Incremental global asymptotic stability equals
incremental global exponential stability - but at
equilibria.

Rui Kato, Daniele Astolfi, Vincent Andrieu, Laurent Praly

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Rui Kato, Daniele Astolfi, Vincent Andrieu, Laurent Praly. Incremental global asymptotic stability equals incremental global exponential stability - but at equilibria.. 13th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2025), IFAC, Jul 2025, Reykjavik, Iceland. hal-05152528

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# Incremental global asymptotic stability equals incremental global exponential stability—but at equilibria*

Rui Kato* Daniele Astolfi* Vincent Andrieu*
Laurent Praly**

* Université Claude Bernard Lyon 1, CNRS, LAGEPP, UMR 5007,
Villeurbanne 69100, France (e-mail: rui.kato@univ-lyon1.fr;
daniele.astolfi@univ-lyon1.fr; vincent.andrieu@univ-lyon1.fr).

** Mines Paris, PSL, CAS, Paris 75006, France (e-mail:
laurent.praly@minesparis.psl.eu)

**Abstract:** Incremental stability has become increasingly common in the context of nonlinear control. However, there is a lack of understanding of the difference between several notions of incremental stability. The objective of this paper is to show that the notion of incremental global asymptotic stability is very similar to that of incremental global exponential stability. In particular, nonexponential relaxation of the convergence rate appears only “locally” but not “globally.” Some examples are provided to justify our theoretical arguments.

**Keywords:** Incremental global asymptotic stability.

## 1. INTRODUCTION

Incremental stability describes the long time behavior of the distance between any two motions. More specifically, the family of motions is said to be incrementally asymptotically stable if all motions converge to each other with a uniform rate. This is stronger than the classical concept of Lyapunov stability, which focuses on the distance with a particular motion. Although incremental stability has long been studied in the literature, many significant tools were recently introduced in Angeli (2002, 2009). A similar notion studied in the literature is convergence in the sense of Demidovich (see Pavlov et al. (2004) for a brief history). The family of motions is said to be convergent if all motions converge to a bounded motion. The relation between these two notions was considered in Rüffer et al. (2013), where it was shown that the two notions are equivalent for any motions on a compact set.

A notable difference between incremental and nonincremental stability concepts lies in the distinction between local and global notions. It can be shown that incremental local asymptotic stability and incremental global asymptotic stability are equivalent (Angeli (2002)). In particular, we can analyze the distance between two faraway motions by summing the distances between two nearby motions. This requires a certain uniformity in the characterization of asymptotic convergence. In the case of a smooth flow, the equivalence between incremental and differential notions of stability can be established although it requires some severe assumptions (Andrieu et al. (2016, 2021); Tran et al. (2019); Wu and Duan (2022); Kawano and Besselink (2024)).

This paper investigates the difference between incremental global asymptotic stability ($\delta$GAS) and incremental global exponential stability ($\delta$GES). Generally speaking, asymptotic (resp., Lyapunov) stability and exponential (resp., Krasovskii) stability are not equivalent for nonlinear systems. In particular, global exponential stability is a very strong notion as it means that a nonlinear system can be transformed to a linear one by a diffeomorphism (see Lan and Mezić (2013)). Also, an obstacle to obtain instability criteria without distinguishing Lyapunov and Krasovskii stability notions is discussed in Leonov (2008). It is thus important to focus on the difference between asymptotic and exponential stability notions. On the other hand, it would be worth mentioning that global asymptotic stability and global exponential stability are equivalent under a nonlinear change of variables which is not a diffeomorphism (Grüne et al. (1999)). This observation suggests that asymptotic and exponential stability notions are, in some sense, closely related. In this paper, we demonstrate that, in the incremental context, asymptotic and exponential properties are even more closely aligned. In particular, we show that the notion of $\delta$GAS is equivalent to that of $\delta$GES at infinity.

