Positively invariant set explained

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

Suppose

• x = f(x) is a dynamical system, x(t,x _{0)} is a trajectory, and x _{0} is the initial point. Let l{O} : = \left \lbrace x \in R ^{n\mid} \varphi (x) = 0 \right \rbrace where \varphi is a real-valued function. The set l{O} is said to be positively invariant if x _{0} \in l{O} implies that x(t,x _{0)} \in l{O} \forall t \ge 0

In other words, once a trajectory of the system enters

l{O} , it will never leave it again.

References

• Dr. Francesco Borrelli http://www.mpc.berkeley.edu/mpc-course-material
• A. Benzaouia. book of "Saturated Switching Systems". chapter I, Definition I, Springer 2012. https://www.springer.com/in/book/9781447128991.