Radially unbounded function: Difference between revisions

Revision as of 08:29, 2 August 2015

Revision as of 17:05, 22 September 2015

Line 1:

{{nomore footnotes|date=October 2010}}

{{Multiple issues|

{{Underlinked|date=February 2015}}

{{no footnotes|date=October 2010}}

}}

In mathematics, a '''radially unbounded function''' is a function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}</math> for which <ref name="Terrell2009">{{Citation | last1=Terrell | first1=William J. | title=Stability and stabilization | publisher=[[Princeton University Press]] | isbn=978-0-691-13444-4 |mr=2482799 | year=2009}}</ref>

Line 8 ⟶ 5:

:<math>\|x\| \to \infty \Rightarrow f(x) \to \infty. \, </math>

Such functions are applied in [[control theory]] and required in [[Mathematical_optimizationMathematical optimization|optimization]] for determination of [[compact_space|compact spacesspace]]s.

Notice that the norm used in the definition can be any norm defined on <math> \mathbb{R}^n </math>, 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:

Line 14 ⟶ 11:

:<math>\|x\| \to \infty \, </math>

For example, the functions

:<math>\ f_1(x)= (x_1-x_2)^2 \, </math>