TY - JOUR

T1 - Frequency-sparse optimal quantum control

AU - Friesecke, Gero

AU - Henneke, Felix

AU - Kunisch, Karl

N1 - Publisher Copyright:
© 2018, American Institute of Mathematical Sciences. All rights reserved.

PY - 2018/3

Y1 - 2018/3

N2 - 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.

AB - 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.

KW - Frequency sparsity

KW - Multi-level Schrödinger system

KW - Optimal quantum control

UR - http://www.scopus.com/inward/record.url?scp=85042541437&partnerID=8YFLogxK

U2 - 10.3934/mcrf.2018007

DO - 10.3934/mcrf.2018007

M3 - Article

AN - SCOPUS:85042541437

SN - 2156-8472

VL - 8

SP - 155

EP - 176

JO - Mathematical Control and Related Fields

JF - Mathematical Control and Related Fields

IS - 1

ER -