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Revision as of 18:24, 16 December 2013

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

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

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+In [[mathematical analysis]], a '''positively invariant set''' is a [[Set (mathematics)|set]] with the following properties:
+Given a [[dynamical system]] <math>\dot{x}=f(x)</math> and [[trajectory]] <math> x(t,x_0) \, </math> where <math> x_0 \, </math> is the initial point. Let <math> \mathcal{O} \triangleq \left \lbrace x \in \mathbb{R}^n| \phi (x) = 0 \right \rbrace</math> where <math>\phi</math> is a real valued [[Function (mathematics)|function]]. The set <math>\mathcal{O}</math> is said to be positively invariant if <math>x_0 \in \mathcal{O}</math> implies that <math>x(t,x_0) \in \mathcal{O} \ \forall \ t \ge 0 </math>
+Intuitively, this means that once a trajectory of the system enters <math>\mathcal{O}</math>, it will never leave it again.
+==References==
+*Dr. Francesco Borrelli [http://www.mpc.berkeley.edu/mpc-course-material]
+{{DEFAULTSORT:Positive Invariant Set}}
+[[Category:Mathematical analysis]]
+{{mathanalysis-stub}}