# Optimal Control Problems with Symmetry Breaking Cost Functions

arXiv: https://arxiv.org/abs/1701.06973 · HTML: https://mlsanity.com/paper/1701.06973/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jan 2017.

## Built on by (most cited first, 2 of 2)

- [Geometric Fault-Tolerant Neural Network Tracking Control of Unknown Systems on Matrix Lie Groups](https://mlsanity.com/paper/2505.04725/) (2025)
- [Latent Lie-Poisson Neural Networks (LLPNNs): Discovering the motion of Lie-Poisson systems through observable data and latent dynamics](https://mlsanity.com/paper/2607.28939/) (2026)

## Abstract (arXiv)

We investigate symmetry reduction of optimal control problems for left-invariant control systems on Lie groups, with partial symmetry breaking cost functions. Our approach emphasizes the role of variational principles and considers a discrete-time setting as well as the standard continuous-time formulation. Specifically, we recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the Euler--Poincar'e equations from a variational principle. By applying a Legendre transformation to it, we recover the Lie-Poisson equations obtained by A. D. Borum [Master's Thesis, University of Illinois at Urbana-Champaign, 2015] in the same context. We also discretize the variational principle in time and obtain the discrete-time Lie-Poisson equations. We illustrate the theory with some practical examples including a motion planning problem in the presence of an obstacle.