Positive invariant set
（重定向自Positive invariance）
In mathematical analysis, a positively invariant set is a set with the following properties:
Given a dynamical system ![\dot{x}=f(x)](/Images/godic/202502/03/68f8b894bdbb55f9307bebe38693d20d0805.png")
and trajectory ![x(t,x_0) \,](/Images/godic/202502/03/6d6fed926415caa70ced410049e011460805.png")
where ![x_0 \,](/Images/godic/202502/03/3bedb02bf0f9a563e6d18e45ab96d2bf0805.png")
is the initial point. Let ![\mathcal{O} \triangleq \left \lbrace x \in \mathbb{R}^n| \phi (x) = 0 \right \rbrace](/Images/godic/202502/03/09529407b7f6b66ea59a3707f7d6364a0805.png")
where ![\phi](/Images/godic/202502/03/7f20aa0b3691b496aec21cf356f63e040805.png")
is a real valued function. The set ![\mathcal{O}](/Images/godic/202502/03/aeb29ed2e1ec2353da1748944aef3c2f0805.png")
is said to be positively invariant if ![x_0 \in \mathcal{O}](/Images/godic/202502/03/09cc2857a568d91e1c6252f22ee6be1a0806.png")
implies that ![x(t,x_0) \in \mathcal{O} \ \forall \ t \ge 0](/Images/godic/202502/03/52270fe6c44184467cf32a05f52dfcb70806.png")