Positively invariant set

In mathematical analysis, a positively (or positive) invariant set is a set with the following properties: Suppose [math]\displaystyle{ \dot{x}=f(x) }[/math] is a dynamical system, [math]\displaystyle{ x(t,x_0) }[/math] is a trajectory, and [math]\displaystyle{ x_0 }[/math] is the initial point. Let [math]\displaystyle{ \mathcal{O} := \left \lbrace x \in \mathbb{R}^n\mid \varphi (x) = 0 \right \rbrace }[/math] where [math]\displaystyle{ \varphi }[/math] is a real-valued function. The set [math]\displaystyle{ \mathcal{O} }[/math] is said to be positively invariant if [math]\displaystyle{ x_0 \in \mathcal{O} }[/math] implies that [math]\displaystyle{ x(t,x_0) \in \mathcal{O} \ \forall \ t \ge 0 }[/math]

In other words, once a trajectory of the system enters [math]\displaystyle{ \mathcal{O} }[/math], 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].