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

2017 ◽  
Vol 7 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Valery Y. Glizer ◽  
◽  
Oleg Kelis ◽  
◽  
Keyword(s):  
Saddle Point ◽  
Differential Game ◽  
Quadratic Differential ◽  
Infinite Horizon ◽  
Linear Quadratic ◽  
Zero Sum ◽  
Saddle Point Equilibrium ◽  
Equilibrium Sequence

scholarly journals Nash Equilibrium Sequence in a Singular Two-Person Linear-Quadratic Differential Game

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 132
Author(s):  
Valery Y. Glizer
Keyword(s):  
Nash Equilibrium ◽  
Differential Game ◽  
Quadratic Functional ◽  
Linear Quadratic ◽  
Feedback Controls ◽  
Cheap Control ◽  
Optimal Values ◽  
Definition Of ◽  
Zero Sum ◽  
Equilibrium Sequence

A finite-horizon two-person non-zero-sum differential game is considered. The dynamics of the game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are singular in both functionals is studied. Hence, the game under the consideration is singular. A novel definition of the Nash equilibrium in this game (a Nash equilibrium sequence) is proposed. The game is solved by application of the regularization method. This method yields a new differential game, which is a regular Nash equilibrium game. Moreover, the new game is a partial cheap control game. An asymptotic analysis of this game is carried out. Based on this analysis, the Nash equilibrium sequence of the pairs of the players’ state-feedback controls in the singular game is constructed. The expressions for the optimal values of the functionals in the singular game are obtained. Illustrative examples are presented.

scholarly journals Feedback saddle point equilibria for soft-constrained zero-sum linear quadratic descriptor differential game

2013 ◽  
Vol 23 (4) ◽  
pp. 473-493
Author(s):  
Muhammad Wakhid Musthofa ◽  
Jacob C. Engwerda ◽  
Ari Suparwanto ◽  
Keyword(s):  
Saddle Point ◽  
Sufficient Conditions ◽  
Planning Horizon ◽  
Descriptor Systems ◽  
Linear Quadratic ◽  
Linear Quadratic Differential Games ◽  
Zero Sum ◽  
Necessary And Sufficient ◽  
Saddle Point Equilibrium ◽  
Infinite Planning Horizon

Abstract In this paper the feedback saddle point equilibria of soft-constrained zero-sum linear quadratic differential games for descriptor systems that have index one will be studied for a finite and infinite planning horizon. Both necessary and sufficient conditions for the existence of a feedback saddle point equilibrium are considered

Berge and Nash Equilibria in a Linear-Quadratic Differential Game

2020 ◽  
Vol 81 (11) ◽  
pp. 2108-2131
Author(s):  
V. I. Zhukovskiy ◽  
A. S. Gorbatov ◽  
K. N. Kudryavtsev
Keyword(s):  
Differential Game ◽  
Quadratic Differential ◽  
Nash Equilibria ◽  
Linear Quadratic

scholarly journals An Analytic and Numerical Investigation of a Differential Game

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 66
Author(s):  
Aviv Gibali ◽  
Oleg Kelis
Keyword(s):  
Differential Game ◽  
Equation Approach ◽  
Linear Quadratic ◽  
Control Cost ◽  
Dual Representation ◽  
Cost Functional ◽  
Variational Derivatives ◽  
Zero Sum ◽  
The Cost ◽  
Derivatives Of

In this paper we present an appropriate singular, zero-sum, linear-quadratic differential game. One of the main features of this game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot be solved either by applying the Isaacs MinMax principle, or the Bellman–Isaacs equation approach. As an application, we introduced an interception differential game with an appropriate regularized cost functional and developed an appropriate dual representation. By developing the variational derivatives of this regularized cost functional, we apply Popov’s approximation method and show how the numerical results coincide with the dual representation.

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