An inverse problem for a nonlinear Schrödinger equation

Abstract

We study the dependence on the control $q$ of the interval of definition of the solution $u$ of the Cauchy problem $\iota u^{\prime }+\Delta u=-\lambda {\left|u\right|}^{2}u-\iota qu$ in ${ℝ}^2\times \left(0,T\right),u\left(x,0\right)=\omega$ in ${ℝ}^2$, and we prove a version of Fibich′s conjecture. Feedback laws for an inverse problem of the above equation with experimental data, measured on a portion of the boundary of an open, bounded subset of ${ℝ}^2$ are established.