Open-loop Nash equilibria in the non-cooperative infinite-planning horizon LQ game

In this paper we reconsider Nash equilibria for the affine linear quadratic differential game for an infinite planning horizon. We consider an open-loop information structure. In the standard literature this problem is solved under the assumption that every player can stabilize the system on his own. In this note we relax this assumption and provide both necessary and sufficient conditions for the existence of Nash equilibria for this game under the assumption that the system as a whole is stabilizable. Basically, it is shown that to see whether equilibria exist for all initial states one has to restrict the analysis to the controllable modes of the different players.