Bi-objective and hierarchical control for the Burgers equation (Q6570516)

This paper considers some bi-objective and hierarchical control problems for the Burgers equation. More precisely, Burgers equation is controlled by three distributed controls, each of them being assigned to a different task. To the first control, called the leader, there are associated two other controls called the followers. The triplet formed by these controls solves an hierarchical Stackelberg-Nash control problem. The main contribution of this paper is the formulation and the solution of Nash and Stackelberg-Nash control problems in a non-linear setting. These results may be seen as a first step for similar control problem in more complex nonlinear situations such as the Navier-Stokes equation. In a first step the authors show the existence of a Nash equilibrium for every given leader control. Then, it is shown, under appropriate hypotheses, that there exists a leader control and an associated Nash equilibrium for the two followers such that the associated state is arbitrarily small.