# Technical Report: Convex Optimization of Nonlinear Feedback Controllers via Occupation Measures

arXiv: https://arxiv.org/abs/1305.7484 · HTML: https://mlsanity.com/paper/1305.7484/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 May 2013.

## Built on by (most cited first, 8 of 8)

- [Verification of Neural Reachable Tubes via Scenario Optimization and Conformal Prediction](https://mlsanity.com/paper/2312.08604/) (2023)
- [Learning Density Distribution of Reachable States for Autonomous Systems](https://mlsanity.com/paper/2109.06728/) (2021)
- [Generating Formal Safety Assurances for High-Dimensional Reachability](https://mlsanity.com/paper/2209.12336/) (2022)
- [Statistical Safety and Robustness Guarantees for Feedback Motion Planning of Unknown Underactuated Stochastic Systems](https://mlsanity.com/paper/2212.06874/) (2022)
- [Denoising Diffusion-Based Control of Nonlinear Systems](https://mlsanity.com/paper/2402.02297/) (2024)
- [Case Studies for Computing Density of Reachable States for Safe Autonomous Motion Planning](https://mlsanity.com/paper/2209.08073/) (2022)
- [Occupation Kernel Hilbert Spaces for Fractional Order Liouville Operators and Dynamic Mode Decomposition](https://mlsanity.com/paper/2102.13266/) (2021)
- [Control Occupation Kernel Regression for Nonlinear Control-Affine Systems](https://mlsanity.com/paper/2106.00103/) (2021)

## Abstract (arXiv)

In this paper, we present an approach for designing feedback controllers for polynomial systems that maximize the size of the time-limited backwards reachable set (BRS). We rely on the notion of occupation measures to pose the synthesis problem as an infinite dimensional linear program (LP) and provide finite dimensional approximations of this LP in terms of semidefinite programs (SDPs). The solution to each SDP yields a polynomial control policy and an outer approximation of the largest achievable BRS. In contrast to traditional Lyapunov based approaches which are non-convex and require feasible initialization, our approach is convex and does not require any form of initialization. The resulting time-varying controllers and approximated reachable sets are well-suited for use in a trajectory library or feedback motion planning algorithm. We demonstrate the efficacy and scalability of our approach on five nonlinear systems.