A compendium of comparison function results

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 input-to-state 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.

In his 1959 manuscript, Wolfgang Hahn [11] termed such functions as class-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-L and class-KL, but also made significant use of these comparison functions and their properties.
Following Hahn's text, functions of class-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 input-to-state 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-KL function first proposed in [12,Equation 26.2]. To study systems under the influence of inputs, Sontag then augmented the class-KL formulation of global asymptotic stability with a class-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 well-known 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-K functions, termed monotone aggregation functions, that are defined on R n ≥0 rather than R ≥0 . For the purpose of illustrating some of the proof techniques, proofs of selected results are included in the Appendix.

Definitions and obvious facts
In what follows, we denote integers by Z and real numbers by R. Restrictions to subsets of Z or R will be denoted by subscripts such as R ≥0 ⊂ R for the nonnegative real half-line or by the standard notation [a, b) ⊂ R for the half-open interval. We denote n-dimensional Euclidean space by R n and use | · | to denote the norm. We will use both parenthesis and the symbol • to denote function composition; i.e., for functions α 1 , α 2 : R ≥0 → R ≥0 , we will use either α 1 (α 2 (s)) or α 1 • α 2 (s) for all s ∈ R ≥0 , where the choice is always made to improve readability.
As previously mentioned, Hahn 3 first defined functions of class-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 α : R ≥0 → R ≥0 is said to be of class-K (α ∈ K) if it is continuous, zero at zero, and strictly increasing. For some a ∈ R >0 , a function α : [0, a) → R ≥0 is said to be of class-K [0,a) , if it is continuous, zero at zero, and strictly increasing.
As cited in the introduction, the first use of class-K functions was to characterize the concept of a locally positive definite function. To see that this is indeed the case, we follow the same argument as [12]. A positive definite function is understood as a function that is zero at the origin, and (strictly) positive at every other point. Given, then, a continuous locally positive definite function ρ : R n → R ≥0 on a closed ball of radius c (denoted by B c ⊂ R n ), we can define a continuous nondecreasing function α : R ≥0 → R ≥0 byα Sinceα(s) is strictly positive for (0, c], it is possible to lower bound this nondecreasing function by one that is strictly increasing to obtain the desired result. Note that globally positive definite continuous functions that satisfy lim s→∞ ρ(s) = 0 cannot be bounded from below by a class-K function. A comparison function lower bound for globally positive definite continuous functions is presented in Lemma 18.
Hahn [11] introduced the term radially unbounded and in [12] characterized a radially unbounded function as one that is lower bounded by a class-K function that approaches infinity as its argument approaches infinity. Hahn cites [4] for introducing the same notion under the terminology "α becomes infinitely large". Sontag [35] introduced the now standard notation K ∞ for such functions. 4 One of the most useful properties of class-K and class-K ∞ functions is their invertibility. First consider α ∈ K\K ∞ ; i.e., α ∈ K but α / ∈ K ∞ . Since α ∈ K\K ∞ is continuous and strictly increasing there exists a constant a ∈ R >0 such that lim s→∞ α(s) = a, the inverse function exists on [0, a), and the inverse function is also continuous and strictly increasing; i.e., the function α −1 : [0, a) → R ≥0 is continuous, zero at zero, strictly increasing, and hence of class-K [0,a) . The inverse function also satisfies lim s→a α −1 (s) = ∞. In the case where α ∈ K ∞ , we observe that the inverse function will be defined globally and consequently α −1 ∈ K ∞ . We thus observe that class-K ∞ functions are the set of homeomorphisms on the nonnegative real half-line, R ≥0 .
Hahn [12] observed that if α 1 , . This is straightforward to see by considering r = α 1 (s) in the first inequality. This property also holds for two class-K functions, though only on [0, a) where a = lim s→∞ min{α 1 (s), α 2 (s)}.
Three further useful and straightforward properties are related to the sum, maximum, or minimum of class-K functions. Suppose we have a finite number of func- for all s ∈ R ≥0 is of class-K. Furthermore, for the summation or maximum, if α i ∈ K ∞ for any i ∈ {1, . . . , N }, then α ∈ K ∞ . Note that maximization and summation are specific instances of the monotone aggregation functions discussed in Sect. 8. Hahn introduced class-L functions in [12, Defn 2.6] as a precursor to defining attractivity of an equilibrium point.
Note that, similar to class-K functions, functions of class-L are invertible on their range and the inverse is itself (nearly) class-L on the range of the original function; i.e., for σ ∈ L, the inverse σ −1 : (0, σ (0)] → R ≥0 is continuous, strictly decreasing, and lim s→0 σ −1 (s) = +∞.
In [12,Section 24], Hahn noted two somewhat obvious facts about the composition of comparison functions. Suppose α, α 1 , α 2 ∈ K and σ ∈ L. Then In addition, we observe that for σ 1 , σ 2 ∈ L and with c .
As a final definition, Hahn [12,Defn 24.2] introduced the class of KL functions: if it is class-K in its first argument and class-L in its second argument. In other words, β ∈ KL if for each fixed t ∈ R ≥0 , β(·, t) ∈ K and for each fixed s ∈ R ≥0 , β(s, ·) ∈ L.
For consistency, we will generally use α or ϕ for functions of class-K or class-K ∞ , σ for functions of class-L, β for functions of class-KL, ρ for positive definite functions, and a, b, c, or λ for positive constants in R >0 . For functions α ∈ K that are differentiable, we will denote the derivative by α . We will denote the identity function by id; i.e., id(s) = s for all s ∈ R ≥0 .

