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 ${\mathcal {O}}\triangleq \left\lbrace x\in \mathbb {R} ^{n}\mid \varphi (x)=0\right\rbrace$ 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 $x(t,x_{0})\in {\mathcal {O}}\ \forall \ t\geq 0$

In other words, once a trajectory of the system enters ${\mathcal {O}}$, it will never leave it again.

A set is said to be positively invariant for a system of dynamical equations if the initial solution of the system of equations exits in the set and all other solutions continue to be in the same set.

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].