On the construction of partially non-anticipative multiselector and its application to dynamic optimization problems

D. A. Serkov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 34 (2024) no. 3, pp. 410-434

Résumé

Let sets of functions $Z$ and $\Omega$ on the time interval $T$ be given, let there also be a multifunction (m/f) $\alpha$ acting from $\Omega$ to $Z$ and a finite set $\Delta$ of moments from $T$. The work deals with the following questions: the first one is the connection between the possibility of stepwise construction (specified by $\Delta$) of a selector $z$ of $\alpha(\omega)$ for an unknown step-by-step implemented argument $\omega\in\Omega$ and the existence of a multiselector (m/s) $\beta$ of the m/f $\alpha$ with a non-anticipatory property of special kind (we call it partially or $\Delta$-non-anticipated); the second question is when and how non-anticipated m/s could be expressed by means of partially non-anticipated one; and the last question is how to build the above $\Delta$-non-anticipated m/s $\beta$ for a given pair $(\alpha,\Delta)$. The consideration of these questions is motivated by the presence of such step-by-step procedures in the differential game theory, for example, in the alternating integral method, in pursuit–evasion problems posed with use of counter-strategies, and in the method of guide control. It is shown that the step-by-step construction of the value $z\in\alpha(\omega)$ can be carried out for any steps-implemented argument $\omega$ if and only if the above m/s $\beta$ is non-empty-valued. The key point of the work is the description of finite-step procedure for calculation of this $\Delta$-non-anticipated m/s $\beta$. Conditions are given that guarantee the m/s $\beta$ be a non-anticipative one. Illustrative examples are considered that include, in particular, control problems with disturbance.

Détail

Keywords: non-anticipative multi-selectors, set-valued strategies, optimization of guarantee

Bibliographie

[1] Fleming W.H., “The convergence problem for differential games”, Journal of Mathematical Analysis and Applications, 3:1 (1961), 102–116 | DOI | MR | Zbl

[2] Pontryagin L.S., “Linear differential games. II”, Soviet Mathematics. Doklady, 8 (1967), 910–912 | MR | Zbl | Zbl

[3] Blagodatskikh A.I., Petrov N.N., “Simultaneous multiple capture of rigidly coordinated evaders”, Dynamic Games and Applications, 9:3 (2019), 594–613 | DOI | MR | Zbl

[4] Chernov A.V., “On Volterra functional operator games on a given set”, Automation and Remote Control, 75:4 (2014), 787–803 | DOI | MR | MR | Zbl

[5] Petrosyan L.A., Zenkevich N.A., Game theory, World Scientific, Singapore, 2016 | DOI | MR | Zbl

[6] Khlopin D.V., Chentsov A.G., “On a control problem with incomplete information: quasistrategies and control procedures with a model”, Differential Equations, 41:12 (2005), 1727–1742 | DOI | MR | Zbl

[7] Gomoyunov M.I., Serkov D.A., “On guarantee optimization in control problem with finite set of disturbances”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 31:4 (2021), 613–628 | DOI | MR | Zbl

[8] Chentsov A.G., “Nonanticipating multimappings and their construction by the method of program iterations: I”, Differential Equations, 37:4 (2001), 498–509 | DOI | MR | Zbl

[9] Chentsov A.G., “Nonanticipating multimappings and their construction by the method of program iterations: II”, Differential Equations, 37:5 (2001), 713–723 | DOI | MR | Zbl

[10] Serkov D.A., “Unlocking of predicate: application to constructing a non-anticipating selection”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 27:2 (2017), 283–291 | DOI | MR | Zbl

[11] Chentsov A.G., Selections of multivalued strategies in differential games, Deposited in VINITI 26.09.1978, No 3101–78, IMM USC of the USSR Academy of Sciences, Sverdlovsk, 45 pp. (in Russian)

[12] Cardaliaguet P., Plaskacz S., “Invariant solutions of differential games and Hamilton–Jacobi–Isaacs equations for time-measurable Hamiltonians”, SIAM Journal on Control and Optimization, 38:5 (2000), 1501–1520 | DOI | MR | Zbl

[13] Serkov D.A., Chentsov A.G., “On the construction of a nonanticipating selection of a multivalued mapping”, Proceedings of the Steklov Institute of Mathematics, 309:suppl. 1 (2020), S125–S138 | DOI | DOI | MR | MR

[14] Chentsov A.G., “The iterative realization of nonanticipating multivalued mappings”, Doklady Mathematics, 56:3 (1997), 927–930 | MR | Zbl

[15] Subbotin A.I., Chentsov A.G., Optimization of guarantee in control problems, Nauka, Moscow, 1981 | MR

[16] Serkov D.A., “Step-by-step construction of optimal motion and non-anticipative multi-selectors”, Control Theory and Mathematical Modeling (Proceedings of the All-Russian Conference with International Participation, Dedicated to the Memory of Professor N.V. Azbelev and Professor E.L. Tonkov), Udmurt State University, Izhevsk, 2022, 219–223 (in Russian)

[17] Engelking R., General topology, Panstwowe Wydawnictwo Naukowe, Warszawa, 1985 | MR

[18] Kuratowski K., Topology, v. 1, Academic Press, New York–London, 1966 | DOI | Zbl

[19] Krasovskij N.N., Subbotin A.I., Ushakov V.N., “A minimax differential game”, Soviet Mathematics. Doklady, 13 (1972), 1200–1204 | MR | Zbl | Zbl

[20] Pshenichnyj B.N., “The structure of differential games”, Soviet Mathematics. Doklady, 10 (1969), 70–72 | Zbl | Zbl