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

TL;DR

This paper investigates the hierarchical null controllability of a semilinear degenerate parabolic equation with a gradient term, employing a Stackelberg-Nash strategy and advanced inequalities to establish controllability results.

Contribution

It introduces a novel approach to handle non-convex functionals in semilinear degenerate parabolic equations using Nash quasi-equilibrium and Carleman inequalities.

Findings

Established existence and uniqueness of Nash quasi-equilibrium.

Proved null controllability of the linearized system.

Extended controllability results to the nonlinear system using fixed point theorem.

Abstract

In this paper, we apply the hierarchical strategy to a semilinear weakly degenerate parabolic equation involving a gradient term. We use the Stackelberg-Nash strategy with one leader which tries 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 Nash quasi-equilibrium and the 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, due to the Kakutani's fixed point Theorem, we obtain the null controllability of the main system.