A compendium of comparison function results
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Abstract
The use of comparison functions has become standard in systems and control theory, particularly for the purposes of studying stability properties. The use of these functions typically allows elegant and succinct statements of stability properties such as asymptotic stability and inputtostate stability and its several variants. Furthermore, over the last 20 years several inequalities involving these comparison functions have been developed that simplify their manipulation in the service of proving more significant results. Many of these inequalities have appeared in the body of proofs or in appendices of various papers. Our goal herein is to collect these inequalities in one place.
Keywords
Comparison functions Stability theory Nonlinear systems1 History
Jose Massera appears to have been the first scholar to introduce comparison functions to the study of stability theory in 1956 [26]. In particular, to elegantly capture the notion of (local) positive definiteness he relied on a function \(a: \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) with
“\(a(r)\) being continuous and increasing when \(r > 0\), \(a(0) = 0\).”
To further describe a function having an infinitely small upper bound, he then used a function \(b: \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) “having the same properties of \(a(r)\)”.^{1}
In his 1959 manuscript, Wolfgang Hahn [11] termed such functions as class\(\mathcal {K}\) functions^{2} and also introduced the terminology “decrescent” for what Lyapunov had originally termed “an infinitely small upper limit” [24]. However, in what remains a remarkably modern text, it was in his 1967 manuscript [12] where Hahn not only introduced the additional function classes of class\(\mathcal {L}\) and class\(\mathcal {KL}\), but also made significant use of these comparison functions and their properties.
Following Hahn’s text, functions of class\(\mathcal {K}\) appeared occasionally in works through the 1970s and 1980s. However, the use of these functions was largely limited to the characterization of positive definite and decrescent functions and the properties that Hahn elucidated in [12] were essentially ignored [21, 32, 34, 41, 42].
Sontag’s seminal 1989 paper [35] introducing the notion of inputtostate stability represented a return to the elegant formulations first presented in [12]. In particular, Sontag took as the definition of global asymptotic stability the formulation involving a class\(\mathcal {KL}\) function first proposed in [12, Equation 26.2]. To study systems under the influence of inputs, Sontag then augmented the class\(\mathcal {KL}\) formulation of global asymptotic stability with a class\(\mathcal {K}\) function of the input.
Since the appearance of [35], the use of comparison functions in the analysis of stability and robustness for nonlinear systems has become standard. Many useful inequalities and other relationships have been developed for comparison functions. Unfortunately, these results tend to be reported in appendices or within the context of proving some larger results relating to stability or robustness. Our goal here is a modest one: to collect some of the most useful comparison function inequalities and relationships in one place.
The manuscript is organized as follows: in Sect. 2 we provide the standard comparison function definitions and review some wellknown facts. In Sect. 3, we provide lemmas that give upper bounds on a given comparison function. By contrast, in Sect. 4, we provide lemmas that give lower bounds on a given function. Section 5 provides lemmas on differential inequalities involving comparison functions. In an overlap of terminology that should cause no confusion, these lemmas are frequently referred to as comparison lemmas. In Sect. 6, we provide some other useful relationships involving comparison functions. In Sect. 7, we briefly mention a few recently defined function classes that we believe are likely to be more widely used in the future. Finally, in Sect. 8 we discuss a generalization of class\(\mathcal {K}\) functions, termed monotone aggregation functions, that are defined on \(\mathbb {R}^n_{\ge 0}\) rather than \(\mathbb {R}_{\ge 0}\). For the purpose of illustrating some of the proof techniques, proofs of selected results are included in the Appendix.
2 Definitions and obvious facts
In what follows, we denote integers by \(\mathbb {Z}\) and real numbers by \(\mathbb {R}\). Restrictions to subsets of \(\mathbb {Z}\) or \(\mathbb {R}\) will be denoted by subscripts such as \(\mathbb {R}_{\ge 0} \subset \mathbb {R}\) for the nonnegative real halfline or by the standard notation \([a,b) \subset \mathbb {R}\) for the halfopen interval. We denote \(n\)dimensional Euclidean space by \(\mathbb {R}^n\) and use \(\cdot \) to denote the norm. We will use both parenthesis and the symbol \(\circ \) to denote function composition; i.e., for functions \(\alpha _1,\alpha _2 : \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\), we will use either \(\alpha _1(\alpha _2(s))\) or \(\alpha _1 \circ \alpha _2(s)\) for all \(s \in \mathbb {R}_{\ge 0}\), where the choice is always made to improve readability.
