Optimal bilinear control of Gross-Pitaevskii equations (Q2848596)

The goal of the important paper is to consider quantum control systems within the framework of optimal control from a partial differential equation constrained point of view. The objective of the control process is thereby quantified through an objective functional \(J=J(\Psi, \alpha)\), which is minimized subject to the condition that the time-evolution of the quantum state is governed by the cubically nonlinear Schrödinger equation (the Gross-Pitaevskii equation)NEWLINENEWLINE(1) \(ih \partial_t \Psi = -\frac{h^2}{2m} \Delta \Psi + U(x) \Psi + N_g|\Psi|^2 \Psi + W(t,x)\Psi, x\in \mathbb R^3, t\in \mathbb R\).NEWLINENEWLINEThe function \(U(x)\) describes an external trapping potential which is necessary for the experimental realization of Bose-Einstein Condensates (BECs). Typically, \(U(x)\) is assumed to be a harmonic confinement. The condensate is consequently manipulated via a time-dependent control potential \(W(t,x)\) which the authors presented in the following form: \(W(t,x)= \alpha(t) V(x)\), where \(\alpha (t)\) denotes the control parameter.NEWLINENEWLINESuch objective functionals \(J(\Psi, \alpha)\) as mentioned above consist of two parts, one being the desired physical quantity (observable) to be minimized and the other describing the cost it takes to obtain the desired outcome through the control process.NEWLINENEWLINEThe work clarifies existence of a minimizer for the proposed control problem. The authors prove that the corresponding optimal solution \(\Psi_* (t,x)\) is indeed a mild (and not only a weak) solution of (1) depending continuously on the initial data \(\Psi_0\). The adjoint equation is derived and analyzed with respect to existence and uniqueness of a solution. A gradient- and Newton-type descent method are defined, respectively, and then used for computing numerical solutions for several illustrative quantum control problems. The optimal shifting of a linear wave package, splitting of a linear wave package, and splitting of BEC, are considered.