Properties of feedback Nash equilibria in scalar LQ differential games

In this paper we study the scalar linear quadratic differential game with state feedback information structure. Using a geometric approach, we present a complete characterization when this game will have no, one or multiple equilibria. Furthermore, we investigate the effect on this solution structure of some characteristics of the game, i.e., the number of players; the entrance of new players; the level of asymmetry; and the impact entrance of an additional player has on the closed-loop stability of the game. For that purpose we distinguish three types of the game: the economic game; the regulator game and the mixed game. The analysis is restricted to the case the involved cost depend only on the state and control variables.