Radially unbounded function

In mathematics, a radially unbounded function is a function $f:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}$ for which ^[1]

\|x\|\to \infty \Rightarrow f(x)\to \infty .\,

Such functions are applied in control theory and required in optimization for determination of compact spaces.

Notice that the norm used in the definition can be any norm defined on $\mathbb {R} ^{n}$ , and that the behavior of the function along the axes does not necessarily reveal that it is radially unbounded or not; i.e. to be radially unbounded the condition must be verified along any path that results in:

\|x\|\to \infty \,

For example, the functions

\ f_{1}(x)=(x_{1}-x_{2})^{2}\,

\ f_{2}(x)=(x_{1}^{2}+x_{2}^{2})/(1+x_{1}^{2}+x_{2}^{2})+(x_{1}-x_{2})^{2}\,

are not radially unbounded since along the line $x_{1}=x_{2}$ , the condition is not verified even though the second function is globally positive definite.

References

↑ Terrell, William J. (2009), Stability and stabilization, Princeton University Press, ISBN 978-0-691-13444-4, MR 2482799

This article is issued from Wikipedia - version of the 9/22/2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.