Equivalences between differential games and optimal controls

It is shown that linear differential pursuit games with linear targets, if player controls are required to be piecewise constant or if player controls areL ₁-functions (but pursuer control is bang-bang whenever quarry control is), are equivalent to a linear, autonomous control problem. As a byproduct, a sufficient condition for terminating the game, in Pontryagin's sense, is obtained.The present paper has been influenced by Prof. O. Hajek's work in differential games; the converse parts of the proofs presented here are very similar to those in Ref. 5. The author wishes to thank Dr. Hajek for his suggestions, comments, and critique. The Brasilian Government BNDE provided partial financial support.