As previously mentioned, Hahn^{3} first defined functions of class\(\mathcal {K}\) in [11], though such functions had been previously used in stability analysis by Massera in [26]. In [12, Defn 2.5] this definition is immediately used to define the notion of a stable equilibrium point rather than as a precursor to using such functions to characterize positive definiteness (as done in [11, 26, 32, 41]).
Definition 1
A function \(\alpha : \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is said to be of class\(\mathcal {K}\) (\(\alpha \in \mathcal {K}\)) if it is continuous, zero at zero, and strictly increasing. For some \(a \in \mathbb {R}_{>0}\), a function \(\alpha : [0,a) \rightarrow \mathbb {R}_{\ge 0}\) is said to be of class\(\mathcal {K}_{[0,a)},\) if it is continuous, zero at zero, and strictly increasing.
Hahn [11] introduced the term radially unbounded and in [12] characterized a radially unbounded function as one that is lower bounded by a class\(\mathcal {K}\) function that approaches infinity as its argument approaches infinity. Hahn cites [4] for introducing the same notion under the terminology “\(\alpha \) becomes infinitely large”. Sontag [35] introduced the now standard notation \(\mathcal {K}_{\infty }\) for such functions.^{4}
Definition 2
A function \(\alpha : \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is said to be of class\(\mathcal {K}_{\infty }\) (\(\alpha \in \mathcal {K}_{\infty }\)) if \(\alpha \in \mathcal {K}\) and, in addition, \(\displaystyle \lim \nolimits _{s \rightarrow \infty } \alpha (s) = \infty \).
One of the most useful properties of class\(\mathcal {K}\) and class\(\mathcal {K}_{\infty }\) functions is their invertibility. First consider \(\alpha \in \mathcal {K}\backslash \mathcal {K}_{\infty }\); i.e., \(\alpha \in \mathcal {K}\) but \(\alpha \notin \mathcal {K}_{\infty }\). Since \(\alpha \in \mathcal {K}\backslash \mathcal {K}_{\infty }\) is continuous and strictly increasing there exists a constant \(a \in \mathbb {R}_{>0}\) such that \(\lim _{s \rightarrow \infty } \alpha (s) = a\), the inverse function exists on \([0,a)\), and the inverse function is also continuous and strictly increasing; i.e., the function \(\alpha ^{1}: [0,a) \rightarrow \mathbb {R}_{\ge 0}\) is continuous, zero at zero, strictly increasing, and hence of class\(\mathcal {K}_{[0,a)}\). The inverse function also satisfies \(\lim _{s \rightarrow a} \alpha ^{1}(s) = \infty \). In the case where \(\alpha \in \mathcal {K}_{\infty }\), we observe that the inverse function will be defined globally and consequently \(\alpha ^{1} \in \mathcal {K}_{\infty }\). We thus observe that class\(\mathcal {K}_{\infty }\) functions are the set of homeomorphisms on the nonnegative real halfline, \(\mathbb {R}_{\ge 0}\).
Hahn [12] observed that if \(\alpha _1, \alpha _2 \in \mathcal {K}_{\infty }\) and if \(\alpha _1(s) \le \alpha _2(s)\) for all \(s \in \mathbb {R}_{\ge 0}\), then \(\alpha _1^{1}(s) \ge \alpha _2^{1}(s)\). This is straightforward to see by considering \(r = \alpha _1(s)\) in the first inequality. This property also holds for two class\(\mathcal {K}\) functions, though only on \([0,a)\) where \(a = \lim _{s \rightarrow \infty } \min \{\alpha _1(s), \alpha _2(s)\}\).
Hahn introduced class\(\mathcal {L}\) functions in [12, Defn 2.6] as a precursor to defining attractivity of an equilibrium point.
