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Insensitizing controls for a class of nonlinear Ginzburg-Landau equations

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Abstract

This paper shows the existence of insensitizing controls for a class of nonlinear complex Ginzburg-Landau equations with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. When the nonlinearity in the equation satisfies a suitable superlinear growth condition at infinity, the existence of insensitizing controls for the corresponding semilinear Ginzburg-Landau equation is proved. Meanwhile, if the nonlinearity in the equation is only a smooth function without any additional growth condition, a local result on insensitizing controls is obtained. As usual, the problem of insensitizing controls is transformed into a suitable controllability problem for a coupled system governed by a semilinear complex Ginzburg-Landau equation and a linear one through one control. The key is to establish an observability inequality for a coupled linear Ginzburg-Landau system with one observer.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    MuMing Zhang & Xu Liu

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  1. MuMing Zhang
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  2. Xu Liu
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Correspondence to Xu Liu.

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Zhang, M., Liu, X. Insensitizing controls for a class of nonlinear Ginzburg-Landau equations. Sci. China Math. 57, 2635–2648 (2014). https://doi.org/10.1007/s11425-014-4837-8

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  • DOI: https://doi.org/10.1007/s11425-014-4837-8

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