2018
DOI: 10.3934/eect.2018016
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Optimal nonlinearity control of Schrödinger equation

Abstract: We study the optimal nonlinearity control problem for the nonlinear Schrödinger equation iut = − u + V (x)u + h(t)|u| α u, which is originated from the Fechbach resonance management in Bose-Einstein condensates and the nonlinearity management in nonlinear optics. Based on the global wellposedness of the equation for 0 < α < 4 N , we show the existence of the optimal control. The Fréchet differentiability of the objective functional is proved, and the first order optimality system for N ≤ 3 is presented. 317 31… Show more

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Cited by 3 publications
(8 citation statements)
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References 27 publications
(47 reference statements)
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“…However, when (𝛾, 𝜌) ≠ (∞, 2), we cannot get the uniformly boundedness of ||u 𝜀 || L 𝛾 t Σ 1,𝜌 (0,T) . Indeed, similar to the case of the power-law nonlinearities, see Wang et al, 27,28 we can deduce from Strichartz's estimates and Lemma 2.2 that M is depended by 𝜌, 𝜀, and ||u 𝜀 || L ∞ t Σ(0,T) .…”
Section: Estimates Of U 𝜀supporting
confidence: 63%
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“…However, when (𝛾, 𝜌) ≠ (∞, 2), we cannot get the uniformly boundedness of ||u 𝜀 || L 𝛾 t Σ 1,𝜌 (0,T) . Indeed, similar to the case of the power-law nonlinearities, see Wang et al, 27,28 we can deduce from Strichartz's estimates and Lemma 2.2 that M is depended by 𝜌, 𝜀, and ||u 𝜀 || L ∞ t Σ(0,T) .…”
Section: Estimates Of U 𝜀supporting
confidence: 63%
“…Then, one can prove the theorem similarly as the case of the GP equation; see previous studies 23,24,28 for more details. Here, we omit it.…”
Section: The Approximate Minimizing Problemmentioning
confidence: 84%
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