Definition 3
A function \(\sigma :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{> 0}\) is said to be of class\(\mathcal {L}\) (\(\sigma \in \mathcal {L}\)), if it is continuous, strictly decreasing,^{5} and \(\displaystyle \lim \nolimits _{s \rightarrow \infty } \sigma (s) = 0\).
Note that, similar to class\(\mathcal {K}\) functions, functions of class\(\mathcal {L}\) are invertible on their range and the inverse is itself (nearly) class\(\mathcal {L}\) on the range of the original function; i.e., for \(\sigma \in \mathcal {L}\), the inverse \(\sigma ^{1}:(0,\sigma (0)] \rightarrow \mathbb {R}_{\ge 0}\) is continuous, strictly decreasing, and \(\lim _{s \rightarrow 0} \sigma ^{1}(s) = +\infty \).
Definition 4
A function \(\beta : \mathbb {R}_{\ge 0} \times \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is said to be of class\(\mathcal {KL}\) (\(\beta \in \mathcal {KL}\)) if it is class\(\mathcal {K}\) in its first argument and class\(\mathcal {L}\) in its second argument. In other words, \(\beta \in \mathcal {KL}\) if for each fixed \(t \in \mathbb {R}_{\ge 0}\), \(\beta (\cdot ,t) \in \mathcal {K}\) and for each fixed \(s \in \mathbb {R}_{\ge 0}\), \(\beta (s,\cdot ) \in \mathcal {L}\).
For consistency, we will generally use \(\alpha \) or \(\varphi \) for functions of class\(\mathcal {K}\) or class\(\mathcal {K}_{\infty }\), \(\sigma \) for functions of class\(\mathcal {L}\), \(\beta \) for functions of class\(\mathcal {KL}\), \(\rho \) for positive definite functions, and \(a,b,c\), or \(\lambda \) for positive constants in \(\mathbb {R}_{>0}\). For functions \(\alpha \in \mathcal {K}\) that are differentiable, we will denote the derivative by \(\alpha '\). We will denote the identity function by \(\mathrm{id}\); i.e., \(\mathrm{id}(s) = s\) for all \(s \in \mathbb {R}_{\ge 0}\).
2.1 Example: Lyapunov functions and \(\mathcal {KL}\)stability
Definition 5
It is obvious that the above definition is equivalent to the standard definition of global asymptotic stability as the combination of stability and attractivity (see [22, Proposition 2.5]).
Definition 6
2.2 A word on regularity
Since class\(\mathcal {K}\) and class\(\mathcal {L}\) functions are monotonic, a theorem due to Lebesgue states that these functions must be differentiable almost everywhere (see, for example, [31, Section 2]) and this property holds without the functions necessarily being continuous.
While we have assumed that functions of class\(\mathcal {K}\) or class\(\mathcal {L}\) are continuous functions, it is sometimes useful to require functions with more regularity; for example, when discussing the nonlinear scaling of Lyapunov functions in the previous section we required the nonlinear scaling \(\alpha \in \mathcal {K}_{\infty }\) to be continuously differentiable. In fact, we can always bound comparison functions from above and below by smooth functions on \(\mathbb {R}_{>0}\). We here state [5, Lemma 2.5] as two lemmas to separately state the results for nondecreasing and nonincreasing functions.
Lemma 1
Lemma 2
In contrast to the above results, when we assume that the function we wish to approximate is of class\(\mathcal {K}\) or class\(\mathcal {L}\), we may additionally control how close the smooth function is to the given function.
Lemma 3
Lemma 4
With a further requirement that the given function be of class\(\mathcal {K}_{\infty }\), we obtain the following two smooth approximation lemmas.
Lemma 5
Lemma 6
The proofs of all of the lemmas in this section follow a similar idea where a continuous piecewise linear function is constructed to bound the given function from above or below while giving the additional desired inequalities. Standard techniques can then be used to smooth these functions where they are pieced together. As an example, and because we have not seen the result previously in the literature, we present the proof of Lemma 3 in Appendix A.1 and provide a remark on the necessary modifications to obtain Lemma 4.
3 Upper bounds
Arguably the most useful lemma regarding \(\mathcal {KL}\)functions is widely known as Sontag’s Lemma on \(\mathcal {KL}\)Estimates which originally appeared as [36, Proposition 7]. The version given here is from [19] and provides some nice properties on one of the resulting \(\mathcal {K}_{\infty }\) functions. The proof is provided in Appendix A.2.
