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An inverse problem for a nonlinear Schrödinger equation. (English) Zbl 1009.35077

Summary: We study the dependence on the control \(q\) of the interval of definition of the solution \(u\) of the Cauchy problem \(\iota u'+\Delta u=-\lambda |u|{2}u-\iota qu\) in \({\mathbb{R}}^{2}\times (0,T)\), \(u(x,0)=\omega\) in \({\mathbb{R}}^{2}\), 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 \({\mathbb{R}}^{2}\) are established.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B37 PDE in connection with control problems (MSC2000)
35R30 Inverse problems for PDEs
49J20 Existence theories for optimal control problems involving partial differential equations
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