Small-time approximate controllability for the nonlinear complex Ginzburg-Landau equation with bilinear control
Xingwu Zeng, Can Zhang
TL;DR
This paper establishes small-time approximate controllability for the nonlinear complex Ginzburg-Landau equation on a torus using a novel multiplicative geometric control approach under a saturation hypothesis.
Contribution
It introduces a multiplicative geometric control method to prove small-time controllability for the nonlinear CGL equation with bilinear control.
Findings
Proves small-time global controllability of the nonlinear CGL equation.
Develops a multiplicative version of geometric control theory.
Applies the approach to equations with power-type nonlinearities.
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}.
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Taxonomy
TopicsStability and Controllability of Differential Equations · Control and Stability of Dynamical Systems · Neural Networks Stability and Synchronization