Linear Quadratic Differential Games: Closed Loop Saddle Points

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. The paper specializes to state feedback via Lebesgue measurable affine closed loop strategies with possible non L-integrable singularities. After sharpening our recent results [3] on the characterization of the open loop lower and upper values of the game, it first deals with L-integrable closed loop strategies and then with the larger family of strategies that may have non L-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 L-type and smooth locally bounded ones except at most in the neighborhood of finitely many instant of time. A necessary and sufficient conditions is that the free end problem be normalizable. 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.