Hierarchical null controllability of a semilinear degenerate parabolic equation with a gradient term

Abstract

ABSTRACT In this paper, we apply a hierarchical strategy to a semilinear weakly degenerate parabolic equation with non-linearity that depends on the solution of the system and the spatial derivative of the solution. We use the Stackelberg–Nash strategy with one leader aiming to drive the solution to zero and two followers intended to solve a Nash equilibrium corresponding to a bi-objective optimal control problem. Since the system is semilinear, the functionals are not convex in general. To overcome this difficulty, we first prove the existence and uniqueness of the Nash quasi-equilibrium, which is a weaker formulation of the Nash equilibrium. Next, with additional conditions, we establish the equivalence between the concepts of Nash quasi-equilibrium and Nash equilibrium. We establish a suitable Carleman inequality for the adjoint system and then an observability inequality. Based on this observability inequality, we prove the null controllability of the linearized system. Then, using Kakutani's fixed point Theorem, we demonstrate the null controllability of the main system.