Radially unbounded function

In mathematics, a radially unbounded function is a function for which [1]

Or equivalently,

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 , 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:

For example, the functions are not radially unbounded since along the line , the condition is not verified even though the second function is globally positive definite.

References

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  1. ^ Terrell, William J. (2009), Stability and stabilization, Princeton University Press, ISBN 978-0-691-13444-4, MR 2482799