Krasovskii–Subbotin Approach to Mean Field Type Differential Games

Averboukh, Yurii

doi:10.1007/s13235-018-0282-6

Cited by 13 publications

(11 citation statements)

References 46 publications

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“…Proposition 3.9. Let µ be a generalized solution produced by a control sequence {u k } ⊂ U , and 5 Then there exist…”

Section: Impulsive Control Theory Formalismmentioning

confidence: 99%

“…A significant progress in analysing such problems has been made in the last few years, mainly thanks to modern achievements in geometry and analysis on metric spaces of probability measures [3,39]. The existing works are mainly concentrated in two directions: one collection of studies is devoted to necessary optimality conditions [8-10, 20, 33], another part is focused on the dynamic programming approach [5,6,18,28]. In all cited papers, the driving vector field is assumed to be L ∞ bounded in time variable, which makes the problem relatively regular.…”

Section: Introductionmentioning

confidence: 99%

“…Again, if it is not the case, we can pass to a pointwise converging subsequence of {F (u k ,|u k |) }, whose existence is guaranteed by the Helly's selection principle, and make it right continuous 5. The natural ordering of jump points of V, as well as the triviality of the singular continuous part V sc , are rather artificial assumptions, which are hard to be guaranteed a priori.…”

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confidence: 99%

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Impulsive control of nonlocal transport equations

Pogodaev

Staritsyn

2020

Journal of Differential Equations

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The paper extends an impulsive control-theoretical framework towards dynamical systems in the space of measures. We consider a transport equation describing the time-evolution of a conservative "mass" (probability measure), which represents an infinite ensemble of interacting particles. The driving vector field contains nonlocal terms and is affine in control variable. The control is assumed to be common for all the agents, i.e., it is a function of time variable only. The main feature of the addressed model is the admittance of "shock" impacts, i.e. controls, which can be arbitrary close in their influence on each an agent to Dirac-type distributions. We construct an impulsive relaxation of this system and of the corresponding optimal control problem. For the latter we establish a necessary optimality condition in the form of Pontryagin's Maximum Principle.2000 Mathematics Subject Classification: 49K20, 49J45, 93C20

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“…Proposition 3.9. Let µ be a generalized solution produced by a control sequence {u k } ⊂ U , and 5 Then there exist…”

Section: Impulsive Control Theory Formalismmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Impulsive control of nonlocal transport equations

Pogodaev

Staritsyn

2020

Journal of Differential Equations

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“…There is proved the existence theorem for the value function. The feedback approach to the mean field type differential games is developed in [5]. That paper extends the Krasovskii-Subbotin extremal shift rule to the mean-field type control system.…”

Section: Introductionmentioning

confidence: 99%

Lattice approximations of the first-order mean field type differential games

Averboukh¹

2020

Preprint

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The theory of first-order mean field type differential games examines the systems of infinitely many identical agents interacting via some external media under assumption that each agent is controlled by two players. We study the approximations of the value function of the first-order mean field type differential game using solutions of model finite-dimensional differential games. The model game appears as a mean field type continuous time Markov game, i.e., the game theoretical problem with the infinitely many agents and dynamics of each agent determined by a controlled finite state nonlinear Markov chain. Given a supersolution (resp. subsolution) of the Hamilton-Jacobi equation for the model game, we construct a suboptimal strategy of the first (resp. second) player and evaluate the approximation accuracy using the modulus of continuity of the reward function and the distance between the original and model games. This gives the approximations of the value function of the mean field type differential game by values of the finite-dimensional differential games. Furthermore, we present the way to build a finite-dimensional differential game that approximates the original game with a given accuracy.

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“…Notice that as the authors work on a canonical probabilistic setting, their game can be seen as a game in the form feedback controls vs feedback controls. We mention also the recent paper [3], which considers deterministic mean-field type differential games with feedback controls.…”

Section: Introductionmentioning

confidence: 99%

Zero-sum stochastic differential games of generalized McKean–Vlasov type

Pham

Cosso

2019

Journal de Mathématiques Pures et Appliquées

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We study zero-sum stochastic differential games where the state dynamics of the two players is governed by a generalized McKean-Vlasov (or mean-field) stochastic differential equation in which the distribution of both state and controls of each player appears in the drift and diffusion coefficients, as well as in the running and terminal payoff functions. We prove the dynamic programming principle (DPP) in this general setting, which also includes the control case with only one player, where it is the first time that DPP is proved for openloop controls. We also show that the upper and lower value functions are viscosity solutions to a corresponding upper and lower Master Bellman-Isaacs equation. Our results extend the seminal work of Fleming and Souganidis [15] to the McKean-Vlasov setting.

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Krasovskii–Subbotin Approach to Mean Field Type Differential Games

Cited by 13 publications

References 46 publications

Impulsive control of nonlocal transport equations

Impulsive control of nonlocal transport equations

Lattice approximations of the first-order mean field type differential games

Zero-sum stochastic differential games of generalized McKean–Vlasov type

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