From the point of view of applications, incremental stability notions and related contraction theory (Lohmiller and Slotine (1998)) have attracted great attention in recent years. Of particular interest is the problem of global output regulation (e.g., Pavlov et al. (2006)), or more specifically, global integral action (e.g., Giaccagli et al. (2022)). Other application areas include observer designs (e.g., Sanfelice and Praly (2012)) and synchronization problems (e.g., Giaccagli et al. (2024)). Although many works aim to make the closed-loop system $\delta$GES, in most cases only $\delta$GAS is essential. Moreover, the use of $\delta$GAS is justified by the fact that exponential convergence is in general not

<footnote>* This research is funded by the ANR project Alligator (ANR-22-CE48-0009-01).</footnote>
guaranteed if LaSalle's invariance principle is used (Forni and Sepulchre (2014)).

In this article, we first revise the notion of δGAS for time-invariant nonlinear systems and we obtain some new characterizations. The main contributions are then summarized as follows:

(1) First, we show that δGAS defined without assuming the existence and uniqueness of global solutions implies forward completeness of the vector field. This justifies the usual definition of δGAS on the entire semiaxis.

(2) Then, we show that δGAS implies the existence of a unique equilibrium. This is a generalization of the well-known result in the context of δGES. Our result indicates boundedness of all solutions is necessary for δGAS. Note that global asymptotic stability does not have the same property.

(3) Finally, we show that δGAS must have the exponential decay property as long as two solutions are far from each other. This is surprising because the notions of δGAS and δGES are the same at infinity. We also demonstrate the difference between these two notions by an example known in the literature.

**Notation:** $\mathbb{R}$ is the set of real numbers, $\mathbb{R}_+ = [0, \infty)$ and $\| \cdot \|$ denotes the standard Euclidean norm. A function $\beta: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ is said to be of class $KL$ if (i) $\beta$ is continuous, (ii) $\beta(s, t)$ is strictly increasing in $s$ and $\beta(0, t) = 0$ for each fixed $t$, and (iii) $\beta(s, t)$ is nonincreasing in $t$ and $\lim_{t \to \infty} \beta(s, t) = 0$ for each fixed $s$.

## 2. NOTIONS OF INCREMENTAL STABILITY

Consider an ordinary differential equation

$$ \dot{x} = f(x), \quad (1) $$

where $f: \mathbb{R}^n \to \mathbb{R}^n$ is a continuous function. For the sake of generality, we do not assume forward completeness of $f$ at this moment. Since $f$ is continuous, for each initial state, there exists at least one solution locally in time. In this paper, we assume that each solution to (1) is defined on the maximal interval of existence. For instance, the domain of definition of a solution $x$ is written as $[0, \tau_x)$, where $\tau_x$ is finite or infinite.

The notions of incremental stability are defined as follows (Angeli (2002)).

**Definition 1.** The family of all solutions to (1) is said to be incrementally globally asymptotically stable (for short, $\delta$GAS) if there exists a class-KL function $\beta$ such that any two solutions $x, y$ satisfy

$$ \|x(t) - y(t)\| \le \beta(\|x(0) - y(0)\|, t) $$

for all $t \in [0, \min\{\tau_x, \tau_y\})$. It is said to be incrementally globally exponentially stable (for short, $\delta$GES) if it is $\delta$GAS and $\beta$ can be written as $\beta(s, t) = \kappa e^{-\lambda t} s$ for some $\kappa \ge 1$ and $\lambda > 0$.

Our definition is slightly different from the one used in previous works. We require that the distance between any two solutions are bounded by a class-KL function only on the interval where those solutions exist. If all solutions are defined on $\mathbb{R}_+$, then the definition above is reduced to the usual definition. In this case, $\delta$GAS implies that every two

solutions converge to each other, but the converse is not true. For the solutions to be $\delta$GAS, it is necessary that the convergence is uniform with respect to the choice of initial conditions.

**Remark 2.** Our goal is not to show the equivalence between $\delta$GAS and $\delta$GES under a change of coordinates as in Grüne et al. (1999). For this reason, in this paper, we restrict ourselves to the notion of $\delta$GAS with respect to an Euclidean distance. For a coordinate independent notion, we refer to Zamani and Tabuada (2011).

For convenience, we give the definition of convergence in the sense of Demidovich (Pavlov et al. (2006)).

**Definition 3.** The family of all solutions to (1) is said to be convergent in the sense of Demidovich if

(1) every solution is defined on $\mathbb{R}_+$;

(2) there exists a bounded solution which is globally asymptotically stable.

