# Convex computation of the maximum controlled invariant set for polynomial control systems

arXiv: https://arxiv.org/abs/1303.6469 · HTML: https://mlsanity.com/paper/1303.6469/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Mar 2013.

## Built on by (most cited first, 5 of 5)

- [Safe Reinforcement Learning with Chance-constrained Model Predictive Control](https://mlsanity.com/paper/2112.13941/) (2021)
- [Safe Reinforcement Learning with Dual Robustness](https://mlsanity.com/paper/2309.06835/) (2023)
- [Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization](https://mlsanity.com/paper/2303.01483/) (2023)
- [Provably Safe, Yet Scalable Reinforcement Learning](https://mlsanity.com/paper/2606.14536/) (2026)
- [Safely Learning Dynamical Systems](https://mlsanity.com/paper/2305.12284/) (2023)

## Abstract (arXiv)

We characterize the maximum controlled invariant (MCI) set for discrete- as well as continuous-time nonlinear dynamical systems as the solution of an infinite-dimensional linear programming problem. For systems with polynomial dynamics and compact semialgebraic state and control constraints, we describe a hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations whose optimal values converge to the volume of the MCI set; dual to these LMI relaxations are sum-of-squares (SOS) problems providing a converging sequence of outer approximations to the MCI set. The approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description. A number of numerical examples illustrate the approach.