Hierarchical control for degenerate wave equations under a Stackelberg–Nash strategy

This paper studies the hierarchical exact controllability of a class of boundary-degenerate wave equations under the Stackelberg–Nash strategy. The system contains one leader control acting on the non-degenerate boundary and two follower controls supported in two nonempty open subsets. For the linear case, we establish the existence and uniqueness of a Nash equilibrium associated with a bi-objective optimal control problem. Then, by establishing observability inequalities through Carleman estimates, we prove the hierarchical exact controllability of the coupled system. For the semilinear case, we introduce the notion of Nash quasi-equilibrium and prove the corresponding controllability result by uniform scalar observability and a Schauder fixed-point argument.