Mathematical and Numerical Aspects of Quantum Dynamics

New optimal control problems for quantum systems

Gero Friesecke

Technische Universit√§t M√ľnchen


Optimal control theory (OCT) for quantum systems is traditionally considered with simple ``cost functionals'' (e.g., a goal term promoting closeness to a favoured subspace plus a space-time L2 norm for regularization.) But the general OCT setting offers great freedom to design and compute controls and states with desired features. As a first example (joint with Felix Henneke und Karl Kunisch, AIMS Math. Control and Related Fields 2018, 8(1): 155-176) we show how one can design frequency-sparse quantum controls, by means of penalizing a suitable time-frequency representation of the control field and employing an L1 or measure form with respect to frequency. Simulations show, and a rigorous proof confirms, that the control fields uses only a finite number of frequencies, even for infinite-dimensional Schroedinger dynamics on multiple potential energy surfaces as arising in laser control of chemical reactions. As a second example (joint with Michael Kniely, arXiv 2018) we formulate various desirable properties of photovoltaic materials as mathematical control goals for excitations in the Kohn-Sham equations, show that the resulting problems are well posed, and present illustrative numerical simulations of optimal doping profiles and the resulting excitations for 1D finite nanocrystals.