On guarantee optimization in control problem with finite set of disturbances. (English) Zbl 1483.49046

Gomoyunov, Mikhail Igorevich; Serkov, Dmitriĭ Aleksandrovich

Vestn. Udmurt. Univ., Mat. Mekh. Komp’yut. Nauki 31, No. 4, 613-628 (2021).

Summary: In this paper, we deal with a control problem under conditions of disturbances, which is stated as a problem of optimization of the guaranteed result. Compared to the classical formulation of such problems, we assume that the set of admissible disturbances is finite and consists of piecewise continuous functions. In connection with this additional functional constraint on the disturbance, we introduce an appropriate class of non-anticipative control strategies and consider the corresponding value of the optimal guaranteed result. Under a technical assumption concerning a property of distinguishability of the admissible disturbances, we prove that this result can be achieved by using control strategies with full memory. As a consequence, we establish unimprovability of the class of full-memory strategies. A key element of the proof is a procedure of recovering the disturbance acting in the system, which allows us to associate every non-anticipative strategy with a full-memory strategy providing a close guaranteed result. The paper concludes with an illustrative example.

MSC:

49N30	Problems with incomplete information (optimization)
49N35	Optimal feedback synthesis
93C15	Control/observation systems governed by ordinary differential equations

Keywords:

control problem under disturbances; optimal guaranteed result; non-anticipative strategy; full-memory strategy; recovery of disturbances; unimprovability

References:

[1]	Isaacs R., Differential games, John Wiley and Sons, New York, 1965 · Zbl 0125.38001
[2]	Pontrjagin L. S., “Linear differential games of pursuit”, Mathematics of the USSR-Sbornik, 40:3 (1981), 285-303 · Zbl 0468.90099
[3]	Krasovskii N. N., Subbotin A. I., Game-theoretical control problems, Springer, New York, 1988 · Zbl 0649.90101
[4]	Krasovskii A. N., Krasovskii N. N., Control under lack of information, Birkh{ä}user, Boston, 1995 · Zbl 0827.93001
[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 · Zbl 0752.93022
[6]	Serkov D. A., “On a dynamic game problem with an indecomposable set of disturbances”, Ural Mathematical Journal, 5:2 (2019), 72-79 · Zbl 1448.91054
[7]	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, 30:4 (2020), 553-571 · Zbl 1479.49085
[8]	Barron E. N., “Differential games with Lipschitz control functions and fixed initial control positions”, Journal of Differential Equations, 26:2 (1977), 161-180 · Zbl 0415.90092
[9]	Nikol’skij M. S., “A crossing problem with possible stop of engine”, Differential Equations, 29:11 (1993), 1681-1684 · Zbl 0816.90147
[10]	Ledyaev Yu. S., “Program-predictive feedback control for systems with evolving dynamics”, IFAC-PapersOnLine, 51:32 (2018), 723-726
[11]	Ukhobotov V. I., “On a control problem under disturbance and possible breakdown”, Proceedings of the Steklov Institute of Mathematics, 307, suppl. 1 (2019), 159-171 · Zbl 1443.93060
[12]	Ushakov V. N., Ershov A. A., “On recovering of unknown constant parameter by several test controls”, Ufa Mathematical Journal, 12:4 (2020), 99-113 · Zbl 1474.93145
[13]	Roxin E., “Axiomatic approach in differential games”, Journal of Optimization Theory and Applications, 3:3 (1969), 153-163 · Zbl 0175.10504
[14]	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 · Zbl 0244.90046
[15]	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), 157-172 · Zbl 1334.49117
[16]	Vinter R. B., “Minimax optimal control”, SIAM Journal on Control and Optimization, 44:3 (2005), 939-968 · Zbl 1087.49015