Some new results in open and closed-loop linear-quadratic differential games

The object of this paper is to revisit the results of P. Bernhard (J. Optim. Theory Appl. 27 (1979), 51-69) on two-person zero-sum linear quadratic differential games and generalize them to utility functions without positivity assumptions on the matrices acting on the state variable in the utility function and to linear dynamics with bounded measurable data matrices. We consider both open and closed loop strategies. We specialize to state feedback via Lebesgue measurable affine closed loop strategies with possible non L2- integrable singularities. We review recent results in the finite dimensional case and provide a classification of closed loop saddle points in terms of the convexity/concavity properties of the utility function and the open loop lower value, upper value, and value of the game. We single out finite dimensional concepts such as normality and normalizability that do not carry over to evolution equations in infinite dimensional spaces.