  We present a computational method for open-loop minimum-norm control
synthesis for fixed-endpoint transfer of bilinear ensemble systems that are
indexed by two continuously varying parameters. We suppose that one ensemble
parameter scales the homogeneous, linear part of the dynamics, and the second
parameter scales the effect of the applied control inputs on the inhomogeneous,
bilinear dynamics. This class of dynamical systems is motivated by robust
quantum control pulse synthesis, where the ensemble parameters correspond to
uncertainty in the free Hamiltonian and inhomogeneity in the control
Hamiltonian, respectively. Our computational method is based on polynomial
approximation of the ensemble state in parameter space and discretization of
the evolution equations in the time domain using a product of matrix
exponentials corresponding to zero-order hold controls over the time intervals.
The dynamics are successively linearized about control and trajectory iterates
to formulate a sequence of quadratic programs for computing perturbations to
the control that successively improve the objective until the iteration
converges. We use a two-stage computation to first ensure transfer to the
desired terminal state, and then minimize the norm of the control function. The
method is demonstrated for the canonical uniform transfer problem for the Bloch
system that appears in nuclear magnetic resonance, as well as the matter-wave
splitting problem for the Raman-Nath system that appears in ultra-cold atom
interferometry.
