Hierarchic Control For A Parabolic Equation With Interior Degeneracy

Hierarchical control of a parabolic equation with interior degeneracy is studied in this paper.Based on the Stackelberg-Nash strategy,we consider the null controllability problem.In the system,there is one leader and two followers.The followers are dependent on the leader.It is possible to find the corresponding followers to make a two-objective optimal control problem reach Nash equilibrium for any given leader control.The goal of leader control is to make the system null controllable and to pay as little as possible.By using the Lax-Milgram theorem,we can first prove the existence and uniqueness of the Nash equilibrium pairs(followers satisfying some conditions).By constructing a variational problem of the adjoint system,the existence of leader control is proved.Last,Carleman estimates are used to prove the observability inequality required for null controllability.