# Invariant set: sufficient condition

Given a system $\dot x = f(x)$, $x \in \mathbb{R}^n$ with a smooth $f(x)$. Let $D$ be a set in $\mathbb{R}^n$ with a smooth boundary $\partial D$ such that $\left.\langle f(x), n(x) \rangle \right|_{\partial D} \leqslant 0$, where $n(x)$ is an exterior normal vector. Is that true, that $D$ is an invariant set: $x(t,x_0)$ lies in $D$ for every $t$ if $x_0 \in D$?

Smoothness is necessary, without smoothness it is not true.

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"Smoothness is sufficient" = "Smoothness is necessary"? –  Ilya May 28 '12 at 9:20
@Ilya yes, necessary –  Nimza May 28 '12 at 11:53
You can use tangent cones, so it is not necessary to assert that D has a smooth boundary. Check out the Nagumo theorem. –  Pantelis Sopasakis Dec 4 '12 at 0:49
@PantelisSopasakis Thank you for answer. Maybe the Nagumo theorem can be found in some book? Google gives only some articles. –  Nimza Dec 4 '12 at 7:17
@Nimza Here's your book: Franco Blanchini and Stefano Miani, "Set-Theoretic Methods in Control", Birkhauser Boston, ISBN-13: 978-0-8176-3255-7, chapter 4. –  Pantelis Sopasakis Dec 4 '12 at 12:14