math.OCOct 26, 2026

Balanced Active Subspaces for Optimal Control via Trajectory Optimization

Authors: Emmanuel S. Ameh, Samuel E. Otto

Organizations: Department of Chemical and Biomolecular Engineering · Department of Mechanical and Aerospace Engineering

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

Abstract

The curse of dimensionality associated with solving the Hamilton-Jacobi-Bellman equation for the value function remains a key obstacle for real-time optimal feedback control of high-dimensional nonlinear dynamical systems. This is especially severe in aerospace applications such as relaminarizing unsteady turbulent flows and recovering from aerodynamic stall where nonlinear dynamics play a significant role. We introduce Balanced Active Subspaces for Optimal Control (BASOC), a model reduction framework which seeks to find directions in state space along which the value function is most sensitive, and the distribution of states exhibits maximum variance along optimal trajectories. BASOC leverages trajectory optimization and the Pontryagin minimum principle to obtain state and co-state (value function gradient) data along locally optimal trajectories for constructing state and gradient covariance matrices. Balancing these matrices provides an oblique projection for approximating the value function, which can also be used to construct reduced-order models and surrogate feedback control laws. We show that for linear-quadratic optimal control problems, BASOC becomes equivalent to a snapshot method for initial condition balancing of the optimally controlled closed-loop realization. The framework is demonstrated on a nonlinear three-dimensional system where POD and balanced truncation are known to perform poorly, and on the nonlinear complex Ginzburg-Landau equation to demonstrate feasibility for larger-scale systems.

Figures & tables

 X=1NIC[X~(1)⋯X~(NIC)]Y=1NIC[Y~(1)⋯Y~(NIC)].X=\frac{1}{\sqrt{N_{\mathrm{IC}}}}[\widetilde{X}^{(1)}\cdots\widetilde{X}^{(N_{\mathrm{IC}})}]\qquad Y=\frac{1}{\sqrt{N_{\mathrm{IC}}}}[\widetilde{Y}^{(1)}\cdots\widetilde{Y}^{(N_{\mathrm{IC}})}]. 
Algorithm 1 Balanced Active Subspace (BASOC) Projection Algorithm
Figure 1: Comparison of the value function along optimal trajectories plotted in reduced variables selected by different methods for the challenging nonlinear toy model.
Figure 2: In (a) we show the uncontrolled CGL equation exhibiting a large downstream response due to upstream disturbances, and in (b) we show the controlled CGL equation under optimal control forcing upstream.
Figure 3: Remaining fraction ∑ℓ=r+1nσℓ2/∑ℓ=1nσℓ2\sum_{\ell=r+1}^{n}\sigma_{\ell}^{2}/\sum_{\ell=1}^{n}\sigma_{\ell}^{2} of the total squared BASOC singular values as a function of the number rr of retained modes for the CGL equation.
Figure 4: Visualization of the value function for all trajectories on three-dimensional subspaces for the different methods. The colorbar indicates the value function V(x)V(x) .
Figure 5: Closed-loop full-state feedback using surrogate control laws for the challenging nonlinear toy model: (a) non-extrapolated test initial conditions; (b) extrapolated test initial conditions.
Figure 7: ROM output prediction for the nonlinear toy model: (a) prediction for a single test trajectory; (b) normalized squared output prediction errors along optimal trajectories for extrapolated test initial conditions. Similar results are obtained for non-extrapolated initial conditions.
Figure 8: Normalized squared state-forecast error across optimal test trajectories for the nonlinear CGL equation for different reduced dimensions rr .
Figure 9: Open-loop reduced-order state estimation using an EKF-ROM. (a) Challenging nonlinear toy model. (b) CGL equation model.
Figure 10: Output-based closed-loop feedback control for the challenging nonlinear toy model for (a) non-extrapolated and (b) extrapolated test initial conditions.
Figure 11: Output-based closed-loop feedback control using the EKF and surrogate control law for the CGL equation model.
Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 12: Convergence of the cost function and its gradient norm during trajectory optimization for (a) the nonlinear toy model and (b) the CGL model.
More in this topic

Optimal Control

Optimal control studies how to choose actions over time to optimize performance while satisfying system dynamics and constraints.

