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

Abstract

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.

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

“…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. It should be noted that the PCCNEG differs considerably from the partial cheap control Nash equilibrium games studied in [33,44].…”

“…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. It should be noted that the PCCNEG differs considerably from the partial cheap control Nash equilibrium games studied in [33,44]. Moreover, in the present paper we are going to analyze the open-loop solution of the PCCNEG, while in [33,44] the feedback solutions of the considered there partial cheap control Nash equilibrium games were analyzed.…”

“…It should be noted that the PCCNEG differs considerably from the partial cheap control Nash equilibrium games studied in [33,44]. Moreover, in the present paper we are going to analyze the open-loop solution of the PCCNEG, while in [33,44] the feedback solutions of the considered there partial cheap control Nash equilibrium games were analyzed.…”

“…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.…”

Asymptotic analysis and open-loop solutions of one class of partial cheap control zero-sum differential games with state and control delays

“…Complete/partial cheap control problems appear in many topics of optimal control, H ∞ control and differential games theories. For example, such problems appear in the following topics: (1) solution of singular optimal control problems by regularization (see, e.g., [5,9,10,11,12,13]); (2) solution of singular H ∞ control problems by regularization (see, e.g., [14,15,16]); (3) solution of singular differential games by regularization (see, e.g., [17,18,19,20,21,37]); (4) limitation analysis for optimal regulators and filters (see, e.g., [7,13,22,32,36,38]); (5) extremal control problems with high gain control in dynamics (see, e.g., [30,44]); (6) inverse optimal control problems (see, e.g., [34]); (7) robust optimal control of systems with uncertainties/disturbances (see, e.g., [39,40]); (8) guidance/interception problems (see, e.g., [38,41,42]).…”