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

In this paper, we apply the hierarchical strategy to a semilinear weakly\ndegenerate parabolic equation involving a gradient term. We use the\nStackelberg-Nash strategy with one leader which tries to drive the solution to\nzero and two followers intended to solve a Nash equilibrium corresponding to a\nbi-objective optimal control problem. Since the system is semilinear, the\nfunctionals are not convex in general. To overcome this difficulty, we first\nprove the existence and uniqueness of the Nash quasi-equilibrium, which is a\nweaker formulation of the Nash equilibrium. Next, with additional conditions,\nwe establish the equivalence between the Nash quasi-equilibrium and the Nash\nequilibrium. We establish a suitable Carleman inequality for the adjoint system\nand then an observability inequality. Based on this observability inequality,\nwe prove the null controllability of the linearized system. Then, due to the\nKakutani's fixed point Theorem, we obtain the null controllability of the main\nsystem.\n