Cost function for optimal control using the GRAPE algorithm. Returns fidelity, gradient and Hessian for a given waveform, specified in polar coordinates. Only the phase channel gradient is returned, the amplitude profile is taken as a given.
amp_profile - set of control pulse amplitudes from an amplitude- phase description. phi_profile - set of control pulse phases from an amplitude-phase description.
fidelity - figure of merit for the overlap of the current state of the system and the desired state(s). When penalty methods are specified, fidelity is returned as an ar- ray separating the penalties from the simulation fidelity. gradient - gradient of the fidelity with respect to the control sequence. When penalty methods are specified, gradi- ent is returned as an array separating penalty gra- dients from the fidelity gradient. hessian - Hessian of the fidelity with respect to the control sequence. When penalty methods are specified, gradi- ent is returned as an array separating penalty Hes- sians from the fidelity Hessian.
A typical call would be from an optimisation function (see examples/optimal_control):
% Run the optimization fminnewton(spin_system,@grape_phase,guess);
The fidelity, the gradient and the Hessian may be supplied to any optimisation routine, including those in the Optimisation Toolbox of Matlab.