Towards optimal control of ensembles of discrete-time systems
Christian Fiedler, Alessandro Scagliotti
TL;DR
This paper develops a theory for optimal control of ensembles of discrete-time systems by formulating the objective as the average finite-horizon cost over an ensemble indexed by a parameter set $\Theta$. It proves the existence of minimisers via the direct method under mild regularity, and introduces a $\Gamma$-convergence framework to justify consistent approximation of the ensemble problem by empirical measures. The analysis covers general nonlinear dynamics on metric spaces and mild regularity on costs, providing conditions under which empirical approximations converge to the true optimal solution. The results lay a solid foundation for data-driven, discrete-time ensemble control and motivate future work on alternative aggregation schemes and infinite-horizon extensions.
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 $Γ$-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.