Stackelberg solutions of feedback type for differential games with random initial data

The paper is concerned with Stackelberg solutions for a differential game with deterministic dynamics but random initial data, where the leading player can adopt a strategy in feedback form: u1 = u1(t, x). The first main result provides the existence of a Stackelberg equilibrium solution, assuming that the family of feedback controls u1(t, ·) available to the leading player are constrained to a finite dimensional space. A second theorem provides necessary conditions for the optimality of a feedback strategy. Finally, in the case where the feedback u1 is allowed to be an arbitrary function, an example illustrates a wide class of systems where the minimal cost for the leading player corresponds to an impulsive dynamics. In this case, a Stackelberg equilibrium solution does not exists, but a minimizing sequence of strategies can be described.