Example: Lyapunov functions and KL-stability
As a brief example of the elegance enabled by the use of comparison functions, we demonstrate the proof that a Lyapunov function implies uniform global asymptotic stability of the origin for an autonomous systeṁ Here, we assume the existence and uniqueness of solutions and denote solutions to the above differential equation from an initial point x 0 = x(0) ∈ R n by x(t) for all t ∈ R ≥0 .

Definition 5
The origin is globally asymptotically stable forẋ = f (x) if there exists a function β ∈ KL such that, for all x 0 ∈ R n , 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 A Lyapunov function
is a continuously differentiable function such that there exist α 1 , α 2 ∈ K ∞ , a continuous positive definite function ρ : R n → R ≥0 and, for all x ∈ R n , Assume the existence of a Lyapunov function forẋ = f (x). Lemma 18 in the sequel yields the existence of α ∈ K ∞ and σ ∈ L such that Since the functions α 1 , α 2 ∈ K ∞ , they are both invertible and we definê Finally, we defineρ : and note that the product of a class-K ∞ function and a class-L function is a positive definite function and henceρ is a positive definite function. Therefore, using (1), (3), and the definitions of functionsα,σ , andρ, we may bound the decrease condition (2) aṡ Using a comparison lemma from the sequel (Lemma 20) we have the existence of a functionβ ∈ KL such that, for all By manipulating the upper and lower bounds in (1) we obtain for all t ∈ R ≥0 and x 0 ∈ R n , proving global asymptotic stability of the origin. Furthermore, we note that any nonlinear scaling of a Lyapunov function by a continuously differentiable functionᾱ ∈ K ∞ satisfyingᾱ (s) > 0 for all s ∈ R >0 yields a Lyapunov function 6 . To see this, given a Lyapunov function V , functions α 1 , α 2 ∈ K ∞ and a continuous positive definite function ρ : R n → R ≥0 so that (1) and (2) hold, define We defineα 1 . =ᾱ • α 1 ∈ K ∞ andα 2 . =ᾱ • α 2 ∈ K ∞ and see that (1) implies that W satisfies the necessary upper and lower boundŝ α 1 (|x|) ≤ W (x) ≤α 2 (|x|), ∀x ∈ R n . 6 Despite the fact that functions α ∈ K are strictly increasing, it is possible that α (s) = 0 for isolated points s ∈ R >0 . For example, α(s) . = s 0 (1 + sin(τ ))dτ is one such function.
Sinceᾱ ∈ K ∞ is continuously differentiable andᾱ (s) > 0 for all s ∈ R >0 , we have for all x ∈ R n \{0}, where the right-hand side is a continuous positive definite function. In other words, the function W (x) =ᾱ(V (x)) is a Lyapunov function for any continuously differentiableᾱ ∈ K ∞ . Finally, let the continuous positive definite functionρ : R ≥0 → R ≥0 be defined as in (4) so that we have the decrease condition (5). Then forρ, Lemma 24 in the sequel yields anᾱ ∈ K ∞ that is continuously differentiable on R >0 such that Repeating the calculations of (5) and (6) we see that, for W (x) .