Lemma 7
A precursor to the above result was presented by Hahn [12]:
Lemma 8
Proof
If there exists a \(c \in \mathbb {R}_{>0}\) so that \(\alpha _1'(s) \le c\) for all \(s \in \mathbb {R}_{>0}\), then the mean value theorem yields the desired result with \(\hat{\alpha }\doteq c \alpha _2\) and \(\hat{\sigma }\doteq \sigma \).
On the other hand, if there exists \(c \in \mathbb {R}_{>0}\) so that \(\alpha _2(s) \le c\) for all \(s \in \mathbb {R}_{>0}\), then the result holds with \(\hat{\alpha }(s) \doteq \sqrt{\alpha _1(\sigma (0)\alpha _2(s))}\) for all \(s \in \mathbb {R}_{\ge 0}\) and \(\hat{\sigma }(t) \doteq \sqrt{\alpha _1(c\sigma (t))}\) for all \(t \in \mathbb {R}_{\ge 0}\). \(\square \)
The following result describes a class of functions that can be bounded from above by a class\(\mathcal {KL}\) function. This result was proved, though not formally stated, in [22].
Lemma 9

for all \(r,\varepsilon \in \mathbb {R}_{>0}\), there exists some \(T = T(r,\varepsilon ) \in \mathbb {R}_{>0}\) such that \(\phi (s,t) < \varepsilon \) for all \(s \le r\) and \(t \ge T\); and

for all \(\varepsilon \in \mathbb {R}_{>0}\) there exists \(\delta \in \mathbb {R}_{>0}\) such that \(\phi (s,t) \le \varepsilon \) for all \(s \le \delta \) and all \(t \in \mathbb {R}_{\ge 0}\).
Lemma 10
It is possible to weakly factor functions of two variables that are jointly of class\(\mathcal {K}\).
Lemma 11
We present a sketch of the proof in Sect. 7.4.
A precursor to the above appeared as [36, Corollary 10] where the relationship between the arguments is a product.
Lemma 12
Lemma 13
Lemma 14
The proof of the above is provided in Appendix A.4.
Lemma 15
Using the Legendre–Fenchel transform, the following general version of Young’s Inequality was presented in [29].
Lemma 16
A useful result for bounding from above the product of two nonnegative numbers based on any given \(\mathcal {K}_{\infty }\) function was presented in [30].
Lemma 17
The proof is straightforward by considering the two cases of \(\alpha ^{1}(pa) \ge b\) and \(\alpha ^{1}(pa) < b\). Alternatively, this can be shown via Young’s inequality (13) by considering \(\alpha \doteq \frac{1}{p}\hat{\alpha }\in \mathcal {K}_{\infty }\).
4 Lower bounds
As comparison functions are most frequently used in gain and stability estimates, they tend to be used to upper bound various quantities. As a consequence, the literature contains many more results for bounding functions from above than for bounding functions from below. Nonetheless, lower bounds have occasionally proved useful.
As previously noted, the initial use of class\(\mathcal {K}\) functions was for the characterization of locally positive definite functions. However, it is not possible to bound every globally positive definite function from below by a function of class\(\mathcal {K}\). A complete characterization of positive definite functions on \(\mathbb {R}_{\ge 0}\) in terms of a lower bound depending on both a class\(\mathcal {K}\) function and a class\(\mathcal {L}\) function was given by in [3, Lemma IV.1]. The version presented here extends the characterization to positive definite functions defined on \(\mathbb {R}^n\). A proof is provided in Appendix A.5.
Lemma 18
The following lower bound on functions of class\(\mathcal {KL}\) was introduced in [37] to separate the effect of the two arguments when considering the decrease condition for an inputtooutput stability Lyapunov function.^{7} The main idea of the proof is discussed in Sect. 7.4.
