+7 (3412) 91 60 92

## Archive of Issues

Russia Yekaterinburg
Year
2020
Volume
30
Issue
4
Pages
553-571
 Section Mathematics Title Non-anticipative strategies in guarantee optimization problems under functional constraints on disturbances Author(-s) Gomoyunov M.I.ab, Serkov D.A.ab Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa, Ural Federal Universityb Abstract For a dynamical system controlled under conditions of disturbances, a problem of optimizing the guaranteed result is considered. A feature of the problem is the presence of functional constraints on disturbances, under which, in general, the set of admissible disturbances is not closed with respect to the operation of “gluing up” of two of its elements. This circumstance does not allow to apply directly the methods developed within the differential games theory for studying the problem and, thus, leads to the necessity of modifying them appropriately. The paper provides a new notion of a non-anticipative control strategy. It is proved that the corresponding functional of the optimal guaranteed result satisfies the dynamic programming principle. As a consequence, so-called properties of $u$- and $v$-stability of this functional are established, which may allow, in the future, to obtain a constructive solution of the problem in the form of feedback (positional) controls. Keywords guarantee optimization, functional constraints, non-anticipative strategies, dynamic programming principle UDC 517.977 MSC 49N70, 49L20, 93C15 DOI 10.35634/vm200402 Received 31 August 2020 Language English Citation Gomoyunov M.I., Serkov D.A. Non-anticipative strategies in guarantee optimization problems under functional constraints on disturbances, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 4, pp. 553-571. References Krasovskii N.N., Subbotin A.I. Game-theoretical control problems, New York: Springer, 1988. https://www.springer.com/gp/book/9781461283188 Krasovskii A.N., Krasovskii N.N. Control under lack of information, Boston: Birkhäuser, 1995. https://doi.org/10.1007/978-1-4612-2568-3 Barron E.N. Differential games with Lipschitz control functions and fixed initial control positions, Journal of Differential Equations, 1977, vol. 26, issue 2, pp. 161-180. https://doi.org/10.1016/0022-0396(77)90188-7 Barron E.N., Jensen R. A nonlinear evolution system with two subdifferentials and monotone differential games, Journal of Mathematical Analysis and Applications, 1983, vol. 97, issue 1, pp. 65-80. https://doi.org/10.1016/0022-247X(83)90238-X Kryazhimskii A.V. The problem of optimization of the ensured result: unimprovability of full-memory strategies, Constantin Caratheodory: An International Tribute, Teaneck, New Jersey: World Scientific, 1991, vol. 1, pp. 636-675. https://doi.org/10.1142/9789814350921_0037 Nikol'skij M.S. A crossing problem with possible stop of engine, Differential Equations, 1993, vol. 29, issue 11, pp. 1681-1684. https://zbmath.org/?q=an:0816.90147 Nikol'skii M.S. On control problems for linear objects with disturbances in the dynamics, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 1995, vol. 3, pp. 132-146 (in Russian). http://mi.mathnet.ru/eng/timm371 Ershov A.A., Ushakov V.N. An approach problem for a control system with an unknown parameter, Sbornik: Mathematics, 2017, vol. 208, no. 9, pp. 1312-1352. https://doi.org/10.1070/SM8761 Ledyaev Yu.S. Program-predictive feedback control for systems with evolving dynamics, IFAC-PapersOnLine, 2018, vol. 51, issue 32, pp. 723-726. https://doi.org/10.1016/j.ifacol.2018.11.461 Ukhobotov V.I. On a control problem under disturbance and possible breakdown, Proceedings of the Steklov Institute of Mathematics, 2019, vol. 307, suppl. 1, pp. S159-S171. https://doi.org/10.1134/S0081543819070137 Serkov D.A. On the unimprovability of full-memory strategies in problems of guaranteed result optimization, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 291, suppl. 1, pp. S157-S172. https://doi.org/10.1134/S0081543815090114 Gomoyunov M.I., Serkov D.A. On a solution of a guarantee optimization problem under a functional constraint on the disturbance, Dynamic Games and Applications, 2019, vol. 9, no. 3, pp. 700-723. https://doi.org/10.1007/s13235-018-0279-1 Ryll-Nardzewski C. A theory of pursuit and evasion, Advances in game theory, Princeton University Press, 1964, vol. 52, pp. 113-126. https://doi.org/10.1515/9781400882014-010 Roxin E. Axiomatic approach in differential games, Journal of Optimization Theory and Applications, 1969, vol. 3, no. 3, pp. 153-163. https://doi.org/10.1007/BF00929440 Varaiya P., Lin J. Existence of saddle points in differential games, SIAM Journal on Control, 1969, vol. 7, issue 1, pp. 141-157. https://doi.org/10.1137/0307011 Elliott R.J., Kalton N.J. The existence of value in differential games of pursuit and evasion, Journal of Differential Equations, 1972, vol. 12, issue 3, pp. 504-523. https://doi.org/10.1016/0022-0396(72)90022-8 Bettiol P., Quincampoix M., Vinter R.B. Existence and characterization of the values of two player differential games with state constraints, Applied Mathematics and Optimization, 2019, vol. 80, issue 3, pp. 765–799. https://doi.org/10.1007/s00245-019-09608-8 Serkov D.A. On a dynamic game problem with an indecomposable set of disturbances, Ural Mathematical Journal, 2019, vol. 5, issue 2, pp. 72-79. https://doi.org/10.15826/umj.2019.2.007 Sousa J.B., Pereira F.L. A differential game with graph constrained switching strategies, IFAC Proceedings Volumes, 2001, vol. 34, issue 6, pp. 887-891. https://doi.org/10.1016/S1474-6670(17)35292-8 Castaing C., Valadier M. Convex analysis and measurable multifunctions, Berlin: Springer, 1977. https://doi.org/10.1007/BFb0087685 Chentsov A.G. Selektory mnogoznachnykh strategii v differentsial’nykh igrakh (Selections of multivalued strategies in differential games), Sverdlovsk, 1978. Available from VINITI, no. 3101-78. Cardaliaguet P. A differential game with two players and one target, SIAM Journal on Control and Optimization, 1996, vol. 34, issue 4, pp. 1441-1460. https://doi.org/10.1137/S036301299427223X Cardaliaguet P., Plaskacz S. Invariant solutions of differential games and Hamilton-Jacobi-Isaacs equations for time-measurable Hamiltonians, SIAM Journal on Control and Optimization, 2000, vol. 38, issue 5, pp. 1501-1520. https://doi.org/10.1137/S0363012998296219 Serkov D.A., Chentsov A.G. On the construction of a nonanticipating selection of a multivalued mapping, Proceedings of the Steklov Institute of Mathematics, 2020, vol. 309, suppl. 1, pp. S125-S138. https://doi.org/10.1134/S008154382004015X Kurzhanskii A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti (Control and observation under conditions of uncertainty), Moscow: Nauka, 1977. Full text