A word on regularity
Since class-K and class-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-K or class-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 α ∈ K ∞ to be continuously differentiable. In fact, we can always bound comparison functions from above and below by smooth functions on R >0 . We here state [5, Lemma 2.5] as two lemmas to separately state the results for nondecreasing and nonincreasing functions.

Lemma 1
Let α : R ≥0 → R ≥0 be continuous, zero at zero, strictly positive on R >0 , and nondecreasing. Then there exist functions α 1 , α 2 ∈ K, smooth on R >0 , so that Furthermore, if lim s→∞ α(s) = ∞ then both functions may be chosen to satisfy α 1 , α 2 ∈ K ∞ . Lemma 2 Let σ : R ≥0 → R >0 be continuous, nonincreasing, and satisfying lim s→∞ σ (s) = 0. Then, there exist functions σ 1 , σ 2 ∈ L, smooth on R >0 , so that In contrast to the above results, when we assume that the function we wish to approximate is of class-K or class-L, we may additionally control how close the smooth function is to the given function.

Lemma 3
For α ∈ K and any ε > 0 there exist α 1 , α 2 ∈ K, smooth on R >0 , so that, Lemma 4 For σ ∈ L and any ε > 0 there exist With a further requirement that the given function be of class-K ∞ , we obtain the following two smooth approximation lemmas.
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.

Upper bounds
Arguably the most useful lemma regarding KL-functions is widely known as Sontag's Lemma on 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 K ∞ functions. The proof is provided in Appendix A.2.
A precursor to the above result was presented by Hahn [12]: Lemma 8 Let α 1 , α 2 ∈ K, σ ∈ L, and either α 1 or α 2 be bounded. Then, there exist α ∈ K andσ ∈ L such that Proof If there exists a c ∈ R >0 so that α 1 (s) ≤ c for all s ∈ R >0 , then the mean value theorem yields the desired result withα The following result describes a class of functions that can be bounded from above by a class-KL function. This result was proved, though not formally stated, in [22].
As in analysis more generally, the existence of a triangle inequality is extremely useful. In [35,Equation (12)], Sontag observed that for any function α ∈ K and any a, b ∈ R ≥0 , A generalization of this was presented in [17] where an additional function ϕ ∈ K ∞ can be used to trade off the relative weighting on the first argument versus the second argument. The following is sometimes referred to as a weak triangle inequality. A proof is provided in Appendix A.3.

Lemma 10 [17, Equation (6)] Given α ∈ K and any function
Clearly, choosing ϕ(s) = 2s for all s ∈ R ≥0 regains the original form of the weak triangle inequality (8). More generally, we can select ϕ(s) = (1 + c)s with c ∈ R >0 , which satisfies Consequently, choosing c < 1 puts a larger weighting on the b term, whereas choosing c > 1 puts a larger weighting on the a term. We note that an alternate, more symmetric, form of (9) can be obtained by choosinĝ This form of the weak triangle inequality was presented in [33, Lemma 1.
It is possible to weakly factor functions of two variables that are jointly of class-K.
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.
Hahn [12] made use of a similar upper bound in a fairly simple case. In particular, given α ∈ K and constants c, λ ∈ R >0 such that α(s) ≤ cs λ for all s ∈ R ≥0 . Then The following lemma provides an upper bound on a given class-K function in terms of the composition of convex and concave functions of class-K. [30,Lemma 14] Given α ∈ K there exist continuously differentiable and convex α v ∈ K and continuously differentiable and concave α c ∈ K such that

