Conditions for optimality and transition surfaces using the reprisal concept†
J. BRADLEY, P. L. Yu · International Journal of Systems Science · 1976
The concept of optimality in zero-sum, two-player differential games introduced by the authors in a previous paper is studied. Furthermore, a number of tools to aid in the study of differential games in which there exist ‘ non-terminating ’ strategies are developed. In the all-terminating case it is shown that for C(z)) strategies or for strategies containing a specified class of discontinuities, satisfying Isaacs' equation is both necessary and sufficient for optimality. In linear, all-terminating games, easily verifiable necessary and sufficient conditions are given and it is shown that, subject to one condition, points of transition surfaces must necessarily be in the zero-set of one of two switching functions. In ‘ non-terminating ’ games, satisfying Isaacs' equation is necessary for optimality but no longer sufficient. Sufficiency conditions for optimality in these games are then proven along with three alternative necessary conditions which points of transition surfaces must satisfy. An example illustrating the application of these results is included together with a heuristic discussion of the ‘ double transition surface ’ phenomenon.