Lemma 19
5 The comparison principle
The comparison principle or comparison lemma makes use of a (usually solvable) scalar differential inequality to make statements about the nature of solutions to a scalar differential equation. An example application of this principle was provided in Sect. 2.1. In the case where the righthand side of the scalar differential inequality is a continuous negative definite function, then solutions are bounded by a function of class\(\mathcal {KL}\). The following result was first demonstrated in a slightly more restrictive form by Hahn [12, Section 24E].
Lemma 20
The above, with \(\lambda =1\), first appeared in [22, Lemma 4.4]. A precursor to this where \(y(t)\) satisfying a differential equation, rather than the inequality above, implies a \(\mathcal {KL}\) bound appeared in [35, Lemma 6.1]. A proof of Lemma 20 is provided in Appendix A.6.
An extension to the above comparison principle allows the righthand side of the differential inequality to also depend on an external input. This is particularly useful when considering robust stability in the context of inputtostate stability and its variants. The following two lemmas allow the input to enter via a maximization and a summation, respectively.
Lemma 21
The proof relies on Lemma 18 applied to the positive definite function \(\rho \) to obtain \(\alpha \in \mathcal {K}_{\infty }\) and \(\sigma \in \mathcal {L}\), so that, with the decreasing property of \(\sigma \), \(\rho (s) \ge \alpha (s)\sigma (2s)\) for all \(s \in \mathbb {R}_{\ge 0}\). Without being overly precise, two time intervals are then considered: times such that \(y(t) \le \sup _{\tau \in [0,t]} v(\tau )\) and times such that \(y(t)\) exceeds this bound. The former set is invariant since \(y(t)\) is decreasing and \(\sup _{\tau \in [0,t]} v(\tau )\) is increasing. In the latter case, we see that \(y(t) > v(t)\) so that \(y(t) \le y(t)+v(t) \le 2y(t)\). Then, since \(\alpha \in \mathcal {K}_{\infty }\), \(\dot{y}(t) \le \alpha (y(t))\sigma (2y(t))\) and we can appeal to Lemma 20 to obtain the desired result. See [3] for a detailed proof.
Lemma 22
6 Other relationships
In addition to the previously presented lemmas providing upper and lower bounds and comparison principles, several results involving comparison functions are available that do not neatly fit into these categories. In this section, we summarize several such results.
Many factorizations of comparison functions are possible, with the derivations frequently relying on the invertibility of class\(\mathcal {K}_{\infty }\) functions. A proof of the following is in Appendix A.7.
Lemma 23
The next two lemmas presented were used to modify decrease conditions of Lyapunov functions in continuous time (Lemma 24) and discrete time (Lemma 25). The following is a combination of [30, Lemmas 11 and 12].
Lemma 24
Lemma 25
Corollary 1
Given a function of class\(\mathcal {K}\), it is possible to find another function of class\(\mathcal {K}\) that upper bounds the given function away from the origin and is linear near the origin.
Lemma 26
The following lemma describes a condition under which the order of composition of two class\(\mathcal {K}\) functions can be reversed whilst maintaining a contraction principle.
Lemma 27
We note that [38, Fact A.2] only proved one direction of the implication. The proof of Lemma 27 is provided in Appendix A.10.
As a corollary to the above, we obtain the following result when both functions are of class\(\mathcal {K}_{\infty }\).
Corollary 2
Finally, in [23], the following characterization of growth rates of some class\(\mathcal {K}_{\infty }\) functions was provided.
Lemma 28
 A function \(\alpha \in \mathcal {K}_{\infty }\) satisfies, for every \(\varepsilon \in \mathbb {R}_{>0}\)$$\begin{aligned} \lim _{s \rightarrow \infty } \left\{ \alpha \left( (1+\varepsilon ) s \right)  \alpha (s) \right\} = \infty ; \end{aligned}$$(19)
 For every \(\varepsilon \in \mathbb {R}_{>0}\) there exists \(\varphi _\varepsilon \in \mathcal {K}_{\infty }\) such that$$\begin{aligned} \alpha (st) \le \alpha \left( (1+\varepsilon )s\right)  \varphi _\varepsilon (t), \quad \forall s\ge t \ge 0. \end{aligned}$$
It was noted in [23] that \(\mathcal {K}_{\infty }\) functions of the form \(\alpha (s) = s^\lambda \) for \(\lambda \in \mathbb {R}_{>0}\) satisfy the growth condition of (19). By contrast, \(\mathcal {K}_{\infty }\) functions of the form \(\alpha (s) = \lambda \log (1+s)\) for \(\lambda \in \mathbb {R}_{>0}\) do not satisfy (19).
