Balanced Active Subspaces for Optimal Control via Trajectory Optimization

Emmanuel S. Ameh; Samuel E. Otto

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

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=\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.

Panel of Figure 2 — 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

\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

r

of retained modes for the CGL equation.

Panel of Figure 4 — Figure 4: Visualization of the value function for all trajectories on three-dimensional subspaces for the different methods. The colorbar indicates the value function $V(x)$ .

Panel of Figure 5 — 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.

Panel of Figure 7 — 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

r

Panel of Figure 9 — Figure 9: Open-loop reduced-order state estimation using an EKF-ROM. (a) Challenging nonlinear toy model. (b) CGL equation model.

Panel of Figure 10 — 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

Panel of Figure 12 — 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.

Optimal Control

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

See all 139 papers

Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs
Jun 22, 2026Matías Gómez-Aedo, Behzad Azmi, Yuyang Huang +2Value Function Estimation PDE Solving
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
Optimal Safety Control using High-Order Control Barrier Functions
Jul 19, 2026Neng Li, Zuodong Pan, Jiaxing Wang +2Lyapunov-Based Control Safety-Critical Control
Key Laboratory of Intelligent Control and Optimization for Industrial Equipment of Ministry of Education, Dalian University of Technology, Dalian 116024, China
Featurized Occupation Measures for Structured Global Search in Numerical Optimal Control
Mar 17, 2026Qi Wei, Jianfeng Tao, Haoyang Tan +1HJB Equations Optimal Control
School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Jun 22, 2026J/K move · Enter open · S save

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

Matías Gómez-Aedo, Behzad Azmi, Yuyang Huang, Dante Kalise+1

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 Estimation PDE Solving Data-Driven Control HJB Equations Optimal Control

Open paper AskPDF arXiv

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.

Abstract

Optimal Control

Explore similar work

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

Optimal Safety Control using High-Order Control Barrier Functions

Featurized Occupation Measures for Structured Global Search in Numerical Optimal Control