Hierarchical control for the semilinear parabolic equations with interior degeneracy

TL;DR

This paper develops a hierarchical control framework for semilinear parabolic equations with interior degeneracy, introducing a novel Carleman estimate to establish null controllability via a Stackelberg-Nash strategy.

Contribution

It introduces a new Carleman estimate for coupled degenerate systems and analyzes a hierarchical control approach with existence and uniqueness results for Nash equilibria.

Findings

Established null controllability for the system.

Proved existence and uniqueness of Nash equilibrium.

Developed a new Carleman estimate for degenerate systems.

Abstract

This paper concerns with the hierarchical control of the semilinear parabolic equations with interior degeneracy. By a Stackelberg-Nash strategy, we consider the linear and semilinear system with one leader and two followers. First, for any given leader, we analyze a Nash equilibrium corresponding to a bi-objective optimal control problem. The existence and uniqueness of the Nash equilibrium is proved, and its characterization is given. Then, we find a leader satisfying the null controllability problem. The key is to establish a new Carleman estimate for a coupled degenerate parabolic system with interior degeneracy.