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