In
applied mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
, comparison functions are several classes of
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s, which are used in
stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differ ...
to characterize the stability properties of control systems as
Lyapunov stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. ...
, uniform asymptotic stability etc.
Let
be a space of continuous functions acting from
to
. 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
-Lemma, named after
Eduardo Sontag. It says that for each
and any
there exist
:
Many further useful properties of comparison functions can be found in.
[C. M. Kellett. A compendium of comparison function results. ''Mathematics of Control, Signals, and Systems'', 26(3):339–374, 2014.]
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
language.
As an example, consider an ordinary differential equation
where
is
locally Lipschitz
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...
. Then:
* () is
globally stable if and only if there is a
so that for any initial condition
and for any
it holds that
* () is
globally asymptotically stable if and only if there is a
so that for any initial condition
and for any
it holds that
The comparison-functions formalism is widely used in
input-to-state stability Input-to-state stability (ISS)Eduardo D. Sontag. Mathematical Control Theory: Finite-Dimensional Systems. Springer-Verlag, London, 1998Hassan K. Khalil. Nonlinear Systems. Prentice Hall, 2002. is a stability notion widely used to study stability o ...
theory.
References
{{Reflist
Types of functions
Stability theory