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

K

{\displaystyle {\mathcal {K}}}

functions belong to this family:

Definition: a continuous function

α

:

[

0

,

a

)

→

[

0

,

∞

)

{\displaystyle \alpha :[0,a)\rightarrow [0,\infty )}

is said to belong to class

K

{\displaystyle {\mathcal {K}}}

if:

it is strictly increasing;

it is s.t.

α

(

0

)

=

0

{\displaystyle \alpha (0)=0}

.

Definition: a continuous function

α

:

[

0

,

a

)

→

[

0

,

∞

)

{\displaystyle \alpha :[0,a)\rightarrow [0,\infty )}

is said to belong to class

K

∞

{\displaystyle {\mathcal {K}}_{\infty }}

if:

it belongs to class

K

{\displaystyle {\mathcal {K}}}

;

it is s.t.

a

=

∞

{\displaystyle a=\infty }

;

it is s.t.

lim

r

→

∞

α

(

r

)

=

∞

{\displaystyle \lim _{r\rightarrow \infty }\alpha (r)=\infty }

.

A nondecreasing positive definite function

β

{\displaystyle \beta }

satisfying all conditions of class

K

{\displaystyle {\mathcal {K}}}

(

K

∞

{\displaystyle {\mathcal {K}}_{\infty }}

) other than being strictly increasi

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