Some properties of extremal solutions of zero-sum differential games

Necessary conditions for infinite-time linear-quadratic two-player differential games with perfect state information are examined in the light of Willems' work on the algebraic Riccati equation [1]. Necessary conditions for the nonzero-sum game do not always reduce precisely to those of the zero-sum game, when the weighting parameters of the cost index are taken as those of a zero-sum game. Formulation of zero-sum game problems in extended L
²
-spaces reveals the existence of nonstabilizing but infimal controls, and explicit frequency-domain conditions for existence of solutions and stabilizing solutions are established through the algebraic Riccati equation (ARE).