7 Other function classes
Functions of class\(\mathcal {K}\), \(\mathcal {K}_{\infty }\), \(\mathcal {L}\), and \(\mathcal {KL}\) have been used in stability theory for over 40 years going back to the original work of Massera and Hahn. Other useful classes have been introduced more recently.
7.1 Functions of class\(\mathcal {KLD}\)
In [9], Grüne defined a subset of class\(\mathcal {KL}\) functions, which he called class\(\mathcal {KL}\mathcal {D}\).
Definition 7
The \(\mathcal {D}\) in \(\mathcal {KL}\mathcal {D}\) above denotes dynamical and refers to the fact that, as a consequence of (20), a function of class\(\mathcal {KL}\mathcal {D}\) defines a dynamical system on \(\mathbb {R}_{\ge 0}\). Class\(\mathcal {KL}\mathcal {D}\) functions were introduced in [9, 10] to characterize the notion of inputtostate dynamical stability (ISDS). ISDS is a robust stability concept equivalent to inputtostate stability and, as such, it is useful to know that any function of class\(\mathcal {KL}\) can be bounded from above by a function of class\(\mathcal {KL}\mathcal {D}\) as follows.
Lemma 29
7.2 Extended realvalued functions
As was earlier noted, functions of class\(\mathcal {K}\) are invertible only on their range. Hence, when using functions of class\(\mathcal {K}\), if their inverses are required, a certain level of notational overhead is necessary. To reduce this notational burden and to provide approximate inverses for functions that are not strictly increasing (see class\(\mathcal {G}\) functions below), Ito [14] considered extended realvalued functions (see also [15]); that is, functions on \(\overline{\mathbb {R}}_{\ge 0} \doteq \mathbb {R}_{\ge 0} \cup \{\infty \}\).
For continuity at infinity of a function \(\alpha : \overline{\mathbb {R}}_{\ge 0} \rightarrow \overline{\mathbb {R}}_{\ge 0}\) we intend that the usual limit condition \(s_n \rightarrow s\) implies \(\alpha (s_n) \rightarrow \alpha (s)\) including for sequences \(s_n \rightarrow \infty \) as well as for any \(s \in \mathbb {R}_{>0}\) such that \(\alpha (s) = \infty \).
Definition 8
A function \(\alpha : \overline{\mathbb {R}}_{\ge 0} \rightarrow \overline{\mathbb {R}}_{\ge 0}\) is said to be of class\(\overline{\mathcal {K}}\) if it is continuous, zero at zero, strictly increasing on the range \(\mathbb {R}_{\ge 0}\) and infinite otherwise.
Lemma 30
Proof
We observe that \(\alpha , \varphi \in \overline{\mathcal {K}}\) from (21) and (22) are pseudoinverses of each other; i.e., \(\alpha ^\ominus = \varphi \) and \(\varphi ^\ominus = \alpha \). Furthermore, \(\alpha \in \overline{\mathcal {K}}\) satisfies (24) while \(\varphi \in \overline{\mathcal {K}}\) satisfies (26).
Definition 9
A function \(\sigma : \overline{\mathbb {R}}_{\ge 0} \rightarrow \overline{\mathbb {R}}_{\ge 0}\) is said to be of class\(\overline{\mathcal {L}}\) if it is continuous, strictly decreasing, and \(\sigma (\infty ) = 0\).
For \(\sigma : \overline{\mathbb {R}}_{\ge 0} \rightarrow \overline{\mathbb {R}}_{\ge 0}\), it is understood that strictly decreasing allows \(\sigma (0) = \infty \) but requires \(\sigma (s) < \infty \) for all \(s \in \overline{\mathbb {R}}_{>0}\).
7.3 Functions of class\(\mathcal {G}\)
The function class\(\mathcal {G}\) was defined in [40] by enlarging the class\(\mathcal {K}\) to include functions that are not strictly increasing.^{8}
Definition 10
A function \(\alpha : \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is said to be of class\(\mathcal {G}\) if it is continuous, nondecreasing, and \(\alpha (0) = 0\).
