In this note, we consider the non-cooperative linear feedback Nash quadratic differential game with an infinite planning horizon. The performance function is assumed to be indefinite and the underlying system affine. We derive both necessary and sufficient conditions under which this game has a Nash equilibrium. As a special case, we derive existence conditions for the multi-player zero-sum game.
Engwerda, J. C., & Salmah, S. (2013). Necessary and sufficient conditions for feedback nash equilibria for the affine-quadratic differential game. Journal of Optimization Theory and Applications, 157(2), 552-563. https://doi.org/10.1007%2fs10957-012-0188-1#