Optimal bilinear control of nonlinear Schrödinger equations with singular potentials. (English) Zbl 1297.49005
To investigate the optimal bilinear control problem for the nonlinear Schrödinger equation one uses the same cost functional as in the paper of M. Hintermüller et al. [SIAM J. Control Optim. 51, No. 3, 2509–2543 (2013; Zbl 1277.49005)]. This objective functional is quadratic in the derivatives of energy and control. Under some special assumptions on the parameters and the potential in the equation, the existence of a minimizer is proved. Thanks to the global well-posedness of the Schrödinger equation for any given initial data in \(H^1\), the unconstrained cost functional is defined. This functional is Gâteaux differentiable and one obtains the first order optimality system. Comments and comparisons to results in the above mentioned article are done.
Reviewer: Claudia Simionescu-Badea (Wien)
MSC:
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 49K40 | Sensitivity, stability, well-posedness |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
Keywords:
optimal bilinear control; nonlinear Schrödinger equation; optimality conditions; singular potentialsCitations:
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