deletions | additions       diff --git a/Optimal control.tex b/Optimal control.tex  index 2c86cd0..bccd254 100644  --- a/Optimal control.tex  +++ b/Optimal control.tex 
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field has rapidly evolved since then. Broadly speaking, QOCT attempts to  answer the following question: given a quantum process governed by a  Hamiltonian that depends on a set of parameters, what are the values of those  parameters that maximize a given observable that depends on the behaviour behavior  of the system? In mathematical terms: let a set of parameters $u_1,\dots,u_M \equiv u$ determine the Hamiltonian of a system $\hat{H}[u,t]$, so that the  evolution of the system also depends on the value taken by those parameters:  \begin{align} 
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The usual formulation of QOCT assumes the linearity of quantum mechanics. However, the time-dependent KS equations are not linear, making both the theory and the numerics more complicated. We have extended the basic theory previously described to handle the TDDFT equations, and implemented the resulting equations in Octopus~\cite{Castro2012a}.  We conclude this section briefly describing some of the applications of the QOCT machinery included in Octopus, that can give an idea of the range of possibilities that can be attempted. The study presented in Ref.~\cite{Rasanen2007} demonstrates the control of single-electron states in a two-dimensional semiconductor quantum ring model. The states whose transitions are manipulated are the current carrying states, that can be populated or de-populated with the help of circulary circularly  polarized light. Ref.~\cite{Rasanen2008} studies double quantum dots, and shows how the electron state of these systems can be manipulated with the help of electric fields tailored by QOCT.