K类函数和KL类函数
Class K \mathcal{K} K function- K \mathcal{K} K类函数
Definition: A continuous function α : [ 0 , a ) → [ 0 , ∞ ) \alpha:[0,a)\rightarrow[0,\infin) α:[0,a)→[0,∞) is said belong to class K \mathcal{K} K if it strictly increasing and α ( 0 ) = 0 \alpha(0)=0 α(0)=0. It is said to belong to class K ∞ \mathcal{K_{\infin}} K∞ if a = ∞ a=\infin a=∞ and α ( r ) → ∞ \alpha(r)\rightarrow\infin α(r)→∞ as r → ∞ r\rightarrow \infin r→∞.
译：如果连续函数 α : [ 0 , a ) → [ 0 , ∞ ) \alpha:[0,a)\rightarrow[0,\infin) α:[0,a)→[0,∞) 严格递增且满足 α ( 0 ) = 0 \alpha(0)=0 α(0)=0，则 α \alpha α 属于 K \mathcal{K} K 类函数。进一步，如果 a = ∞ a=\infin a=∞ 且当 r → ∞ r\rightarrow \infin r→∞ 时有 α ( r ) → ∞ \alpha(r)\rightarrow\infin α(r)→∞ 成立，则 α \alpha α 属于 K ∞ \mathcal{K_\infin} K∞ 类函数。
Class K L \mathcal{KL} KL function- K L \mathcal{KL} KL类函数
Definition: A continuous function β : [ 0 , a ) × [ 0 , ∞ ) → [ 0 , ∞ ) \beta:[0,a)\times[0,\infin)\rightarrow[0,\infin) β:[0,a)×[0,∞)→[0,∞) is said belong to class K L \mathcal{KL} KL if, for each fixed s s s, the mapping β ( r , s ) \beta(r,s) β(r,s) belongs to class K \mathcal{K} K with respected to r r r and, for each fixed r r r, the mapping β ( r , s ) \beta(r,s) β(r,s) is decreasing with respected to s s s and β ( r , s ) → 0 \beta(r,s)\rightarrow 0 β(r,s)→0 as s → ∞ s\rightarrow \infin s→∞.
译：对于连续函数 β : [ 0 , a ) × [ 0 , ∞ ) → [ 0 , ∞ ) \beta:[0,a)\times[0,\infin)\rightarrow[0,\infin) β:[0,a)×[0,∞)→[0,∞)，如果对于每个固定的 s s s，映射 β ( r , s ) \beta(r,s) β(r,s) 都是关于 r r r 的 K \mathcal{K} K 类函数, 而对于每个固定的 r r r，映射 β ( r , s ) \beta(r,s) β(r,s) 是关于 s s s 的递减函数，且当 s → ∞ s\rightarrow \infin s→∞ 时 β ( r , s ) → 0 \beta(r,s)\rightarrow 0 β(r,s)→0，则 β \beta β属于 K L \mathcal{KL} KL 类函数。
参考文献
【1】Hassan K. Khalil, Nonlinear Systems, 3 edition, page 144.