It is clear that any class\(\mathcal {G}\) function can be bounded from above by a function of class\(\mathcal {K}\). However, it is not the case that every class\(\mathcal {G}\) function can be bounded from below by a function of class\(\mathcal {K}\). For example, a function which is identically zero for all \(s \in [0,1]\) and equal to \(s1\) for all \(s >1\) is of class\(\mathcal {G}\), but cannot be bounded from below by a function of class\(\mathcal {K}\).
Since class\(\mathcal {G}\) functions are not strictly increasing, they may fail to be invertible. This shortcoming can typically be circumvented by allowing setvalued inverses or by the introduction of class\(\mathcal {K}\) upper and lower bounds, where especial care is required to properly deal with any lower bound.
An alternative approach is to make use of the extended realvalued functions defined in the previous section to construct approximate inverses. Define class\(\overline{\mathcal {G}}\) as the enlargement of class\(\overline{\mathcal {K}}\) to include functions that are not strictly increasing.
Lemma 31
The proof of the above lemma is provided in Appendix A.11.
7.4 Functions of class\(\mathcal {N}\)
The nomenclature of class\(\mathcal {N}\) has been used to denote two different function classes^{9}. We here adopt the definition in [3].
Definition 11
A function \(\nu : \mathbb {R}\rightarrow \mathbb {R}\) is said to be of class\(\mathcal {N}\) if it is continuous, nondecreasing, and unbounded below (that is, \(\inf \nu = \infty \)).
The following lemma is the key element in the proofs of Lemma 11 and Lemma 19.
Lemma 32
8 Monotone aggregation functions
To compare two vectors \(x,y \in \mathbb {R}^n_{\ge 0}\), we use \(x > \!\! > y\) if and only if \(x_i > y_i\) for all \(i \in \{1, \ldots , n\}\). In other words, each element of \(x \in \mathbb {R}^n_{\ge 0}\) must be greater than the corresponding element in \(y \in \mathbb {R}^n_{\ge 0}\).
A generalization of class\(\mathcal {K}\) functions allowing a domain of \(\mathbb {R}^n_{\ge 0}\), rather than a domain of \(\mathbb {R}_{\ge 0}\), was introduced in [33] (see also [6]). These socalled monotone aggregation functions are defined as follows:
Definition 12
A continuous function \(\mu : \mathbb {R}^n_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is a monotone aggregation function (MAF \(_n\)) if it is positive definite and strictly monotone; i.e., for any \(x, y \in \mathbb {R}^n_{\ge 0}\), if \(x > \!\! > y\) then \(\mu (x) > \mu (y)\). The class of unbounded monotone aggregation functions on \(\mathbb {R}^n_{\ge 0}\) is denoted by MAF\(_{n,\infty }\).
Together with gain matrices describing the interconnection structure of largescale systems, monotone aggregation functions were used in [33] and subsequent works to provide results on stability of largescale systems, particularly through the use of smallgain theorems.

The function classes \(\mathcal {K}\) and \(\mathrm{MAF}_1\) are equivalent.

If \(\alpha \in \mathcal {K}\) and \(\mu \in \mathrm{MAF}_n\), then \(\alpha \circ \mu \in \mathrm{MAF}_n\).
 If \(\alpha _i \in \mathcal {K}\) for \(i=1, \ldots , n\), and \(\mu \in \mathrm{MAF}_n\), then the functionis in \(\mathrm{MAF}_n\).$$\begin{aligned} \mu _\alpha (s_1, \ldots , s_n) \doteq \mu (\alpha _1(s_1),\ldots , \alpha _n(s_n)) \end{aligned}$$

Denote the vector of all ones in \(\mathbb {R}^n\) by \(\mathbf {1}_n\). If \(\mu \in \mathrm{MAF}_n\) then \(\mu (s \mathbf {1}_n)\), \(s \in \mathbb {R}_{\ge 0}\), is a function of class\(\mathcal {K}\).
