Hierarchical exact controllability of the fourth-order parabolic equations

This paper is concerned with the application of Stackelberg-Nash strategies to control fourth-order linear and semilinear parabolic equations. We assume that the system is acted through a hierarchy of distributed controls: one main control (the leader) that is responsible for an exact controllability property; and a couple of secondary controls (the followers) that minimize two prescribed cost functionals and provide a pair of Nash equilibria for the two prescribed cost functionals. We first prove the existence of an associated Nash equilibrium pair corresponding to a hierarchical bi-objective optimal control problem for each leader by Lax-Milgram theorem. Then, we establish observability inequalities of fourth-order coupled parabolic equations by global Carleman inequalities and energy methods. Based on these results, we obtain the existence of a leader that drives the controlled system exactly to a prescribed (but arbitrary) trajectory. Furthermore, we also give the second-order sufficient conditions of optimality for the secondary controls.