Dynamical low-rank approximations for the optimal control of matrix ODEs
Abstract
We consider the open-loop optimal control of a dynamical system whose state is described by a time-dependent matrix X(t)∈ℝm× n. Gradient-based optimization methods may incur substantial computational and memory costs, particularly for large-scale systems and fine temporal discretizations. To alleviate this computational burden, we develop a mathematical framework to use Dynamical Low Rank Approximations (DLRA) in optimal control problems. We study two distinct paradigms: the first one, called Optimize-then-Project (OtP), first derives the optimality system and then approximates both the state and adjoint equations using DLRA. The second one, called Project-then-Optimize (PtO), first replaces the state equation with its DLR approximation, and then derives the optimality conditions for the resulting optimization problem. The two approaches differ fundamentally concerning the role of the adjoint variable. In OtP, the adjoint variable evolves on the manifold of fixed rank matrices, whereas in PtO, the adjoint variable belongs to the tangent space to the fixed-rank manifold at the evolving state variable. This latter formulation leads to a covariant ODE describing the dynamics of the adjoint variable as the underlying tangent space evolves in time. For both approaches we establish rigorous error estimates for the approximated gradients, identifying how their accuracy depends on the low-rank approximations of initial and terminal conditions, the components of the state and adjoint vector fields normal to the relevant tangent spaces, the stability properties of the dynamics, and the manifold curvature. PtO further requires tailored time discretization strategies for the adjoint covariant ODE, which we show to be robust against small singular values of the state variable. To illustrate the framework, we apply these methods to the optimal control of a random parabolic PDE.
Related
- Structure-Preserving Dynamical Low-Rank Approximations for Stochastic Vlasov--Poisson Equations
- Numerical Methods for Dynamical Low-Rank Approximations of Stochastic Differential Equations -- Part II: Stochastic discretization
- Robust preconditioners for finite element approximations of control-state constrained optimal control problems governed by elliptic equations
- Rank-Optimal Sample Complexity of Riemannian Gauss-Newton for Low-Rank Matrix Recovery
- Unbiased Low-Rank Approximation for Repeated Matrix-Vector Multiplication: Stability and Optimal Sampling
BibTeX
@misc{carrel2026dynamical,
title = {{Dynamical low-rank approximations for the optimal control of matrix ODEs}},
author = {Benjamin Carrel and Tommaso Vanzan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29554},
eprint = {2610.29554},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29554},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dynamical low-rank approximations for the optimal control of matrix ODEs
AU - Benjamin Carrel
AU - Tommaso Vanzan
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29554
UR - https://arxiv.org/abs/2610.29554
AB - We consider the open-loop optimal control of a dynamical system whose state is described by a time-dependent matrix X(t)∈ℝ^m× n. Gradient-based optimization methods may incur substantial computational and memory costs, particularly for large-scale systems and fine temporal discretizations. To alleviate this computational burden, we develop a mathematical framework to use Dynamical Low Rank Approximations (DLRA) in optimal control problems. We study two distinct paradigms: the first one, called Optimize-then-Project (OtP), first derives the optimality system and then approximates both the state and adjoint equations using DLRA. The second one, called Project-then-Optimize (PtO), first replaces the state equation with its DLR approximation, and then derives the optimality conditions for the resulting optimization problem. The two approaches differ fundamentally concerning the role of the adjoint variable. In OtP, the adjoint variable evolves on the manifold of fixed rank matrices, whereas in PtO, the adjoint variable belongs to the tangent space to the fixed-rank manifold at the evolving state variable. This latter formulation leads to a covariant ODE describing the dynamics of the adjoint variable as the underlying tangent space evolves in time. For both approaches we establish rigorous error estimates for the approximated gradients, identifying how their accuracy depends on the low-rank approximations of initial and terminal conditions, the components of the state and adjoint vector fields normal to the relevant tangent spaces, the stability properties of the dynamics, and the manifold curvature. PtO further requires tailored time discretization strategies for the adjoint covariant ODE, which we show to be robust against small singular values of the state variable. To illustrate the framework, we apply these methods to the optimal control of a random parabolic PDE.
ER -