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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 $\mathcal{𝒪}\triangleq \{x\in {\mathbb{ℝ}}^n|\varphi (x)=0\}$ where $\varphi$ is a real valued function. The set $\mathcal{𝒪}$ is said to be positively invariant if $x_0\in \mathcal{𝒪}$ implies that $x(t,x_0)\in \mathcal{𝒪}\mathrm{\forall }t\ge 0$

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

References

• Dr. Francesco Borrelli [1]