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

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