Optimal bilinear control for Gross-Pitaevskii equations with singular potentials
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We study the optimal bilinear control problem of the generalized Gross-Pitaevskii equation
i∂tu = -∆u + U(x)u + φ(t) 1/|x|α u + λ|u|2σu, x ∈ ℝ3,
where U(x) is the given external potential, φ(t) is the control function. The existence of an optimal control and the optimality condition are presented for suitable α and σ. In particular, when 1 ≤ α < 3/2, the Fréchet-differentiability of the objective functional is proved for two cases: (i) λ < 0, 0 < σ < 2/3; (ii) λ > 0, 0 < σ < 2. Comparing with the previous studies in , the results fill the gap for σ ∈ (0, 1/2).
CitationWang, K., & Zhao, D. (2019). Optimal bilinear control for Gross-Pitaevskii equations with singular potentials. Electronic Journal of Differential Equations, 2019(115), pp. 1-13.
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