Gradient flow for controlling quantum ensemble

We propose in this paper a gradient-type dynamical system to solve the problem of maximizing quantum observables for finite dimensional closed quantum ensembles governed by the controlled Liouville-von Neumann equation. The asymptotic behavior is analyzed: we show that under the regularity assumption on the controls the dynamical system almost always converges to a solution of the maximization problem; we also detail the difficulties related to the occurrence of singular controls.