Towards optimal control of ensembles of discrete-time systems

TL;DR

This paper develops a theoretical foundation for optimal control of ensembles of discrete-time nonlinear systems, establishing existence of solutions and approximation methods for complex ensemble control problems.

Contribution

It introduces the first general framework for ensemble optimal control in discrete time, including existence proofs and a $ ext{Gamma}$-convergence approximation approach.

Findings

Proved existence of minimizers for ensemble control problems.

Established $ ext{Gamma}$-convergence for approximation of ensemble control.

Provided a foundation for future research in discrete-time ensemble control.

Abstract

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.