Linear-quadratic differential games: Open and Closed Loop Strategies with or without Singularities

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 and to linear dynamics with bounded measurable data matrices. This is done in the light of the recent work of P. Zhang [SIAM J. Control Optim., 43 (2005), pp. 2157-2165] who established the equivalence between the flniteness of the open loop value of a game and the finiteness of its open loop lower and upper values. 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 make the connection with the normality and normalizability of the problem and provide a complete 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.