Positively invariant set

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 $\phi$ is a real valued function that characterizes ${\mathcal {O}}$. The set ${\mathcal {O}}$ is said to be positively invariant if $x_{0}\in {\mathcal {O}}$. Positive invariance implies $x(t,x_{0})\in {\mathcal {O}}\ \forall \ t\geq 0$

References

• Dr. Arun D. Mahindrakar [www.ee.iitm.ac.in/~arun_dm/reports/positive_invariant.pdf]