Lemma 13
The key idea of the proof involves the fact that the integral of a class-K function is convex. On the one hand, the convex function required is directly defined by where s * . = min{1, a 2 } and a = lim s→∞ α(s). These latter points guarantee that the inverse α −1 is well defined. On the other hand, the concave function is defined by specifying its inverse as the convex function An upper bound for class-L functions was presented in [12] as follows: The proof of the above is provided in Appendix A.4. Finally, we introduce the Legendre-Fenchel transform to present a general version of Young's Inequality. Suppose α ∈ K ∞ is continuously differentiable and that, in addition, α ∈ K ∞ . To ease the notation, denote The Legendre-Fenchel transform appears to have first been used in conjunction with comparison functions in [29]. Several interesting properties of the Legendre-Fenchel transform were provided in [20].
Using the Legendre-Fenchel transform, the following general version of Young's Inequality was presented in [29].
with equality if and only if The original version of Young's inequality [13,Theorem 156] bounds the product of two nonnegative numbers as follows: for anyα ∈ K ∞ and a, b with equality if and only if b =α(a). The relationship of (13) to Lemma 16 is straightforward to see by consideringα = α . A useful result for bounding from above the product of two nonnegative numbers based on any given K ∞ function was presented in [30].

Lemma 17
For any α ∈ K ∞ and any p ∈ R >0 , The proof is straightforward by considering the two cases of α −1 ( pa) ≥ b and α −1 ( pa) < b. Alternatively, this can be shown via Young's inequality (13) by considering α .

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-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-K. A complete characterization of positive definite functions on R ≥0 in terms of a lower bound depending on both a class-K function and a class-L function was given by in [3, Lemma IV.1]. The version presented here extends the characterization to positive definite functions defined on R n . A proof is provided in Appendix A.5.

Lemma 18
Let ρ : R n → R ≥0 be a continuous positive definite function. Then, there exist functions α ∈ K ∞ and σ ∈ L such that The following lower bound on functions of class-KL was introduced in [37] to separate the effect of the two arguments when considering the decrease condition for an input-to-output stability Lyapunov function. 7 The main idea of the proof is discussed in Sect. 7.4.

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 right-hand side of the scalar differential inequality is a continuous negative definite function, then solutions are bounded by a function of class-KL.
The above, with λ = 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 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 right-hand side of the differential inequality to also depend on an external input. This is particularly useful when considering robust stability in the context of input-to-state stability and its variants. The following two lemmas allow the input to enter via a maximization and a summation, respectively.
Footnote 7 continued second argument. However, this definition admits functions such that β(s, t) = 0 for some s, t ∈ R >0 finite, while the right-hand side of (14) is strictly positive. The second is that it is necessary that α 2 ∈ K ∞ whereas in [37,Lemma A.2] it is only stated that α 2 ∈ K. However, if α 2 ∈ K is not of class-K ∞ , then the right-hand side of (14) may remain bounded away from zero as t → ∞ while lim t→∞ β(s, t) = 0. continuous function and y : [0,t) → R is a (locally) absolutely continuous function with y(0) = y 0 ∈ R ≥0 , and iḟ holds for almost all t ∈ [0,t) then The proof relies on Lemma 18 applied to the positive definite function ρ to obtain α ∈ K ∞ and σ ∈ L, so that, with the decreasing property of σ , ρ(s) ≥ α(s)σ (2s) for all s ∈ R ≥0 . Without being overly precise, two time intervals are then considered: times such that y(t) ≤ sup τ ∈[0,t] |v(τ )| and times such that y(t) exceeds this bound. The former set is invariant since y(t) is decreasing and sup τ ∈[0,t] |v(τ )| is increasing. In the latter case, we see that y(t) > v(t) so that y(t) ≤ y(t) + v(t) ≤ 2y(t). Then, since α ∈ K ∞ ,ẏ(t) ≤ α(y(t))σ (2y(t)) and we can appeal to Lemma 20 to obtain the desired result. See [3] for a detailed proof.

Lemma 22 [3, Corollary IV.3] Given any continuous positive definite function
is a measureable, locally essentially bounded function and y : [0,t) → R ≥0 is (locally) absolutely continuous function with y(0) = y 0 ∈ R ≥0 , and iḟ for almost all t ∈ [0,t), then The proof follows by defining certain auxiliary functions and showing that these functions satisfy the requirements of Lemma 21. In particular, let w(t) be the solution to the initial value probleṁ The function w 1 then plays the role of y in Lemma 21. As before, see [3] for a detailed proof.

