Radially unbounded function: Difference between revisions

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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>

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

Or equivalently,

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

<math display="block">\forall c > 0:\exists r > 0 : \forall x \in \mathbb{R}^n: [\Vert x \Vert > r \Rightarrow f(x) > c]</math>

Such functions are applied in [[control theory]] and required in [[Mathematical optimization|optimization]] for determination of [[compact space]]s.

Such functions are applied in [[control theory]].

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:

:<math display="block">\|x\| \to \infty \, </math>

For example, the functions

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

<math display="block">\begin{align}

f_1(x) &= (x_1-x_2)^2 \\

For example the functions

:<math>\ f_1 f_2(x) &= (x_1^2+x_2^2)/(1+x_1^2+x_2^2)+(x_1-x_2)^2 \, </math>

\end{align} </math>

:<math>\ f_2(x)= (x_1^2+x_2^2)/(1+x_1^2+x_2^2)+(x_1-x_2)^2 \, </math>

are not radially unbounded since along the line <math> x_1 = x_2 </math>, the condition is not verified even though the second function is globally positive definite.

==References==

{{Reflist}}

*{{Citation | last1=Terrell | first1=William J. | title=Stability and stabilization | publisher=[[Princeton University Press]] | isbn=978-0-691-13444-4 | id={{MathSciNet | id = 2482799}} | year=2009}}

[[Category:Real analysis]]