# Stability Analysis of Complementarity Systems with Neural Network Controllers

arXiv: https://arxiv.org/abs/2011.07626 · HTML: https://mlsanity.com/paper/2011.07626/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Nov 2020.

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

- [Assured Neural Network Architectures for Control and Identification of Nonlinear Systems](https://mlsanity.com/paper/2109.10298/) (2021)
- [Revisiting PGD Attacks for Stability Analysis of Large-Scale Nonlinear Systems and Perception-Based Control](https://mlsanity.com/paper/2201.00801/) (2022)

## Abstract (arXiv)

Complementarity problems, a class of mathematical optimization problems with orthogonality constraints, are widely used in many robotics tasks, such as locomotion and manipulation, due to their ability to model non-smooth phenomena (e.g., contact dynamics). In this paper, we propose a method to analyze the stability of complementarity systems with neural network controllers. First, we introduce a method to represent neural networks with rectified linear unit (ReLU) activations as the solution to a linear complementarity problem. Then, we show that systems with ReLU network controllers have an equivalent linear complementarity system (LCS) description. Using the LCS representation, we turn the stability verification problem into a linear matrix inequality (LMI) feasibility problem. We demonstrate the approach on several examples, including multi-contact problems and friction models with non-unique solutions.