Sparse Control of Quantum Systems
Abstract
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 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 \textit{ finitely many} frequencies. These results cover important systems of physical interest, including (infinite-dimensional) Schr\"odinger 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 costs, concentrate on just a few frequencies, even in the infinite-dimensional case of laser-controlled chemical reactions.
Cite
@article{arxiv.1507.00768,
title = {Sparse Control of Quantum Systems},
author = {Gero Friesecke and Felix Henneke and Karl Kunisch},
journal= {arXiv preprint arXiv:1507.00768},
year = {2016}
}