Frequency-sparse optimal quantum control

Gero Friesecke, Felix Henneke, Karl Kunisch

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


A new class of cost functionals for optimal control of quantum systems which produces controls which are sparse in frequency and smooth in time is proposed. This is achieved by penalizing a suitable time-frequency representation of the control field, rather than the control field itself, and by employing norms which are of L1 or measure form with respect to frequency but smooth with respect to time. We prove existence of optimal controls for the resulting nonsmooth optimization problem, derive necessary optimality conditions, and rigorously establish the frequency-sparsity of the optimizers. More precisely, we show that the time-frequency representation of the control field, which a priori admits a continuum of frequencies, is supported on only finitely many frequencies. These results cover important systems of physical interest, including (infinite-dimensional) Schrödinger dynamics on multiple potential energy surfaces as arising in laser control of chemical reactions. Numerical simulations confirm that the optimal controls, unlike those obtained with the usual L2 costs, concentrate on just a few frequencies, even in the infinite-dimensional case of laser-controlled chemical reactions.

Original languageEnglish
Pages (from-to)155-176
Number of pages22
JournalMathematical Control and Related Fields
Issue number1
StatePublished - Mar 2018


  • Frequency sparsity
  • Multi-level Schrödinger system
  • Optimal quantum control


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