Open-Loop Saddle Points for Irregular Linear-Quadratic Two-Person Zero-Sum Games

In this paper, we consider the feedback representation of open-loop saddle points for irregular linear-quadratic (LQ) two-person zero-sum games, where the control weighting matrices in the cost functional are only semidefinite. The existence of an open-loop saddle point is characterized by the solvability of a system of constrained linear forward-backward differential equations (FBDEs), together with a convexity-concavity condition. In classical zero-sum games, the feedback representation is obtained by decoupling the FBDEs through the regular solution to a Riccati differential equation. But the associated Riccati equation cannot be used to decouple the FBDEs to obtain the feedback representation of open-loop saddle points. The essential differences between regular and irregular LQ zero-sum games are investigated. The irregular feedback representation can be derived from two equilibrium conditions in two different layers by using the “two-layer optimization” approach.