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Local exact controllability of a one-dimensional nonlinear Schrödinger equation. (English) Zbl 1326.35330
The purpose of this article is to study a one-dimensional nonlinear Schrödinger equation. This is equivalent to a control system in which the state is the wave function \(\Psi\) and the control is the length of the box. First new variables are introduced, which impose a constraint on the control \(u\). This implies that a certain map is surjective. The well-posedeness, Cauchy problem and existence and uniqueness of positive solutions are studied, as well as asymptotic estimates. The Banach fixed point theorem, Galiardo-Nirenberg, Cauchy-Schwartz and Gronvalls inequalities are used in the proofs.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q93 PDEs in connection with control and optimization
93C20 Control/observation systems governed by partial differential equations
35B09 Positive solutions to PDEs
Software:
OCTBEC
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