Positive Invariant Set

In mathematical analysis, a positively (or positive) invariant set is a set with the following properties:

Suppose $\dot=f(x)$ is a dynamical system, $x(t,x_0)$ is a trajectory, and $x_0$ is the initial point. Let $\mathcal := \left \lbrace x \in \mathbb^n\mid \varphi (x) = 0 \right \rbrace$ where $\varphi$ is a real-valued function. The set $\mathcal$ is said to be positively invariant if $x_0 \in \mathcal$ implies that $x(t,x_0) \in \mathcal \ \forall \ t \ge 0$

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

References

*Dr. Francesco Borrell

* A. Benzaouia. book of "Saturated Switching Systems". chapter I, Definition I, Springer 2012.

Mathematical analysis

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