir, M., 1969, 309 pp.

[22] Dugundji J., “An extension of {Tietze's} theorem”, Pacific J. Math., 1:3 (1951), 353–367 | DOI | MR | Zbl

[23] Chentsov A.G., “Maksiminnoe uklonenie v differentsialnoi igre”, Differents. uravneniya, 12:5 (1976), 848–856 | MR | Zbl

[24] Dedonne Zh., Osnovy sovremennogo analiza, Mir, M., 1964, 430 pp.

[25] Constantin A., “A uniqueness criterion for ordinary differential equations”, J. Diff. Eq., 342:5 (2023), 179–192 | DOI | MR | Zbl