Author’s recent concepts and results are extended here to the more general case of
(i.e., time-dependent) differential games.
The main idea is to introduce first a concept of
admissible pair of
to which one may associate a
and which reduce the differential game to a pair of symmetric, non-smooth optimal control problems for differential inclusions; secondly, one introduces the concepts of
pairs of feedback strategies and one proves an
abstract verification theorem
containing necessary and sufficient optimality conditions; next, this approach is made more realistic by the proof of several “practical verification theorems” containing corresponding differential inequalities and regularity hypotheses on the value function that imply the optimality.
As a method for constructing simultaneously, the value function, the pair of optimal feedback strategies and also the optimal trajectories, certain natural extensions of Cauchy’s Method of Characteristics to non-smooth Hamilton-Jacobi equations are suggested.