A key point in the definition above is the existence of a bounded solution. If an unbounded solution is globally asymptotically stable, then all solutions are unbounded. Thus, convergence in the sense of Demidovich indicates that the motions are dissipative in some sense. In contrast with $\delta$GAS, convergence in the sense of Demidovich does not require the uniformity of convergence.

**Remark 4.** When the vector field is time-varying, then we need the additional condition that at least one solution is bounded uniformly in the initial time.

## 3. EXAMPLES: GAS VS. GES

In this section, by using simple examples, we discuss the difference between global asymptotic stability and global exponential stability.

**Example 5.** (Example 3.5 in Khalil (2002)). Consider

$$ \dot{x} = -x^3. $$

The existence and uniqueness of global solutions are guaranteed, and the flow $\phi: \mathbb{R}_+ \times \mathbb{R} \to \mathbb{R}$ is given by

$$ \phi(t, x) = \frac{x}{\sqrt{1 + 2x^2t}}. $$

It is clear that the origin is globally asymptotically stable. Yet, it is not globally exponentially stable due to the lack of local exponential stability. Is the family of all solutions $\delta$GAS? The answer is affirmative, and it can be confirmed as follows. The goal is to find a class-KL function $\beta$ such that

$$ \phi(t, x + s) - \phi(t, x) \le \beta(s, t) $$

for all $s, t \ge 0$ and all $x \in \mathbb{R}$. Note that

$$ \frac{\partial}{\partial x}[\phi(t, x + s) - \phi(t, x)] = \left( \frac{1}{\sqrt{1 + 2(x + s)^2t}} \right)^3 - \left( \frac{1}{\sqrt{1 + 2x^2t}} \right)^3. $$

This becomes zero when $x = -s/2$, and hence,

$$ \phi(t, x + s) - \phi(t, x) \le \phi(t, s/2) - \phi(t, -s/2) $$

for all $s, t \ge 0$ and all $x \in \mathbb{R}$. We set

$$ \begin{aligned} \beta(s, t) &= \phi(t, s/2) - \phi(t, -s/2) \\ &= \frac{\sqrt{2s}}{\sqrt{2 + s^2t}}. \end{aligned} $$
The function $\beta$ is of class $KL$, and thus, the family of all solutions is $\delta$GAS.

**Example 6.** Consider

$$ \dot{x} = -\arctan(x). $$

The right-hand side is globally Lipschitz, and therefore is forward complete. Clearly, the origin is globally asymptotically stable and locally exponentially stable. However, it is not globally exponentially stable due to the lack of exponential decay at infinity. Based on the result which will be given in the next section, we can deduce that the family of all solutions is not $\delta$GAS. Indeed, exponential decay at infinity is necessary for $\delta$GAS.

**Remark 7.** (incremental Lyapunov function). Assume that there exist a $C^1$-function $V$ and class-$K_\infty$ functions $\underline{\alpha}, \overline{\alpha}$ such that

$$ \underline{\alpha}(\|x - y\|) \leq V(x, y) \leq \overline{\alpha}(\|x - y\|) \quad \forall (x, y) $$

and

$$ \frac{\partial V}{\partial x}(x, y)f(x) + \frac{\partial V}{\partial y}(x, y)f(y) < 0 \quad \forall (x, y) : x \neq y. $$

Then, all solutions converge to each other. However, this does not necessarily imply $\delta$GAS. As stated in the converse theorem in Angeli (2002), it is necessary that the decreasing rate of $V$ along solutions is uniform. For example, in Example 6, the function $V(x, y) = (x - y)^2$ satisfies the above two conditions. However, it is not $\delta$GAS.

We have seen that the lack of local exponential stability or exponential decay at infinity prevents global exponential stability. In this paper, we derive some properties of $\delta$GAS in contrast with $\delta$GES. For the discussion on the difference between exponential stability and incremental exponential stability, we refer to, e.g., Jouffroy and Slotine (2004).

# 4. MAIN RESULTS

We derive a set of necessary conditions that the property of $\delta$GAS implies. Our main results are stated below. The details follow in the proceeding subsections.

**Theorem 8.** Suppose that the family of all solutions to (1) is $\delta$GAS. Then, the following assertions are true:

(i) The vector field $f$ is forward complete.