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-K ∞ functions. A proof of the following is in Appendix A.7.
Furthermore, if α ∈ K ∞ then α 1 and α 2 can be chosen of class-K ∞ as well.
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].  It is possible to derive a corollary to Lemma 25 to allow for positive constants less than one. The proof is provided in Appendix A.9.
Given a function of class-K, it is possible to find another function of class-K that upper bounds the given function away from the origin and is linear near the origin.
The following lemma describes a condition under which the order of composition of two class-K functions can be reversed whilst maintaining a contraction principle.
As a corollary to the above, we obtain the following result when both functions are of class-K ∞ . Corollary 2 Let α 1 , α 2 ∈ K ∞ . Then A simple example shows that, for more than two functions, we cannot change the order of composition and maintain a contraction condition. For all s ∈ R ≥0 , let Then, for all s ∈ R >0 , [23], the following characterization of growth rates of some class-K ∞ functions was provided.

Other function classes
Functions of class-K, K ∞ , L, and 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.

Functions of class-KLD
In [9], Grüne defined a subset of class-KL functions, which he called class-KLD.
Definition 7 A function μ : R ≥0 × R → R ≥0 is said to be of class-KLD if its restriction to R ≥0 × R ≥0 is class-KL and, in addition, for all r ∈ R ≥0 , s, t ∈ R.
The D in KLD above denotes dynamical and refers to the fact that, as a consequence of (20), a function of class-KLD defines a dynamical system on R ≥0 . Class-KLD functions were introduced in [9,10] to characterize the notion of input-to-state dynamical stability (ISDS). ISDS is a robust stability concept equivalent to input-tostate stability and, as such, it is useful to know that any function of class-KL can be bounded from above by a function of class-KLD as follows.

Extended real-valued functions
As was earlier noted, functions of class-K are invertible only on their range. Hence, when using functions of class-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-G functions below), Ito [14] considered extended real-valued functions (see also [15]); that is, For continuity at infinity of a function α : R ≥0 → R ≥0 we intend that the usual limit condition s n → s implies α(s n ) → α(s) including for sequences s n → ∞ as well as for any s ∈ R >0 such that α(s) = ∞. Definition 8 A function α : R ≥0 → R ≥0 is said to be of class-K if it is continuous, zero at zero, strictly increasing on the range R ≥0 and infinite otherwise.
By strictly increasing on the range R ≥0 we mean that on the set {s ∈ R ≥0 : α(s) = ∞} the function α is strictly increasing. An example of a function α ∈ K is Another example of a function ϕ ∈ K is Ito defined the pseudo-inverse of a function α ∈ K as α (s) sup{s ∈ R ≥0 : α(s) < ∞}, otherwise.
We note that, with these definitions, if a function α ∈ K then α ∈ K. Furthermore, we may state the following result.

Definition 9
A function σ : R ≥0 → R ≥0 is said to be of class-L if it is continuous, strictly decreasing, and σ (∞) = 0.
We define the pseudo-inverse of a function σ ∈ L as Similar to the fact that the inverse of a function σ ∈ L is nearly of class-L (see Section 2), the pseudo-inverse of a function σ ∈ L is nearly of class-L; i.e., σ † : R ≥0 → R ≥0 is continuous, strictly decreasing for s ∈ [0, σ (0)), and σ † (∞) = 0. We observe that, for all s ∈ R ≥0 , Constraining σ ∈ L so that σ (0) = ∞ yields that σ † ∈ L is a true inverse of σ ∈ L. For example, the function is of class-L and is its own pseudo-inverse.

