Non-anticipative strategies in guarantee optimization problems under functional constraints on disturbances

M. I. Gomoyunov   ; D. A. Serkov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 30 (2020) no. 4, pp. 553-571

Résumé

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.

Détail

Keywords: guarantee optimization, functional constraints, non-anticipative strategies, dynamic programming principle.

Bibliographie

[1] Krasovskii N. N., Subbotin A. I., Game-theoretical control problems, Springer, New York, 1988 https://www.springer.com/gp/book/9781461283188 | Zbl

[2] Krasovskii A. N., Krasovskii N. N., Control under lack of information, Birkhäuser, Boston, 1995 | DOI | MR

[3] Barron E. N., “Differential games with Lipschitz control functions and fixed initial control positions”, Journal of Differential Equations, 26:2 (1977), 161–180 | DOI | Zbl

[4] Barron E. N., Jensen R., “A nonlinear evolution system with two subdifferentials and monotone differential games”, Journal of Mathematical Analysis and Applications, 97:1 (1983), 65–80 | DOI | MR | Zbl

[5] Kryazhimskii A. V., “The problem of optimization of the ensured result: unimprovability of full-memory strategies”, Constantin Caratheodory: An International Tribute, v. 1, World Scientific, Teaneck, New Jersey, 1991, 636–675 | DOI | MR

[6] Nikol'skij M. S., “A crossing problem with possible stop of engine”, Differential Equations, 29:11 (1993), 1681–1684 | MR | Zbl

[7] Nikol'skii M. S., “On control problems for linear objects with disturbances in the dynamics”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 3, 1995, 132–146 (in Russian)

[8] Ershov A. A., Ushakov V. N., “An approach problem for a control system with an unknown parameter”, Sbornik: Mathematics, 208:9 (2017), 1312–1352 | DOI | DOI | MR | Zbl

[9] Ledyaev Yu.S., “Program-predictive feedback control for systems with evolving dynamics”, IFAC-PapersOnLine, 51:32 (2018), 723–726 | DOI

[10] Ukhobotov V. I., “On a control problem under disturbance and possible breakdown”, Proceedings of the Steklov Institute of Mathematics, 307, suppl. 1 (2019), S159–S171 | DOI | DOI | MR | Zbl

[11] Serkov D. A., “On the unimprovability of full-memory strategies in problems of guaranteed result optimization”, Proceedings of the Steklov Institute of Mathematics, 291, suppl. 1 (2015), S157–S172 | DOI | MR | Zbl

[12] 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, 9:3 (2019), 700–723 | DOI | MR | Zbl

[13] Ryll-Nardzewski C., “A theory of pursuit and evasion”, Advances in game theory, 52, Princeton University Press, 1964, 113–126 | DOI | MR

[14] Roxin E., “Axiomatic approach in differential games”, Journal of Optimization Theory and Applications, 3:3 (1969), 153–163 | DOI | MR | Zbl

[15] Varaiya P., Lin J., “Existence of saddle points in differential games”, SIAM Journal on Control, 7:1 (1969), 141–157 | DOI | Zbl

[16] Elliott R. J., Kalton N. J., “The existence of value in differential games of pursuit and evasion”, Journal of Differential Equations, 12:3 (1972), 504–523 | DOI | MR | Zbl

[17] Bettiol P., Quincampoix M., Vinter R. B., “Existence and characterization of the values of two player differential games with state constraints”, Applied Mathematics Optimization, 80:3 (2019), 765–799 | DOI | MR | Zbl

[18] Serkov D. A., “On a dynamic game problem with an indecomposable set of disturbances”, Ural Mathematical Journal, 5:2 (2019), 72–79 | DOI | MR | Zbl

[19] Sousa J. B., Pereira F. L., “A differential game with graph constrained switching strategies”, IFAC Proceedings Volumes, 34:6 (2001), 887–891 | DOI

[20] Castaing C., Valadier M., Convex analysis and measurable multifunctions, Springer, Berlin, 1977 | DOI | MR | Zbl

[21] Chentsov A. G., Selections of multivalued strategies in differential games, Available from VINITI, no. 3101-78, Sverdlovsk, 1978

[22] Cardaliaguet P., “A differential game with two players and one target”, SIAM Journal on Control and Optimization, 34:4 (1996), 1441–1460 | DOI | MR | Zbl

[23] 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

[24] 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

[25] Kurzhanskii A. B., Control and observation under conditions of uncertainty, Nauka, M., 1977