In applied mathematics, comparison functions are several classes of continuous functions, which are used in stability theory to characterize the stability properties of control systems as Lyapunov stability, uniform asymptotic stability etc.
be a space of continuous functions acting from
The most important classes of comparison functions are: Functions of class
are also called positive-definite functions.
One of the most important properties of comparison functions is given by Sontag’s
{\displaystyle {\mathcal {KL}}}
-Lemma,[1] named after Eduardo Sontag.
Many further useful properties of comparison functions can be found in.
[2][3] Comparison functions are primarily used to obtain quantitative restatements of stability properties as Lyapunov stability, uniform asymptotic stability, etc.
These restatements are often more useful than the qualitative definitions of stability properties given in
As an example, consider an ordinary differential equation
is locally Lipschitz.
Then: The comparison-functions formalism is widely used in input-to-state stability theory.