Functions of class-G
The function class-G was defined in [40] by enlarging the class-K to include functions that are not strictly increasing. 8

Definition 10
A function α : R ≥0 → R ≥0 is said to be of class-G if it is continuous, nondecreasing, and α(0) = 0.
It is clear that any class-G function can be bounded from above by a function of class-K. However, it is not the case that every class-G function can be bounded from below by a function of class-K. For example, a function which is identically zero for all s ∈ [0, 1] and equal to s − 1 for all s > 1 is of class-G, but cannot be bounded from below by a function of class-K.
Since class-G functions are not strictly increasing, they may fail to be invertible. This shortcoming can typically be circumvented by allowing set-valued inverses or by the introduction of class-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 real-valued functions defined in the previous section to construct approximate inverses. Define class-G as the enlargement of class-K to include functions that are not strictly increasing. [15,Proposition 1] Suppose γ ∈ G is positive definite and α ∈ K. Let c ∈ R ≥0 be given by c = sup{s ∈ R ≥0 : α(s) < ∞} and assume

Lemma 31
Then, for all s ∈ R ≥0 , The proof of the above lemma is provided in Appendix A.11.
As an example, suppose we wish to derive an approximate inverse for the positive definite function γ ∈ G defined by A well-defined inverse exists on [0, 1) but we encounter an obvious difficulty at the value of one. One possible approximation of an inverse for γ is given by α ∈ K defined as for some a > 1. We see that this extended real-valued function is the expected inverse on [0, 1] and goes to infinity arbitrarily close to one. It is straightforward to see that, for all s ∈ R ≥0 , both α • γ (s) ≤ s and γ • α(s) ≤ s, indicating that Lemma 31 captures a property of reasonable approximations of inverses of class-G functions.

Functions of class-N
The nomenclature of class-N has been used to denote two different function classes 9 .
We here adopt the definition in [3].

Definition 11
A function ν : R → R is said to be of class-N if it is continuous, nondecreasing, and unbounded below (that is, inf ν = −∞).
The following lemma is the key element in the proofs of Lemma 11 and Lemma 19.

Lemma 32 [3, Proposition IV.4]
Suppose φ : R 2 → R is class-N in each of its arguments; i.e., φ(·, y) ∈ N for each fixed y ∈ R and φ(x, ·) ∈ N for each fixed x ∈ R. Then there exists κ ∈ N such that To prove Lemma 11, given α : R 2 ≥0 → R ≥0 of class-K in each of its arguments independently, the function where log denotes the natural logarithm, can be shown to be of class-N . Applying Lemma 32, Lemma 1, and taking an exponential on both sides of the resulting inequality gives the desired result withα ∈ K bounding ϕ(s) . = e k(log(s)) from above.
The core idea of the proof of Lemma 19 similarly requires defining an appropriate function φ. Define |x| + . = max{0, x} and for a functionβ ∈ KL that is a lower bound for the given β ∈ KL with certain desirable properties. Again we apply Lemma 32, Lemma 1, and take exponentials on both sides of the resulting inequality. Some straightforward manipulations then yield the desired result.

Monotone aggregation functions
To compare two vectors x, y ∈ R n ≥0 , we use x > > y if and only if x i > y i for all i ∈ {1, . . . , n}. In other words, each element of x ∈ R n ≥0 must be greater than the corresponding element in y ∈ R n ≥0 . A generalization of class-K functions allowing a domain of R n ≥0 , rather than a domain of R ≥0 , was introduced in [33] (see also [6]). These so-called monotone aggregation functions are defined as follows: Definition 12 A continuous function μ : R n ≥0 → R ≥0 is a monotone aggregation function (MAF n ) if it is positive definite and strictly monotone; i.e., for any x, y ∈ R n ≥0 , if x > > y then μ(x) > μ(y). The class of unbounded monotone aggregation functions on R n ≥0 is denoted by MAF n,∞ .
Together with gain matrices describing the interconnection structure of large-scale systems, monotone aggregation functions were used in [33] and subsequent works to provide results on stability of large-scale systems, particularly through the use of small-gain theorems.
Examples of monotone aggregation functions include maximization and all pnorms on R n . As such, they provide a general and elegant formulation encapsulating and expanding the standard input-to-state stability estimates for systems with inputs. In particular, forẋ = f (x, w), we can subsume both the estimate There are several useful and straightforward relationships between functions of class-K and monotone aggregation functions: -The function classes K and MAF 1 are equivalent.
is in MAF n . -Denote the vector of all ones in R n by 1 n . If μ ∈ MAF n then μ(s1 n ), s ∈ R ≥0 , is a function of class-K.
Despite the equivalence of class-K and MAF 1 , monotone aggregation functions are in fact different to functions α : R n ≥0 → R ≥0 , n ∈ Z >1 , that are of class-K in each argument, or what might be termed jointly class-K. For example, in the case of a jointly class-K function of two arguments, α ∈ K 2 , we see that α(s, 0) = 0 for all s ∈ R ≥0 . On the other hand, a function μ ∈ MAF 2 is positive definite; i.e., μ(s, 0) > 0 for all s ∈ R >0 .
That μ 1 ∈ MAF n 1 and μ 2 ∈ MAF n 2 follow from the fact that the sum of n functions of class-K is a function of class-MAF n .
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 K ∞ ∪ {0} those functions that are either class-K ∞ or identically zero.
for all s ∈ [α −1 (R i−1 ), α −1 (R i )). Thatᾱ 2 is continuous at the origin follows from the fact that i → −∞ when s → 0 and both lim i→−∞ R i = 0 and It is straightforward to calculate A standard regularization technique applied toᾱ 2 then yields a function α 2 ∈ K that is smooth on R >0 and satisfies Finally, the smooth function α 1 ∈ K can be obtained by a similar procedure where a strictly increasing continuous function similar to (34) is defined bȳ ,

