Local exact controllability of a one-dimensional nonlinear Schrödinger equation.

*(English)*Zbl 1326.35330The 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.

Reviewer: Thomas Ernst (Uppsala)

##### 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 |

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\textit{K. Beauchard} et al., SIAM J. Control Optim. 53, No. 5, 2781--2818 (2015; Zbl 1326.35330)

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