Linear closed-loop no-memory strategies for the LQ Nash game are considered. We exhibit a class of such problems with the property that the solution exists for any finite time interval; for the infinite time case, there exist none or a unique or many solutions, depending on the choice of the parameters. In addition, the limit of the finite time solution as the time interval increases does not have to yield the infinite time case solution. A geometric formulation of the coupled algebraic Riccati equation is given. This formulation seems to be an interesting starting point for a thorough study of these equations.
Papavassilopulos, G. P., & Olsder, G. J. (1984). On the linear-quadratic, closed-loop, no-memory Nash game. Journal of optimization theory and applications, 42(4), 551-560. https://doi.org/10.1007/BF00934566