(ii) There exists a unique $x_* \in \mathbb{R}^n$ such that $f(x_*) = 0$.

(iii) For any $R > 0$, there exist $\kappa \ge 1$ and $\lambda > 0$ such that

$$ \begin{gathered} \|\phi(t, x) - \phi(t, y)\| \le \max\{\kappa e^{-\lambda t}\|x - y\|, R\} \\ \text{for all } t \ge 0 \text{ and all } x, y \in \mathbb{R}^n. \end{gathered} $$

## 4.1 Forward Completeness

We prove the item (i) of Theorem 8.

**Proposition 9.** Suppose that the family of all solutions to (1) is $\delta$GAS. Then, the vector field $f$ is forward complete.

**Proof.** Let $x: [0, \tau_x) \to \mathbb{R}^n$ and $y: [0, \tau_y) \to \mathbb{R}^n$ be two solutions. If $x(0) = y(0)$, then we have

$$ \|x(t) - y(t)\| \le \beta(\|x(0) - y(0)\|, t) = 0 $$

for all $t \in [0, \min\{\tau_x, \tau_y\})$. Hence, all solutions are unique whenever they exist. We assume that $\tau_x$ is finite and set

$$ y(0) = x(\tau_x/2). $$

By the semigroup property, we have $\tau_y = \tau_x/2$. Since $x$ is bounded on $[0, \tau_x/2)$ while $y$ is unbounded on $[0, \tau_x/2)$,

$$ \lim_{t \to \tau_x/2-0} \|x(t) - y(t)\| = \infty. $$

However, $\delta$GAS yields

$$ \lim_{t \to \tau_x/2-0} \|x(t) - y(t)\| \le \lim_{t \to \tau_x/2-0} \beta(\|x(0) - y(0)\|, t) < \infty. $$

So, we have a contradiction and $\tau_x = \infty$. This completes the proof.

We have shown that forward completeness comes directly from the property of $\delta$GAS in Definition 1. Therefore, all solutions to (1) define a smooth (semi)flow. Let $\phi: \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{R}^n$ be the flow generated by $f$, that is,

$$ \frac{\partial}{\partial t} \phi(t, x) = f(\phi(t, x)). $$

## 4.2 Existence of Unique Equilibria

We start with the following lemma.

**Lemma 10.** Suppose that the family of all solutions to (1) is $\delta$GAS. Then, all solutions are bounded.

**Proof.** See Appendix A.1.

The above lemma indicates that $\delta$GAS is stronger than convergence in the sense of Demidovich for autonomous systems. In particular, the difference between $\delta$GAS and convergence is just uniformity of convergence.

**Remark 11.** For nonautonomous systems, boundedness of solutions is not necessarily required for incremental stability (see Rüffer et al. (2013)).

The item (ii) of Theorem 8 can then be established directly from the above result.

**Proposition 12.** Suppose that the family of all solutions to (1) is $\delta$GAS. Then, there exists a unique $x_* \in \mathbb{R}^n$ such that $f(x_*) = 0$.

**Proof.** Recall that any bounded solution has a nonempty $\omega$-limit set. This set can be reduced to a singleton because of $\delta$GAS. Thus, every solution converges to an equilibrium. From $\delta$GAS, the equilibrium must be unique.

While the existing result involving $\delta$GES is based on Banach's fixed-point theorem (see Appendix A in Giaccagli et al. (2022)), our proof follows a different approach. Thus, our result has a potential to extend many existing results involving the contraction mapping principle. In particular, exponential decay is not necessary to ensure convergence to a unique equilibrium.

**Remark 13.** We remark that Theorem 4.3 Bullo (2022) shows that a weakly contracting vector field has at least one equilibrium if there exists a compact invariant set. However, to the best of the authors' knowledge, there is no result showing that $\delta$GAS implies boundedness of all solutions, by which the existence of a unique equilibrium is followed.

## 4.3 Exponential Decay at Infinity

Finally, we prove the item (iii) of Theorem 8. The following result highlights that the notions of $\delta$GAS and $\delta$GES are
in fact very similar and that the nonexponential relaxation
occurs only “locally” but not “globally.”

