Reducing a Differential Game to a Pair of Optimal Control Problems

Author’s recent concepts and results are extended here to the more general case of

non-autonomous

(i.e., time-dependent) differential games.

The main idea is to introduce first a concept of

admissible pair of

(

multivalued

)

feedback strategies

to which one may associate a

value function

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

bilaterally-optimal

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.