Nash equilibrium in a singular two-person linear-quadratic differential game: A regularization approach

Glizer, Valery Y.

doi:10.1109/med.2016.7535851

Cited by 6 publications

(4 citation statements)

References 18 publications

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“…Zero-sum differential games with a complete/partial cheap control of at least one player were studied in many works (see, e.g., [11,12,13,20,21,23,24,25,42,43] ). Non zero-sum differential games with a complete cheap control of one player were considered only in few works (see [4,31,32]). However, to the best of our knowledge, a non zero-sum differential game with a partial cheap control of at least one player has been considered only in two works [33,44] in the literature.…”

Section: Problem Formulationmentioning

confidence: 99%

“…For this singular game, a Nash equilibrium set of open-loop controls was obtained in the class of regular functions. In [31] and [32,33], finite horizon and infinite horizon versions of two-person Nash equilibrium differential game with n-order linear dynamics and vector-valued unconstrained players' controls were considered. The cost functionals of both players in the games are quadratic.…”

Section: Introductionmentioning

confidence: 99%

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Solutions of one class of singular two-person Nash equilibrium games with state and control delays: Regularization approach

2023

Appl. Set-Valued Anal. Optim.

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In this paper, a finite-horizon two-person Nash equilibrium game for a linear time-dependent differential system with delays (point-wise and distributed) in the state variable and the players' control variables is considered. The behaviour of each player is evaluated by its own quadratic functional to be minimized by this player. The feature of this game is that a weight matrix of the control in the functional of one player is singular (but, in general, non-zero). Due to this feature, the game itself is singular. Using the regularization method and the asymptotic analysis of the regularized Nash equilibrium game, an open-loop Nash equilibrium solution to the considered singular game is derived. Illustrative example is presented. Along with this example, two examples on non-uniqueness of open-loop Nash equilibrium solutions to the singular game are presented.

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Section: Problem Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Solutions of one class of singular two-person Nash equilibrium games with state and control delays: Regularization approach

2023

Appl. Set-Valued Anal. Optim.

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“…Finite horizon cheap control differential games (zero-and non zero-sum) were studied in the literature in a number of works (see e.g. [8,11,33,35,38,39] and references therein), while infinite horizon cheap control differential games were studied only in the works [13,29]. Since for any ε > 0 the weight matrix for the minimizer's control cost in the cost functional (19) is positive definite, the CCDG is a regular differential game.…”

Section: 2mentioning

confidence: 99%

“…In [12], an optimal trajectory sequence and a generalized optimal trajectory in the considered singular game also were obtained. In [11], using a regularization approach, a Nash equilibrium sequence was obtained in a two-person singular non zero-sum differential game.…”

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confidence: 99%

Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence

Glizer¹,

Kelis²

2017

Numerical Algebra, Control &Amp; Optimization

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We consider an infinite horizon zero-sum linear-quadratic differential game in the case where the cost functional does not contain a control cost of the minimizing player (the minimizer). This feature means that the game under consideration is singular. For this game, novel definitions of the saddle-point equilibrium and game value are proposed. To obtain these saddlepoint equilibrium and game value, we associate the singular game with a new differential game for the same equation of dynamics. The cost functional in the new game is the sum of the original cost functional and an infinite horizon integral of the square of the minimizer's control with a small positive weight coefficient. This new game is regular, and it is a cheap control game. Using the solvability conditions, the solution of the cheap control game is reduced to solution of a Riccati matrix algebraic equation with an indefinite quadratic term. This equation is perturbed by a small parameter. Subject to a proper assumption, an asymptotic expansion of a stabilizing solution to this equation is constructed and justified. Using this asymptotic expansion, the existence of the saddle-point equilibrium and the value of the original game is established, and their expressions are derived. Illustrative example is presented.

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Finite-Horizon $$H_{\infty }$$ Control Problem with Singular Control Cost

Glizer

Kelis

2019

Informatics in Control, Automation and Robotics

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Nash equilibrium in a singular two-person linear-quadratic differential game: A regularization approach

Cited by 6 publications

References 18 publications

Solutions of one class of singular two-person Nash equilibrium games with state and control delays: Regularization approach

Solutions of one class of singular two-person Nash equilibrium games with state and control delays: Regularization approach

Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence

Finite-Horizon $$H_{\infty }$$ Control Problem with Singular Control Cost

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