Radially unbounded function: Difference between revisions

Revision as of 12:15, 21 October 2012

Revision as of 12:25, 21 October 2012

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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. that to be radially unbounded the condition must be verified along any path that results in:

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

For example the functionfunctions

:<math>\ ff_1(x)= (x_1-x_2)^2 \, </math>

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

is not radially unbounded since along the line <math> x_1 = x_2 </math>, the condition is not verified

isare 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==