Positive invariant set

In mathematical analysis, a positively invariant set is a set with the following properties:

Given a dynamical 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. The set ${\mathcal {O}}$ is said to be positively invariant if $x_{0}\in {\mathcal {O}}$ implies that $x(t,x_{0})\in {\mathcal {O}}\ \forall \ t\geq 0$

Intuitively, this means that once a trajectory of the system enters ${\mathcal {O}}$ , it will never leave it again.

References

Dr. Francesco Borrelli
A. Benzaouia. book of "Saturated Switching Systems". chapter I, Definition I, springer 2012. ISBN 978-1-4471-2900-4 .

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