See all 139 papers

Explore similar work

CardsList
Jun 22, 2026

Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs

A data-driven method is developed for approximating value functions in deterministic optimal control problems with nonlinear control-affine dynamics. The Pontryagin Maximum Principle optimality system is solved from multiple initial conditions to generate training data consisting of values, gradients, and Hessians of the value function, where Hessian information is obtained from a matrix Riccati equation along optimal trajectories. These quantities augment a weighted least-squares regression over sparse polynomial bases on hyperbolic cross index sets, with gradients and Hessians contributing additional linear equations per sample and substantially reducing sample complexity compared to value-only regression. Feedback laws are recovered analytically from the learned value function. In high dimensions, a partial Hessian strategy controls the cost of data generation. The approach is validated on problems of increasing state dimension, where second-order data augmentation is shown to improve approximation accuracy and closed-loop performance, with up to an order-of-magnitude reduction in the number of training samples required relative to lower-order methods.

Matías Gómez-Aedo, Behzad Azmi, Yuyang Huang +2

Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ London, UK · Department of Mathematics and Statistics, University of Konstanz, Germany · RICAM and Institute of Mathematics and Scientific Computing, University of Graz, Austria

Value Function EstimationPDE SolvingData-Driven ControlHJB EquationsOptimal Control
AskPDF
Jul 19, 2026

Optimal Safety Control using High-Order Control Barrier Functions

This paper investigates the optimal safety control problem of nonlinear control systems by proposing novel high-order control barrier functions (HOCBFs). Different from zeroing HOCBFs, two novel HOCBFs are derived and the safety controllers are designed in an explicit way. Next, we implement vector Lyapunov function approach to propose a novel high-order control Lyapunov function (HOCLF) for the stabilization control problem. The relations between the proposed and existing HOCBFs are discussed. Afterwards, the compatibility of the proposed HOCLF and HOCBF is addressed to guarantee the stabilization and safety control objectives simultaneously, and thus the optimal controller is established. Finally, a numerical example from the navigation problem of quadrotors is presented to illustrate the efficacy of the derived results.

Neng Li, Zuodong Pan, Jiaxing Wang +2

Key Laboratory of Intelligent Control and Optimization for Industrial Equipment of Ministry of Education, Dalian University of Technology, Dalian 116024, China

Lyapunov-Based ControlSafety-Critical ControlControl Barrier FunctionsControl Lyapunov Functions
AskPDF
Mar 17, 2026

Featurized Occupation Measures for Structured Global Search in Numerical Optimal Control

Numerical optimal control is split between globally structured but dimensionally intractable Hamilton--Jacobi--Bellman (HJB) methods and scalable but local trajectory optimization. We introduce Featurized Occupation Measures (FOM), a finite-dimensional primal--dual interface where HJB subsolutions guide primal search and primal occupation data can inform certificate refinement. Explicit and implicit realizations, based on finite Liouville tests and rollout-sampled residuals, are proved asymptotically consistent with the exact occupation-measure LP. FOM also identifies when HJB-type certificates avoid full state-space representation. For factor graphs induced by compatible passivity-based interconnections, blockwise HJB inequalities yield block-additive OM-dual certificates, including blockwise approximate variants. The curse of dimensionality is then shifted from state dimension to interconnection topology. Approximate certificates remain reusable under time shifts and bounded model perturbations, with explicit degradation bounds. A static obstacle-avoidance benchmark illustrates how tighter certificates improve sample-based global search.

Qi Wei, Jianfeng Tao, Haoyang Tan +1

School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

HJB EquationsOptimal ControlGlobal Optimization
AskPDF