Proposition 14. Suppose that the family of all solutions to (1) is δGAS. Then, for any R > 0, there exist κ ≥ 1 and λ > 0 such that

$$
\|\phi(t, x) - \phi(t, y)\| \le \max \{\kappa e^{-\lambda t} \|x - y\|, R\}
$$

for all $t \ge 0$ and all $x, y \in \mathbb{R}^n$.

**Proof.** See Appendix A.2.

We remark that if the family of all solutions is δGAS, then
there exists a compact set containing the equilibrium such
that it is forward invariant and globally exponentially sta-
ble. The following corollary implies that δGAS is actually
a strong notion as local and global exponential stability
become equivalent.

Corollary 15. Suppose that the family of all solutions to (1) is δGAS. If the equilibrium is locally exponentially stable, then it is globally exponentially stable.

**Proof.** Since the equilibrium $x_*$ is locally exponentially stable, there exist $\delta > 0$, $\kappa \ge 1$, and $\lambda > 0$ such that

$$
\|\phi(t, x) - x_*\| \le \kappa e^{-\lambda t} \|x - x_*\|
$$

for all $t \ge 0$ and all $x \in \mathcal{B}_\delta(x_*)$. For such $\delta$, we have

$$
\|\phi(t, x) - x_*\| \le \max \{\tilde{\kappa}e^{-\tilde{\lambda}t} \|x - x_*\|, \delta\}
$$

with $\tilde{\kappa} \ge \kappa$ and $\tilde{\lambda} \le \lambda$. Therefore,

$$
\|\phi(t, x) - x_*\| \le \tilde{\kappa} e^{-\tilde{\lambda}t} \|x - x_*\|
$$

for all $t \ge 0$ and all $x \in \mathbb{R}^n$.

The following example is from Angeli (2002). Based on our
result, we show that a claim in that paper is wrong.

Example 16. (Example 2 in Angeli (2002)). Consider

$$
\dot{x} = - \frac{x}{(1 + x^2)^{1/3}}.
$$

Assume that the family of all solutions is δGAS. Because
the origin is locally exponentially stable, it must be glob-
ally exponentially stable. Hence, there exists λ > 0 such
that

$$
|x(t)| \le e^{-\lambda t} |x(0)| \quad \forall t \ge 0.
$$

To show a contradiction, we consider the differential equa-
tion

$$
\dot{u} = -u^{1/3}.
$$

If $u(0) > 0$, the solution $u$ is given by

$$
u(t) = \begin{cases} \left(u(0)^{2/3} - \frac{2}{3}t\right)^{3/2}, & t \in [0, \frac{3}{2}u(0)^{2/3}) \\ 0, & t \ge \frac{3}{2}u(0)^{2/3} \end{cases},
$$

Set $x(0) = c^{3/2}$ for some $c > 0$. Since

$$
-\frac{x}{(1+x^2)^{1/3}} \geq -x^{1/3},
$$

the comparison theorem gives

$$
x(t) \geq \left(c - \frac{2}{3}t\right)^{3/2} \quad \forall t \in \left[0, \frac{3}{2}c\right).
$$

Thus, we have

$$
\left(c - \frac{2}{3}t\right)^{3/2} \leq c^{3/2} e^{-\lambda t} \quad \forall t \in \left[0, \frac{3}{2}c\right).
$$

By letting $t = \frac{3}{4}c$ one can verify that the previous inequal-
ity is violated for $c > \frac{2}{\lambda} \ln 2$, hence a contradiction. As
a consequence, the family of all solutions cannot be $\delta$GAS
on the contrary of what is claimed in Angeli (2002).

5. CONCLUSION

We revised the notion of δGAS and we derived some necessary conditions in the context of autonomous nonlinear systems. The first consideration is that assuming forward completeness of solutions in the incremental context has no loss of generality. Furthermore, we showed that δGAS implies the existence of a unique equilibrium with a proof which does not need classical contraction theorems (such as Banach’s fixed point theorem). Finally, we proved that the notion of δGAS implies exponential convergence at large. To summarize, our conclusion is that the notions of δGAS and δGES are equivalent at infinity. There are several open problems regarding δGAS. We will extend the results in this paper to the time-varying case. For example, entrainment to periodic orbits has been studied only in the context of δGES (Duvall and Sontag (2024)). Our goal is to relax the requirement of δGES assumed in the literature to δGAS.

