Iterative ensemble control synthesis for bilinear systems

An emerging area in mathematical control theory called Ensemble Control constitutes a class of problems that require the manipulation of an uncountably infinite collection of structurally identical dynamical systems, which are indexed by a parameter set, by applying a common open-loop control. Our investigation is motivated by compelling engineering problems in areas including quantum control and sensorless robotic manipulation that involve ensembles of bilinear systems for which analytical control laws are infeasible in practice or do not exist. While controllability of such systems has been investigated, constructive control design methods remain elusive. We introduce an iterative fixed-point method for optimization-free synthesis of ensemble controls for these systems. At each step, the bilinear ensemble system is approximated by a time-varying linear ensemble system, and the next control iterate is synthesized using a method for linear systems based on the singular value decomposition (SVD). The procedure converges in the given examples to a control function that accomplishes the desired state transfer.