On various equilibrium solutions for linear quadratic noncooperative games

by Xu Wang

Institution: The Ohio State University
Department: Electrical Engineering
Degree: PhD
Year: 2007
Record ID: 1793367
Full text PDF: http://rave.ohiolink.edu/etdc/view?acc_num=osu1196223332


Game theory has been widely used to model decision making processes because it can capture the nature of multi-player problems: the determination of one player's control strategy is not only subject to the system state evolution but is also tightly coupled to the determination of the other players' strategies and vice versa. In this dissertation, we categorize linear quadratic (LQ) games into three groups: definite, singular and indefinite. For singular LQ games: 1) a new equilibrium concept: asymptotic e-Nash equilibrium is proposed for a two-player nonzero-sum game where each player has a control-free cost functional quadratic in the system states over an infinite horizon and each player's control strategy is constrained to be continuous linear state feedback; 2) a group of algebraic equations of system coefficients is found whose solution can constitute the partial state feedback asymptotic e-Nash equilibrium for the singular LQ games. Conditions on initial states and the parameter e are provided such that the asymptotic e-Nash equilibrium will be an e-Nash equilibrium or a Nash equilibrium; 3) for a class of 2nd-order singular LQ games, the closed-form asymptotic e-Nash equilibrium is explicitly found in terms of system coefficients. Robust equilibrium solutions for two-player asymmetric games with an additive uncertainty are studied: 1) regarding the uncertainty as the third player, a three-player non-cooperative nonzero-sum game is formed and each player’s cost functional value resulting from the output feedback Nash equilibrium of this three-player game is not as conservative as his/her individual rationality; 2) regarding the coalition of the original two players as one player and the uncertainty as another player, a two-player non-cooperative nonzero-sum game is formed to find un-improvable robust a equilibrium for the original game. Inverse problems for indefinite games are investigated (for which weighting matrices in the cost functional such that the given linear state feedback control strategy can constitute a Nash equilibrium solution): 1) a necessary and sufficient condition for the inverse problem is provided using a group of algebraic equations linear in the variables and the weighting matrices; 2) the inverse problem for a class of 2<sup>nd</sup>-order two-player LQ game is thoroughly discussed.