Positive invariant set

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A positive invariant set is a set with the following properties:

Given a system $\dot{x}=f(x)$ and trajectory $x(t,x_0) \,$ where $x_0 \,$ is the initial point. Let $\mathcal{O} \triangleq \left \lbrace x \in \mathbb{R}^n| \phi (x) = 0 \right \rbrace$ where $φ$ is a real valued function that characterizes $\mathcal{O}$ . The set $\mathcal{O}$ is said to be positively invariant if $x_0 \in \mathcal{O}$ implies that $x(t,x_0) \in \mathcal{O} \ \forall \ t \ge 0$