Distributional Control of Ensemble Systems

TL;DR

The paper proposes distributional control for ensemble systems, shifting from state-centric control to steering output-measure distributions induced by aggregated measurements. It develops a dynamic moment kernelization that connects the evolution of output measures on $\mathcal{P}(N)$ to an infinite sequence of moments, enabling a moment-based representation $m(t)$ and a corresponding moment dynamics. By coupling optimal transport with these moment representations, the authors formulate OT-guided distributional control and provide finite-dimensional truncations that guarantee convergence to the true OT trajectory under mild conditions. The resulting framework supports data-driven, measure-focused control of large populations and offers practical methodologies for pattern shaping, synchronization, and other ensemble tasks with aggregated data. This approach broadens ensemble control theory by incorporating distributional objectives and OT-based optimization, with potential impact in robotics, neuroscience, and quantum information sciences.

Abstract

Ensemble control offers rich and diverse opportunities in mathematical systems theory. In this paper, we present a new paradigm of ensemble control, referred to as distributional control, for ensemble systems. We shift the focus from controlling the states of ensemble systems to controlling the output measures induced by their aggregated measurements. To facilitate systems-theoretic analysis of these newly formulated distributional control challenges, we establish a dynamic moment kernelization approach, through which we derive the distributional system and its corresponding moment system for an ensemble system. We further explore optimal distributional control by integrating optimal transport concepts and techniques with the moment representations, creating a systematic computational distributional control framework.

Figures (4)

• Figure 1: Sample aggregated measurements $Y_{t_k}$ at time $t_k$ for $k=0,1,\ldots,N$ of an ensemble system.
• Figure 2: Functional control of the linear ensemble system $\Sigma_2$ in \ref{['eq:ECS_linear']}. (a) plots the system trajectory $x_t$ (blue solid curves) and OT trajectory $x_t^*$ (red dashed curves) for $t=0,0.2,0.4,0.6,0.8,1$ (top panel) as well as the optimal tracking controls (bottom panel) for $q=8$ (moment truncation order) and $p=8$ (number of control inputs). (b) shows the final ($x_1$, blue solid curve) and the desired ($x_1^*$, red dashed curve) state (top panel), and the calculated control inputs (bottom panel) for $q=8$ and $p=4$.
• Figure 3: Distributional control of the linear ensemble system $\Sigma_2$ in \ref{['eq:ECS_linear']}. (a) shows the empirical distribution approximation of the controlled output measure $\mu_t$ at $t=0,0.2,0.4,0.6,0.8,1$ by using 1000 randomly selected individual systems in the ensemble, and (b) plots the control inputs for $q=8$ (moment truncation order) and $p=8$ (number of control inputs).
• Figure 4: Distributional control for synchronization of the Kuramoto oscillator network in \ref{['eq:kuramoto']}. The initial and target distributions are the uniform distribution and a point mass on $\mathbb{S}^1$, respectively, with the moment truncation order chosen to be $q=10$. (a) shows the empirical distribution approximation of the initial (red) and final (blue) output measures generated by 1000 oscillators in the network. (b) plots the control input (top panel) and the approximated probability density functions of the initial and final distributions using the order-10 truncated moment sequences (bottom panel).

Theorems & Definitions (26)

• Definition 1: Ensemble controllability
• Example 1: Probability distributions induced by aggregated measurements
• Definition 2: Output measure
• Definition 3: Pattern controllability
• Remark 1
• Theorem 1
• proof
• Remark 2
• Theorem 2
• proof