Risk-Averse Ensemble Control for Control-Affine Systems

TL;DR

This paper develops a mathematical framework for risk-averse ensemble control of control-affine systems, ensuring existence of solutions and optimality conditions, with validation in quantum control applications.

Contribution

It provides a rigorous analysis of risk-averse ensemble control, including existence, differentiability, and optimality conditions, for control-affine systems.

Findings

Established existence of optimal solutions under risk-averse criteria.

Proved regularity properties of the control-to-state mapping.

Validated theoretical results with a quantum control numerical experiment.

Abstract

A number of important modern applications in optimal control can be formulated as open loop control problems in which the underlying dynamical systems are subject to random inputs. These so-called ensemble control problems require the corresponding optimal control to be deterministic, as it must be computed before the realization of uncertainty and the passage of time. Practical applications of ensemble control include quantum control and the training of Neural ODEs. However, the standard approach to ensemble control treats the uncertainty in the objective function via the expectation, which provides optimal controls that only work well on average while ignoring critical outlier phenomena. This study provides a comprehensive mathematical treatment of risk-averse ensemble control. Within this setting, we adopt a control-affine structure that ensures the lower semi-continuity needed for proving the existence of optimal solutions. The central analytical contribution of this paper is a rigorous characterization of the control-to-state mapping in which we establish weak-to-strong continuity, continuous Fr\'echet differentiability, and weak-to-strong continuity of the derivative operator. Furthermore, this regularity yields primal and dual first-order optimality conditions characterized by an adjoint state of bounded variation, and it fulfills the functional prerequisites required for the convergence of infinite dimensional optimization algorithms. We conclude by validating these theoretical developments through a numerical experiment in quantum control.