Positively invariant set

In mathematical analysis, a positively (or positive) invariant set is a set with the following properties:

Suppose $\dot{x}=f(x)$ is a dynamical system, $x(t,x_0)$ is a trajectory, and $x_0$ is the initial point. Let $𝒪:=\{x\in {ℝ}^n\mid \phi (x)=0\}$ where $\phi$ is a real-valued function. The set $𝒪$ is said to be positively invariant if $x_0\in 𝒪$ implies that $x(t,x_0)\in 𝒪\mathrm{\forall }t\ge 0$

In other words, 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. [2].