Riemannian Density-Driven Optimal Control: Tangent-Space LQR for Second-Order Multi-Agent Systems on Curved Manifolds
Abstract
Density-Driven Optimal Control (D2OC) provides an effective framework for steering multi-agent systems toward prescribed spatial distributions. However, existing D2OC formulations are primarily developed for Euclidean domains and do not directly account for intrinsic manifold geometry. This paper extends D2OC to second-order multi-agent systems evolving on Riemannian manifolds. The proposed Riemannian D2OC (R-D2OC) constructs a local distribution objective in the tangent space of each agent through logarithmic maps and uses its weighted center as the reference for a finite-horizon LQR. The resulting control is executed on the manifold through intrinsic second-order dynamics and parallel transport within a receding-horizon scheme. We establish local curvature-dependent bounds that quantify the approximation introduced by the tangent-space reduction and characterize the resulting target bias. Furthermore, we derive a conditional discrete-descent result showing that the closed-loop objective decreases when a local velocity-alignment condition is satisfied. Numerical simulations on a 3D ellipsoidal manifold demonstrate distribution-level control and empirically support the proposed approximation and descent results.
Related
- Kernel Density Regression Estimation with Covariates on Riemannian Manifolds
- Multi-marginal optimal transport on sub-Riemannian manifolds
- The pseudo-Riemannian spectral zeta function density on scattering manifolds
- Stability of optimal transport on sub-Riemannian manifolds
- A Lagrange-Newton method for solving optimal control problems on manifolds
BibTeX
@misc{lee2026riemannian,
title = {{Riemannian Density-Driven Optimal Control: Tangent-Space LQR for Second-Order Multi-Agent Systems on Curved Manifolds}},
author = {Kooktae Lee and Ruchika Singh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22678},
eprint = {2609.22678},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22678},
}
RIS
TY - UNPB
TI - Riemannian Density-Driven Optimal Control: Tangent-Space LQR for Second-Order Multi-Agent Systems on Curved Manifolds
AU - Kooktae Lee
AU - Ruchika Singh
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22678
UR - https://arxiv.org/abs/2609.22678
AB - Density-Driven Optimal Control (D2OC) provides an effective framework for steering multi-agent systems toward prescribed spatial distributions. However, existing D2OC formulations are primarily developed for Euclidean domains and do not directly account for intrinsic manifold geometry. This paper extends D2OC to second-order multi-agent systems evolving on Riemannian manifolds. The proposed Riemannian D2OC (R-D2OC) constructs a local distribution objective in the tangent space of each agent through logarithmic maps and uses its weighted center as the reference for a finite-horizon LQR. The resulting control is executed on the manifold through intrinsic second-order dynamics and parallel transport within a receding-horizon scheme. We establish local curvature-dependent bounds that quantify the approximation introduced by the tangent-space reduction and characterize the resulting target bias. Furthermore, we derive a conditional discrete-descent result showing that the closed-loop objective decreases when a local velocity-alignment condition is satisfied. Numerical simulations on a 3D ellipsoidal manifold demonstrate distribution-level control and empirically support the proposed approximation and descent results.
ER -