Open Loop Robust Equilibria in Uncertain Discrete Time Games

다국어 초록 (Multilingual Abstract)

This paper tackles the problem of a discrete time N players game affected by some sort of time-varying uncertain perturbation. The philosophy of the uncertainty worst case scenario with respect to the i-player is developed to derive necessary and sufficient conditions for the existence of an Open Loop Robust Nash Equilibria. Such conditions are presented in terms of the solvability of a set of discrete time Riccati type equations with some boundary conditions. As an illustration of the solution, a simulation of the coordination of a two-echelon supply chain with uncertain seasonal fluctuations in the demand is developed.

참고문헌 (Reference)

1 M. Jiménez, "ε-equilibrium LQ differential games with bounded uncertain disturbance : Robustness of standard strategies and new strategies with adaptation" 79 (79): 786-797, 2006

2 J. C. Engwerda, "Robust open-loop Nash equilibria in the noncooperative LQ game revisited" 38 (38): 795-813, 2017

3 W. A. van den Broek, "Robust equilibria in indefinite linear-quadratic differential games" 19 (19): 565-595, 2003

4 G. Barbaraci, "Robust control for a suspended rotating shaft by radial active magnetic bearing" 1 (1): 4-11, 2011

5 M. Jiménez-Lizárraga, "Robust Nash equilibrium in multi-model LQ differential games : analysis and extraproximal numerical procedure" 28 (28): 117-141, 2007

6 I. Ross, "Primer on Pontryagin’s Principle in Optimal Control" Collegiate Publishers 2015

7 S. Jorgensen, "Optimal production, purchasing and pricing : A differential game approach" 24 (24): 64-76, 1986

8 G. Jank, "Optimal control of disturbed linearquadratic differential games" 8 (8): 152-162, 2002

9 S. Sethi, "Optimal Control Theory: Applications to Management Science and Economics" Springer 2005

10 P. V. Reddy, "Open-loop Nash equilibria in a class of linear-quadratic difference game with constraints" 60 (60): 2559-2564, 2015

11 X. Chen, "Multiobjective H2=H1 control design" 40 (40): 628-660, 2002

12 Y. Wu, "Linear dynamic games with polytope strategy sets" 11 (11): 2146-2151, 2017

13 J. Engwerda, "LQ Dynamic Optimization and Differential Games" John Wiley and Sons Ltd 2005

14 G. Barbaraci, "Influence of uncertainties on rotating flexible shaft suspended by 4-axis radial active magnetic bearings" 3 (3): 18-25, 2012

15 G. Barbaraci, "Influence of uncertainties on PD tuning" 211-219, 2013

16 P. V. Reddy, "Feedback Nash equilibria in linear-quadratic difference games with constraints" 62 (62): 590-604, 2017

17 G. Jank, "Existence and uniqueness of open-loop Nash equilibria in linear-quadratic discrete time games" 48 (48): 267-271, 2003

18 A. E. Bryson Jr., "Dynamic Optimization" AddisonWesley 1999

19 T. Basar, "Dynamic Noncooperative Game Theory" Academic Press 1995

20 H. Kebriaei, "Discrete-time robust hierarchical linear-quadratic dynamic games" 63 (63): 902-909, 2018

21 M. Pachter, "Discrete-time linearquadratic dynamic games" 146 : 151-179, 2010

22 M. Jungers, "Bounded Nash type controls for uncertain linear systems" 44 (44): 1874-1879, 2008

23 A. S. Poznyak, "Advanced Mathematical Tools for Automatic Control Enginers, Volume 1: Deterministic Techniques" Elsevier Ltd. 2008