LINEAR QUADRATIC DIFFERENTIAL GAMES: CLOSED LOOP SADDLE POINTS.

The object of this paper is to revisit the results of Bernhard [J. Optim. Theory Appl., 27 (1979), pp. 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. Our paper specializes to state feedback via Lebesgue measurable affine closed loop strategies with possible non-L2-integrable singularities. After sharpening the recent results of Delfour [SIAM J. Control Optim., 46 (2007), pp. 750-774] on the characterization of the open loop lower and upper values of the game, it first deals with L2-integrable closed loop strategies and then with the larger family of strategies that may have non-L2-integrable singularities. A new conceptually meaningful and mathematically precise definition of a closed loop saddle point is introduced to simultaneously handle state feedbacks of the L2 type and smooth locally bounded ones, except at most in the neighborhood of finitely many instants of time. A necessary and sufficient condition is that the free end problem be normalizable almost everywhere. This relaxation of the classical notion allows singularities in the feedback law at an infinite number of instants, including accumulation points that are not isolated. A complete classification of closed loop saddle points is given in terms of the convexity/concavity properties of the utility function, and connections are given with the open loop lower value, upper value, and value of the game. [ABSTRACT FROM AUTHOR]