REFERENCES

Andrieu, V., Jayawardhana, B., and Praly, L. (2016). Transverse exponential stability and applications. *IEEE Transactions on Automatic Control*, 61(11), 3396–3411.

Andrieu, V., Jayawardhana, B., and Praly, L. (2021). Characterizations of global transversal exponential stability. *IEEE Transactions on Automatic Control*, 66(8), 3682–3694.

Angeli, D. (2002). A Lyapunov approach to incremental stability properties. *IEEE Transactions on Automatic Control*, 47(3), 410–421.

Angeli, D. (2009). Further results on incremental input-
to-state stability. *IEEE Transactions on Automatic
Control*, 54(6), 1386–1391.

Bullo, F. (2022). *Contraction Theory for Dynamical Systems*. Kindle Direct Publishing.

Duvall, A. and Sontag, E.D. (2024). Global exponential stability or contraction of an unforced system do not imply entrainment to periodic inputs. In *Proceedings of American Control Conference*, 1837–1842.

Forni, F. and Sepulchre, R. (2014). A differential Lyapunov framework for contraction analysis. *IEEE Transactions on Automatic Control*, 59(3), 614–628.

Giaccagli, M., Astolfi, D., Andrieu, V., and Marconi, L. (2022). Sufficient conditions for global integral action via incremental forwarding for input-affine nonlinear systems. *IEEE Transactions on Automatic Control*, 67(12), 6537–6551.

Giaccagli, M., Zoboli, S., Astolfi, D., Andrieu, V., and Casadei, G. (2024). Synchronization in networks of nonlinear systems: Contraction analysis via Riemannian metrics and deep-learning for feedback estimation. *IEEE Transactions on Automatic Control*, 69(11), 8041–8048.

Grüne, L., Sontag, E.D., and Wirth, F.R. (1999). Asymp-
totic stability equals exponential stability, and ISS
equals finite energy gain—if you twist your eyes. *Sys-
tems & Control Letters*, 38(2), 127–134.
Jouffroy, J. and Slotine, J.J.E. (2004). Methodological remarks on contraction theory. In Proceedings of 43rd IEEE Conference on Decision and Control, 2537–2543.

Kawano, Y. and Besselink, B. (2024). Incremental versus differential approaches to exponential stability and passivity. IEEE Transactions on Automatic Control, 69(9), 6450–6457.

Khalil, H.K. (2002). *Nonlinear Systems*. Prentice Hall, 3rd edition.

Lan, Y. and Mezić, I. (2013). Linearization in the large of nonlinear systems and Koopman operator spectrum. Physica D, 242(1), 42–53.

Leonov, G.A. (2008). *Strange Attractors and Classical Stability Theory*. St. Petersburg University Press.

Lohmiller, W. and Slotine, J.J.E. (1998). On contraction analysis for non-linear systems. Automatica, 34(6), 683–696.

Pavlov, A., Pogromsky, A., van de Wouw, N., and Nijmeijer, H. (2004). Convergent dynamics, a tribute to Boris Pavlovich Demidovich. Systems & Control Letters, 52(3–4), 257–261.

Pavlov, A., van de Wouw, N., and Nijmeijer, H. (2006). Uniform Output Regulation of Nonlinear Systems: A Convergent Dynamics Approach. Birkhäuser Boston.

Rüffer, B.S., van de Wouw, N., and Mueller, M. (2013). Convergent systems vs. incremental stability. Systems & Control Letters, 62(3), 277–285.

Sanfelice, R.G. and Praly, L. (2012). Convergence of nonlinear observers on $\mathbb{R}^n$ with a Riemannian metric (part I). IEEE Transactions on Automatic Control, 57(7), 1709–1722.

Tran, D.N., Rüffer, B.S., and Kellett, C.M. (2019). Convergence properties for discrete-time nonlinear systems. IEEE Transactions on Automatic Control, 64(8), 3415–3422.

Wu, D. and Duan, G.R. (2022). Further geometric and Lyapunov characterizations of incrementally stable systems on Finsler manifolds. IEEE Transactions on Automatic Control, 67(10), 5614–5621.

Zamani, M. and Tabuada, P. (2011). Backstepping design for incremental stability. IEEE Transactions on Automatic Control, 56(9), 2184–2189.

