Ensemble Optimal Control of Control-Affine Systems with Uncertain Initial Conditions

We explore optimal control problems in systems with uncertain parameters that follow a probability distribution, aiming to optimize average performance. Known as Riemann-Stieltjes, average, or ensemble optimal control, these problems are essential when parameter uncertainty plays a significant role. We establish necessary conditions and analyze feedback control structures for control-affine systems. By extending the Pontryagin Maximum Principle through a Hilbert space approach, we characterize singular arcs, and numerical examples demonstrate the practical applicability of our findings.