Small-time approximate controllability for the nonlinear complex Ginzburg-Landau equation with bilinear control
Xingwu Zeng, Can Zhang
TL;DR
This work studies small-time global approximate controllability for the nonlinear complex Ginzburg-Landau equation on the torus with bilinear control acting through a low-dimensional forcing, $\partial_t \psi = V\psi + (1+i\nu)\Delta\psi - (1+i\mu)|\psi|^{2\sigma}\psi + (1+iR)\langle u(t),Q(x)\rangle \psi$. It develops a multiplicative, saturating geometric control framework inspired by Agrachev–Sarychev, proving small-time approximate null-controllability and approximate controllability to phase-modulated targets under a saturation condition on $Q$, with $s\ge s_d$. A key technical device is a small-time limit that generates new controllable directions via the limit $e^{(r_1+i r_2)(\mathbb{B}(\varphi)+\langle u,Q\rangle)}\psi_0$, enabling an iterative construction of reachable subspaces $H_j$ whose union densifies $C^r(\mathbb{T}^d;\mathbb{R})$. The results extend bilinear control theory to nonlinear complex PDEs and provide a foundation for phase-control and $L^2$-type approximate controllability in amplitude-phase models arising from fluid dynamics and pattern formation.
Abstract
In this paper, we consider the bilinear approximate controllability for the complex Ginzburg-Landau (CGL) equation with a power-type nonlinearity of any integer degree on a torus of arbitrary space dimension. Under a saturation hypothesis on the control operator, we show the small-time global controllability of the CGL equation. The proof is obtained by developing a multiplicative version of a geometric control approach, introduced by Agrachev and Sarychev in \cite{AS05,AS06}.