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. 317 318 KAI WANG, DUN ZHAO AND BINHUA FENG problem which reads

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We consider the linearly constrained separable convex minimization problem, whose objective function consists of the sum of m individual convex functions in the absence of any coupling variables. While augmented Lagrangian-based decomposition methods have been well developed in the literature for solving such problems, a noteworthy requirement of these methods is that an additional correction step is a must to guarantee their convergence. This note shows that a straightforward Jacobian decomposition of the augmented Lagrangian method is globally convergent if the involved functions are further assumed to be strongly convex.

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