Stackelberg-Nash controllability for a multi-objective Stefan problem

TL;DR

This paper studies a hierarchical control approach combining Stackelberg and Nash strategies for a one-dimensional Stefan problem, establishing local null controllability via Carleman estimates.

Contribution

It introduces the first Stackelberg-Nash control framework for Stefan systems, reducing the problem to null controllability of an optimality system.

Findings

Established local null controllability under geometric conditions.

Derived observability inequalities using Carleman estimates.

Reduced the control problem to a nonlinear optimality system.

Abstract

We investigate a hierarchical control problem for a one-dimensional Stefan system with localized distributed controls. The setting combines a Stackelberg strategy with a Nash equilibrium among multiple followers, yielding a multi-objective free-boundary problem. The interaction between the hierarchical control and the moving interface results in a nonlinear optimality system, and we show that the original problem reduces to the null controllability of this optimality system. Under suitable geometric conditions on the control regions, we establish a local null controllability result. The proof relies on an observability inequality for a linearized system, obtained through Carleman estimates adapted to the presence of a moving boundary. These results constitute, to the best of our knowledge, the first treatment of a Stefan system within a Stackelberg-Nash framework.