On the approximate controllability of Stackelberg-Nash strategies

Let us consider a distributed system, i.e. a system whose state is defined by the solution of a Partial Differential Equation (PDE). We assume that we can act on this system by a hierarchy of controls. There is a “global” control v, which is the leader, and there are N “local” controls, denoted by w₁,…, w _N , which are the followers. The followers, assuming that the leader has made a choice v of its policy, look for a Nash equilibrium of their cost functions (the criteria they are interested in). Then the leader makes its final choice for the whole system. This is the Stackelberg-Nash strategy.