# On Maximal Robust Positively Invariant Sets in Constrained Nonlinear Systems

## TL;DR

This paper investigates the properties of the maximal robust positively invariant set in constrained nonlinear systems with bounded disturbances, revealing its boundary structure and relation to the maximum principle.

## Contribution

It extends barrier theory to characterize the boundary of invariant sets, introducing the invariance barrier concept for nonlinear systems.

## Key findings

- The maximal robust positively invariant set is closed.
- Its boundary includes the invariance barrier defined by maximum principle trajectories.
- The boundary structure aids in understanding system robustness under disturbances.

## Abstract

In this technical communique we study the maximal robust positively invariant set for state-constrained continuous-time nonlinear systems subjected to a bounded disturbance. Extending results from the theory of barriers, we show that this set is closed and that its boundary consists of two complementary parts, one of which we name the invariance barrier, which consists of trajectories that satisfy the maximum principle.