deletions | additions       diff --git a/Optimal control.tex b/Optimal control.tex  index bccd254..c94f041 100644  --- a/Optimal control.tex  +++ b/Optimal control.tex 
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\\  \vert\psi(0)\rangle & = \vert\psi_0\rangle\,,  \end{align}  \textit{i.e.} \textit{i.e.}\  the solution of Schr{\"{o}}dinger's the Schr{\"{o}}dinger  equation determines a map $u \longrightarrow \psi[u]$. Suppose we wish to optimize a  functional of the system $F=F[\psi]$. QOCT is about finding the extrema of  $G(u)=F[\psi[u]]$. Beyond this search, QOCT also studies topics such 
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\\  \vert \chi(T) \rangle & = \frac{\delta F}{\delta \psi^*(T)}\ .  \end{align}  This equation assumes, in order to keep this description short, that the target functional $F$ depends on the state of the system only at the final time of the propagation $T$, i.e. \textit{i.e.}\  it is a functional of $\psi(T)$. Note the presence of a boundary value equation at the final time of the propagation, as opposed to the equation of motion for the ``real'' system $\psi$, that naturally depends on an initial value condition, at time zero. With these simple equations, we may already summarize what is needed from an implementation point of view in order to perform basic QOCT calculations: The first step is the selection of the parameters $u$, that constitute the \emph{search space}. Frequently, these parameters are simply the values that the \emph{control function} (typically, the electric-field amplitude) takes at the time intervals that are used to discretize the propagation interval, \textit{i.e.} it is a ``real-time parametrization''. However, more sophisticated parametrizations allow fine-tuning of the search space, introducing constraints and penalties into the formulation.  Then, one must choose an algorithm for maximizing multi-dimensional  functions such as $G$. One possibility is the family of \emph{gradient-less} algorithms, which only require a procedure to compute the value of the function, and do not need the gradient. In this case, the previous equations are obviously not needed. One only has to propagate the system forwards in time, which is what Octopus can do best. The value of the function $G$ can then be computed from the evolution of $\psi$ obtained with this propagation, and feed it fed  into the optimization procedure. A few gradient-less algorithms are implemented in Octopus. The most efficient optimizations can be obtained if the gradient information is employed. In that case, we can use standard schemes, such as for example the family of conjugate-gradient algorithms, or the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton scheme -- we use the implementation of these algorithms included in the GSL mathematical library~\cite{Galassi2009}. Some ad-hoc algorithms, developed explicitly for QOCT, exist. These may in some circumstances be faster than the general purpose ones. Some of those are implemented in Octopus as well~\cite{Zhu1998,Zhu1998a,Ohtsuki1999}.  In order to compute the gradient, one must implement a \emph{backwards-propagation} scheme for the costate, which does not differ from the ones used for the normal forwards propagation~\cite{Castro_2004}. Note, however however,  that in some cases the backwards propagation does not have the exact same simple linear form than the forwards propagation, and may include inhomogeneous or non-linear terms. The final step is the computation of the gradient from the integral given in eq.~(\ref{eq:qoctgradient}). The previously sketched formulation of QOCT is quite generic; in our case the quantum systems are  those that can be modeled with Octopus (periodic systems are not supported at the moment), and the 
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The feasibility of using the electronic current to define the target functional of the QOCT formalism is considered in Ref~\cite{Kammerlander2011a}.   Finally, a series of works have has  studied the use of  optimal control for photo-chemical control: the tailoring of laser pulses to create or break selected bonds in molecules. The underlying physical model should be based on TDDFT, and on a mixed quantum classical scheme (within octopus, Ehrenfest molecular dynamics). Some first attempts in this area were reported in Refs~\cite{Krieger2011,Castro14}. However However,  these works did not consider a fully consistent optimal control theory encompassing TDDFT and Ehrenfest dynamics. This theory has been recently presented~\cite{Castro2014}, and the first computations demonstrating its feasibility will be reported soon.