Local null controllability of a cubic Ginzburg-Landau equation with dynamic boundary conditions

TL;DR

This paper establishes the local null controllability of a cubic Ginzburg-Landau equation with dynamic boundary conditions using linearization, Carleman estimates, and inverse function theorem techniques.

Contribution

It introduces a novel controllability result for a nonlinear PDE with dynamic boundary conditions, combining linearization, duality, and Carleman estimates.

Findings

Proves local null controllability of the nonlinear system.

Develops Carleman estimates for the linearized system.

Utilizes inverse function theorem for nonlinear controllability.

Abstract

This paper deals with controllability properties of a cubic Ginzburg-Landau equation with dynamic boundary conditions. More precisely, we prove a local null controllability result by using a single control supported in a small subset of the domain. In order to achieve this result, we firstly linearize the system around the origin and we analyze it by the duality approach and an appropriate Carleman estimate. Then, by using an inverse function theorem, the local null controllability of the nonlinear system is proven.