On a solution of a guarantee optimization problem under the functional constraints on the disturbance. (English) Zbl 1431.93043

Gomoyunov, Mikhail; Serkov, Dmitriy

Dyn. Games Appl. 9, No. 3, 700-723 (2019).

Summary: The paper deals with a control problem for a dynamical system under disturbances. A motion of the system is considered on a finite interval of time and described by a nonlinear ordinary differential equation. The control is aimed at minimization of a given quality index. In addition to geometric constraints on the control and disturbance, it is supposed that the disturbance satisfies a compact functional constraint. Namely, all disturbance realizations that can happen in the system belong to some unknown set that is compact in the space \(L_1\). Within the game-theoretical approach, the problem of optimizing the guaranteed result of the control is studied. For solving this problem, we propose a new construction of the optimal control strategy. In the linear-convex case, this strategy can be numerically realized on the basis of the upper convex hulls method. Examples are considered. Results of numerical simulations are given.

MSC:

93C73	Perturbations in control/observation systems
93C15	Control/observation systems governed by ordinary differential equations
49J15	Existence theories for optimal control problems involving ordinary differential equations
91A80	Applications of game theory

Keywords:

control problem; disturbances; functional constraint; optimal guaranteed result; optimal strategy; reconstruction; numerical method

References:

[1]	Gomoyunov MI, Kornev DV (2016) On calculating the value of a differential game in the class of counter strategies. Ural Math J 2(1):38-47 · Zbl 1398.91087
[2]	Gomoyunov MI, Serkov DA (2017) Control with a guide in the guarantee optimization problem under functional constraints on the disturbance. Proc Steklov Inst Math 299(Suppl 1):49-60 · Zbl 1387.49055
[3]	Isaacs R (1965) Differential games. Wiley, New York · Zbl 0125.38001
[4]	Kornev DV (2012) On numerical solution of positional differential games with nonterminal payoff. Autom Rem Control 73(11):1808-1821 · Zbl 1270.91014
[5]	Krasovskii AN, Krasovskii NN (1995) Control under lack of information. Birkhäuser, Boston · Zbl 0827.93001
[6]	Krasovskii NN (1985) Control of a dynamical system. Nauka, Moscow (in Russian)
[7]	Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New York
[8]	Kryazhimskii, AV; Kryazhimskii, AV (ed.), The problem of optimization of the ensured result: unimprovability of full-memory strategies, No. 1, 636-675 (1991), Teaneck · Zbl 0752.93022
[9]	Lukoyanov NYu, Gomoyunov MI (2019) Differential games on minmax of the positional quality index (submitted to Dyn Games Appl) · Zbl 1431.91037
[10]	Osipov YuS, Kryazhimskii AV (1995) Inverse problems for ordinary differential equations: dynamical solutions. Gordon and Breach Science Publishers, London · Zbl 0884.34015
[11]	Roxin E (1969) Axiomatic approach in differential games. J Optim Theory Appl 3(3):153-163 · Zbl 0175.10504
[12]	Ryll-Nardzewski, C.; Dresher, M. (ed.); Shapley, LS (ed.); Tucker, AW (ed.), A theory of pursuit and evasion, 113-126 (1964), Princeton
[13]	Serkov DA (2012) Guaranteed control under functionally restricted disturbances. Mat Teor Igr Prilozh 4(2):71-95 (in Russian) · Zbl 1273.93114
[14]	Serkov DA (2012) Optimal guarantee under the disturbances of Caratheodory type. Vestn Udmurtsk Univ Mat Mekh Komp Nauki 2:74-83 (in Russian) · Zbl 1299.93119
[15]	Serkov DA (2013) Optimal risk control under functionally restricted disturbances. Mat Teor Igr Prilozh 5(1):74-103 (in Russian) · Zbl 1273.91102
[16]	Serkov DA (2014) On the unimprovability of full-memory strategies in the risk minimization problem. Proc Steklov Inst Math 287(Suppl 1):175-184 · Zbl 1312.49022
[17]	Serkov DA (2015) On the unimprovability of full-memory strategies in problems of guaranteed result optimization. Proc Steklov Inst Math 291(Suppl 1):157-172 · Zbl 1334.49117
[18]	Subbotin AI, Chentsov AG (1981) Guarantee optimization in control problems. Nauka, Moscow (in Russian) · Zbl 0542.90106