Remark 1
The proof of Lemma 4 follows a similar approach with the only significant difference being the definition of the partition. This is done by first defining (compare with (32)) and then (compare with (33)). The proof then follows that of Lemma 3, mutatis mutandis, by defining similar continuous, though strictly decreasing, functions.

A.3 Proof of Lemma 10 (Weak triangle inequality)
We here prove the weak triangle inequality in the form of Eq. (10): given α ∈ K, for We consider two cases: ϕ(a) ≥ b and ϕ(a) < b. In the first case, making use of the invertibility of K ∞ functions, we have that a ≥ ϕ −1 (b) so that On the other hand, if ϕ(a) < b, then a < ϕ −1 (b) and we see that Combining (43) and (44) then yields the result.

A.4 Proof of Lemma 14 (Class-L upper bound)
Using the fact that a class-K function of a class-L is again a class-L function, we see that if σ ∈ L then √ σ ∈ L. Furthermore, since σ ∈ L is strictly decreasing, we know that σ (t + τ ) ≤ σ (t) for all t, τ ∈ R ≥0 . Let σ i ∈ L satisfy σ i (t) ≥ √ σ (t) for i = 1, 2 and all t ∈ R ≥0 . Then, for all r, s ∈ R ≥0 we may write

A.5 Proof of Lemma 18 (Positive definite functions)
Without loss of generality we assume ρ(x) → 0 as |x| → ∞. (If this is not the case, since we are interested in deriving a lower bound for ρ(x), we can instead consider any positive definite functionρ : R n → R ≥0 satisfyingρ(x) ≤ ρ(x) for all x ∈ R n andρ(x) → 0 as |x| → ∞.) With this assumption, and the facts that ρ(0) = 0 and ρ(x) > 0 for all x ∈ R n \{0}, we know that the function has a global maximum, x * ∈ R n and that ρ(x * ) > 0.
We note that (46) and the condition that y(t) ≥ 0 for all t ∈ [0, T ] imply that if there exists at ∈ [0, T ] such that y(t) = 0, then y(t) ≡ 0 for all t ∈ [t, T ]. In the event that such a time exists, let t * .

A.10 Proof of Lemma 27
The proof in one direction follows that in [38]. The converse uses a similar argument but illustrates the care that must be taken when using class-K functions rather than class-K ∞ functions.
Case 3: Finally, suppose a 1 < c and s * ∈ [a 1 , c]. Then α 1 • α 2 (s * ) < a 1 ≤ s * which immediately contradicts (57) and thus proves the lemma. We note that the same proof can be used to demonstrate Corollary 2 without needing to account for the values for which α −1 2 is well defined.
Next, consider s * ≥ c. In this case, α(s * ) = ∞ so that which contradicts the condition that γ (∞) < c. This then proves the first implication.