Feedback properties of descriptor systems using matrix projectors and applications to descriptor differential games

In this paper we discuss linear differential games under algebraic constraints. We use the theory of projector chains to decouple algebraic and differential parts of the descriptor system, and then the usual theory of ordinary differential games is applied to derive both necessary and sufficient conditions for the existence of feedback Nash equilibria for linear quadratic differential games. This approach is new in the context of dynamic games. To address this problem the effects of feedback on the behavior of the descriptor system under two informational constraints, namely, the partial and full state feedbacks are analyzed in detail. We show that the partial state feedbacks have a Jordan structure (and thus index) preserving property. Further, for index $1$ descriptor systems, we show that every full state, index preserving state feedback can be realized as a partial state feedback.