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Undid revision 1152058228 by Lucidum Hydra (talk)

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In [[applied mathematics]], '''comparison functions''' are several classes of [[continuous function]]s, which are used in [[stability theory]] to characterize the stability properties of control systems as [[Lyapunov stability]], uniform asymptotic stability etc.

Let <math>C(X,Y)</math> be a space of continuous functions acting from <math>X</math> to <math>Y</math>. The most important classes of comparison functions are:

: <math>
\begin{align}
\mathcal{P} &:= \left\{\gamma \in C({\mathbb{R}}_+,{\mathbb{R}}_+): \gamma(0)=0 \text{ and } \gamma(r)>0 \text{ for } r>0 \right\} \\[4pt]
\mathcal{K} &:= \left\{\gamma \in {\mathcal{P}}: \gamma \text{ is strictly increasing} \right\}\\[4pt]
\mathcal{K}_\infty &:=\left\{\gamma\in{\mathcal{K}}: \gamma\text{ is unbounded}\right\}\\[4pt]
\mathcal{L} &:=\{\gamma\in C({\mathbb{R}}_+,{\mathbb{R}}_+): \gamma\text{ is strictly decreasing with } \lim_{t\rightarrow\infty}\gamma(t)=0 \}\\[4pt]
\mathcal{KL} &:= \left\{\beta \in C({\mathbb{R}}_+\times{\mathbb{R}}_+,{\mathbb{R}}_+): \beta \text{ is continuous, } \beta(\cdot,t)\in{{\mathcal{K}}},\ \forall t \geq 0, \ \beta(r,\cdot)\in {{\mathcal{L}}},\ \forall r > 0\right\}
\end{align}
</math>

Functions of class <math>{\mathcal{P}}</math> are also called [[Positive-definite function|''positive-definite functions'']].

One of the most important properties of comparison functions is given by Sontag’s <math>{\mathcal{KL}}</math>-Lemma,<ref>E. D. Sontag. Comments on integral variants of ISS. ''Systems & Control Letters'', 34(1-2):93–100, 1998.</ref> named after [[Eduardo Sontag]]. It says that for each <math>\beta \in {\mathcal{KL}}</math> and any <math>\lambda>0</math> there exist <math>\alpha_1,\alpha_2 \in {\mathcal{K_\infty}}</math>:

{{NumBlk|:|
<math>
\alpha_1(\beta(s,t)) \leq \alpha_2(s) e^{-\lambda t}, \quad t,s \in \R_+.
</math>
|{{EquationRef|1}}}}

Many further useful properties of comparison functions can be found in.<ref>W. Hahn. ''Stability of motion''. Springer-Verlag, New York, 1967.</ref><ref>C. M. Kellett. A compendium of comparison function results. ''Mathematics of Control, Signals, and Systems'', 26(3):339–374, 2014.</ref>

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 <math>\varepsilon\text{-}\delta</math> language.

As an example, consider an ordinary differential equation
{{NumBlk|:|
<math>
\dot{x} = f(x),
</math>
|{{EquationRef|2}}}}
where <math>f:{\mathbb{R}}^n\to{\mathbb{R}}^n</math> is [[Lipschitz continuity|locally Lipschitz]]. Then:

* ({{EquationNote|2}}) is [[Lyapunov stability|globally stable]] if and only if there is a <math>\sigma\in{\mathcal{K_\infty}}</math> so that for any initial condition <math>x_0 \in{\mathbb{R}}^n</math> and for any <math>t\geq 0</math> it holds that
{{NumBlk|:|
: <math>
|x(t)| \leq \sigma(|x_0|).
</math>
|{{EquationRef|3}}}}
* ({{EquationNote|2}}) is [[Lyapunov stability|globally asymptotically stable]] if and only if there is a <math>\beta\in{\mathcal{KL}}</math> so that for any initial condition <math>x_0 \in{\mathbb{R}}^n</math> and for any <math>t\geq 0</math> it holds that
{{NumBlk|:|
: <math>
|x(t)| \leq \beta(|x_0|,t).
</math>
|{{EquationRef|4}}}}

The comparison-functions formalism is widely used in [[input-to-state stability]] theory.

== References ==
{{Reflist}}

[[Category:Types of functions]]
[[Category:Stability theory]]

Comparison function - Revision history

imported>Dkv1: Undid revision 1152058228 by Lucidum Hydra (talk)