Lemma 33
Proof
The above result leads to the following result from [8] allowing a general monotone aggregation function to be bounded from above by a summation. We denote by \(\mathcal {K}_{\infty }\cup \{ 0 \}\) those functions that are either class\(\mathcal {K}_{\infty }\) or identically zero.
Lemma 34
Proof
We apply Lemma 33 twice to the function \(\mu \in \mathrm{MAF}_{n+1}\) to obtain \(\hat{\mu }_1 \in \mathrm{MAF}_1\), \(\hat{\mu }_2 \in \mathrm{MAF}_1\), and \(\hat{\mu }\in \mathrm{MAF}_{n1}\). With the equivalence of class\(\mathcal {K}\) and \(\mathrm{MAF}_1\), we can define \(\hat{\beta } \doteq \hat{\mu }_1 \circ \beta \in \mathcal {KL}\) and \(\hat{\alpha }_n \doteq \hat{\mu }_2 \circ \alpha _n \in \mathcal {K}\). \(\square \)
Footnotes
 1.
 2.
 3.
It has been speculated that Hahn’s usage of the letter \(\mathcal {K}\) was in reference to Kamke, though Hahn himself never apparently stated this. Functions similar to class\(\mathcal {K}\) functions do appear in [28] and are, in fact, denoted by \(K(\cdot )\). In a similar vein, [27] makes use of a bounding function that is continuous and monotonically decreasing to zero on \((0,\infty )\).
 4.
Observe that class\(\mathcal {K}_{[0,\infty )}\) and class\(\mathcal {K}_{\infty }\) are not the same. In fact, class\(\mathcal {K}_{[0,\infty )}\) is the same as class\(\mathcal {K}\) since the subscript interval refers to the domain of definition of a function. By contrast, the subscript of \(\mathcal {K}_{\infty }\) refers to the behavior of the function in the limit as its argument goes to infinity.
 5.
Note that some authors only assume that functions of class\(\mathcal {L}\) are nonincreasing. In many cases, this is sufficient. However, we prefer the symmetry with the strictly increasing property of class\(\mathcal {K}\) functions and the clean results on class\(\mathcal {L}\) functions that the assumption of strictly decreasing enables. Note that, since functions of class\(\mathcal {L}\) are strictly decreasing and go to zero in the limit, these functions are also strictly positive. This is explicit in the definition in that the range of class\(\mathcal {L}\) functions is \(\mathbb {R}_{>0}\).
 6.
Despite the fact that functions \(\alpha \in \mathcal {K}\) are strictly increasing, it is possible that \(\alpha '(s) = 0\) for isolated points \(s \in \mathbb {R}_{>0}\). For example, \(\alpha (s) \doteq \int \nolimits _0^s (1 + \sin (\tau ))d\tau \) is one such function.
 7.
We note that the statement of Lemma 19 in [37] was not entirely precise in two respects. The first is that the authors explicitly defined class\(\mathcal {KL}\) functions as nonincreasing, rather than strictly decreasing, in their second argument. However, this definition admits functions such that \(\beta (s,t) = 0\) for some \(s,t \in \mathbb {R}_{>0}\) finite, while the righthand side of (14) is strictly positive. The second is that it is necessary that \(\alpha _2 \in \mathcal {K}_{\infty }\) whereas in [37, Lemma A.2] it is only stated that \(\alpha _2 \in \mathcal {K}\). However, if \(\alpha _2 \in \mathcal {K}\) is not of class\(\mathcal {K}_{\infty }\), then the righthand side of (14) may remain bounded away from zero as \(t \rightarrow \infty \) while \(\lim _{t \rightarrow \infty } \beta (s,t) = 0\).
 8.
 9.
In [2], the function class\(\mathcal {N}\) denotes those functions which map \(\mathbb {R}_{\ge 0}\) to \(\mathbb {R}_{\ge 0}\) and are continuous and increasing. In this case, class\(\mathcal {K}\) is the subset of class\(\mathcal {N}\) functions that are also zero at zero.
Notes
Acknowledgments
The author would like to thank Lars Grüne, Björn Rüffer, Andy Teel, and Fabian Wirth for comments on early drafts of this manuscript. The work of the anonymous reviewers is also gratefully acknowledged.
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