Carleman estimate for full-discrete approximations of the complex Ginzburg-Landau equation with dynamic boundary conditions and applications to controllability
Xu Zhu, Wenwen Zhou, Bin Wu
TL;DR
The paper addresses controllability for fully-discrete approximations of a one-dimensional complex Ginzburg-Landau equation with dynamic boundary conditions, formulated as $\partial_t y - (\alpha + i\beta)\partial_x^2 y + (c + i\gamma) y = \mathbf{1}_{\omega} v$. It develops a discrete Carleman estimate for the adjoint system using a weighted framework with $s(t)=\tau\theta(t)$, $r(x,t)=e^{s(t)\varphi(x)}$, and $\rho=r^{-1}$, and derives a relaxed observability inequality from which a $\phi(\Delta x)$-null controllability result follows for the fully-discrete scheme. The analysis separates the large Carleman parameter $\lambda$ from auxiliary constants and carefully handles dynamic boundary terms, showing consistency with discrete parabolic Carleman theory while highlighting limitations for Schrödinger-type limits. The results provide a rigorous route to uniform controllability as the spatial step $\Delta x$ vanishes, under explicit constraints on discretization and weight parameters. $
Abstract
In this paper, we investigate Carleman estimate and controllability result for the fully-discrete approximations of a one-dimensional Ginzburg-Landau equation with dynamic boundary conditions. We first establish a new discrete Carleman estimate for the corresponding adjoint system. Based on this Carleman estimate, we obtain a relaxed observability inequality for the adjoint system, and then a controllability result for the fully-discrete Ginzburg-Landau equation with dynamic boundary conditions.