Towards optimal control of ensembles of discrete-time systems

The control of ensembles of dynamical systems is an intriguing and challenging problem, arising for example in quantum control. We initiate the investigation of optimal control of ensembles of discrete-time systems, focusing on minimising the average finite horizon cost over the ensemble. For very general nonlinear control systems and stage and terminal costs, we establish existence of minimisers under mild assumptions. Furthermore, we provide a $\Gamma$-convergence result which enables consistent approximation of the challenging ensemble optimal control problem, for example, by using empirical probability measures over the ensemble. Our results form a solid foundation for discrete-time optimal control of ensembles, with many interesting avenues for future research.