Positive invariant set

In mathematical analysis, a positively invariant set is a set with the following properties: Given a dynamical system $\dot{x}=f(x)$ and trajectory $x(t,x_0)$ where $x_0$ is the initial point. Let ${𝒪}\triangleq \{x\in {\mathbb{ℝ}}^n|\varphi (x)=0\}$ where $\varphi$ is a real valued function. The set ${𝒪}$ is said to be positively invariant if $x_0\in {𝒪}$ implies that $x(t,x_0)\in {𝒪}\mathrm{\forall }t\ge 0$

Intuitively, this means that once a trajectory of the system enters ${𝒪}$, it will never leave it again.

References

• Dr. Francesco Borrelli [1]
• A. Benzaouia. book of "Saturated Switching Systems". chapter I, Definition I, springer 2012. ISBN 978-1-4471-2900-4 [2].