DISCONTINUOUS DIFFERENTIAL-EQUATIONS .1.

The classical notions of solution, to an ordinary differential equation, are sometimes insufficient. Three generalisations have been proposed (Filippov, Krasovskij, and, implicitly, Hermes). The first part of the paper studies these concepts; it is shown that, in very general circumstances, the last two are equivalent (and sometimes one of the alternative descriptions has a decided advantage over the other). Subsequently the results will be applied to optimal control theory and to differential games. © 1979.