Radially unbounded function

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

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

Such functions are applied in control theory. 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. that to be radially unbounded the condition must be verified along any path that results in:

$\|x\|\to \infty \,$

For example the function

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

is not radially unbounded since along the line $x_{1}=x_{2}$, the condition is not verified

References

• Terrell, William J. (2009), Stability and stabilization, Princeton University Press, ISBN 978-0-691-13444-4, MR2482799