Open-loop Nash strategy for linear-quadratic games via matrix pencil approach

This paper deals with solving non-symmetric algebraic Riccati systems (NARS) and non-symmetric algebraic Riccati equations (NARE), from Nash strategy with an open-loop information structure applied on linear-quadratic games. A matrix pencil method is chosen for its theoretical and numerical efficiency. The main result provides a one-to-one correspondance between disconjugate proper deflating subspaces of a characteristic matrix pencil and the solutions of NARS and NARE. It is shown that this approach is more relevant than ones in the literature, because classical assumptions of some matrix invertibility could be avoided.