In control theory, it is often required to check if a nonautonomous system is stable or not. To cope with this it is necessary to use some special comparison functions. Class functions belong to this family:


Definition: a continuous function is said to belong to class if:

  • it is strictly increasing;
  • it is s.t. .

In fact, this is nothing but the definition of the norm except for the triangular inequality.


Definition: a continuous function is said to belong to class if:

  • it belongs to class ;
  • it is s.t. ;
  • it is s.t. .

A nondecreasing positive definite function satisfying all conditions of class () other than being strictly increasing can be upper and lower bounded by class () functions as follows:

Thus, to proceed with the appropriate analysis, it suffices to bound the function of interest with continuous nonincreasing positive definite functions. In other words, when a function belongs to the () it means that the function is radially unbounded.

See also

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