# Nagumo-Type Characterization of Forward Invariance for Constrained Systems

## TL;DR

This paper introduces a Nagumo-type condition for ensuring forward invariance of constrained systems, extending classical results to more general boundary cases with additional geometric assumptions.

## Contribution

It generalizes Nagumo's invariance condition to differential inclusions on closed sets, providing necessary and sufficient conditions under specific geometric assumptions.

## Key findings

- The proposed condition reduces to classical Nagumo condition on interior boundaries.
- Additional assumptions ensure equivalence between necessary and sufficient conditions.
- Examples demonstrate the importance of geometric assumptions for invariance.

## Abstract

This paper proposes a Nagumo-type invariance condition for differential inclusions defined on closed constraint sets. More specifically, given a closed set to render forward invariant, the proposed condition restricts the system's dynamics, assumed to be locally Lipschitz, on the boundary of the set restricted to the interior of the constraint set. In particular, when the boundary of the set is entirely within the interior of the constraint set, the proposed condition reduces to the well-known Nagumo condition, known to be necessary and sufficient for forward invariance in this case. This being said, the proposed condition is only necessary in the general setting. As a result, we provide a set of additional assumptions relating the constrained system to the set to render forward invariant, and restricting to the geometry at the intersection between the two sets, so that the equivalence holds. The importance of the proposed assumptions is illustrated via examples.