## Appendix A. PROOFS

### A.1 Proof of Lemma 10

We only need to show that one solution is bounded. For a fixed $x$, we show that

$$ \forall \varepsilon > 0, \exists t_0 \ge 0, \forall t_1 \ge 0, \|\phi(t_0 + t_1, x) - \phi(t_0, x)\| < \varepsilon. \quad (A.1) $$

Thus, after some time, the solution $t \mapsto \phi(t, x)$ cannot leave a ball whose radius can be arbitrarily small. This implies that the solution $t \mapsto \phi(t, x)$ is bounded.

Now, we prove the assertion (A.1) by contradiction. To do so, we assume that

$$ \exists \varepsilon > 0, \forall t_1 \ge 0, \exists t_2 \ge 0, \|\phi(t_1 + t_2, x) - \phi(t_1, x)\| \ge \varepsilon. $$

We fix $\varepsilon$ from now on. Let $t_0 > 0$ be such that

$$ \beta(\varepsilon, t_0) = \frac{\varepsilon}{3}. $$

Then, we choose $t_1$ such that

$$ \beta(\|\phi(t_0, x) - x\|, t_1) = \frac{\varepsilon}{6}. $$

By assumption, the set

$$ \{t \ge 0 : \|\phi(t_1 + t, x) - \phi(t_1, x)\| \ge \varepsilon\} $$

is nonempty and we take $t_2$ as its infimum, i.e.,

$$ \|\phi(t_1 + t_2, x) - \phi(t_1, x)\| = \varepsilon. $$

The triangle inequality gives

$$ \begin{aligned} & \|\phi(t_1 + t_2, x) - \phi(t_1, x)\| \\ & \le \|\phi(t_0 + t_1 + t_2, x) - \phi(t_1 + t_2, x)\| \\ & \quad + \|\phi(t_0 + t_1, x) - \phi(t_1, x)\| \\ & \quad + \|\phi(t_0 + t_1 + t_2, x) - \phi(t_0 + t_1, x)\|. \end{aligned} $$

However, we have

$$ \begin{aligned} & \|\phi(t_0 + t_1 + t_2, x) - \phi(t_1 + t_2, x)\| \\ & \le \beta(\|\phi(t_0, x) - x\|, t_1 + t_2) \\ & \le \beta(\|\phi(t_0, x) - x\|, t_1) = \frac{\varepsilon}{6} \end{aligned} $$

and

$$ \|\phi(t_0 + t_1, x) - \phi(t_1, x)\| \le \beta(\|\phi(t_0, x) - x\|, t_1) = \frac{\varepsilon}{6}. $$

Also, we have

$$ \begin{aligned} & \|\phi(t_0 + t_1 + t_2, x) - \phi(t_0 + t_1, x)\| \\ & \le \beta(\|\phi(t_1 + t_2, x) - \phi(t_1, x)\|, t_0) \\ & \le \beta(\varepsilon, t_0) = \frac{\varepsilon}{3}. \end{aligned} $$

These inequalities yield

$$ \begin{aligned} & \varepsilon = \|\phi(t_1 + t_2, x) - \phi(t_1, x)\| \\ & \le \frac{\varepsilon}{6} + \frac{\varepsilon}{6} + \frac{\varepsilon}{3} = \frac{5}{6}\varepsilon. \end{aligned} $$

We do have a contradiction. Therefore, the assertion (A.1) is true and all solution are bounded from $\delta$GAS.

### A.2 Proof of Proposition 14

Given $R > 0$, we choose $K > 1$ and $L > 0$ such that

$$ \beta \left( \frac{1}{L(K-1)}, 0 \right) \le R. $$

If

$$ \|x - y\| \le \frac{1}{L(K - 1)}, $$

then we have

$$ \begin{aligned} & \|\phi(t, x) - \phi(t, y)\| \le \beta \left( \frac{1}{L(K-1)}, t \right) \\ & \le R \end{aligned} $$

for all $t \ge 0$. Thus, we need to consider the case where

$$ \|x - y\| \ge \frac{1}{L(K - 1)}. $$

In this case, Lemma 17 in Appendix B implies that

$$ \|\phi(t, x) - \phi(t, y)\| \le KL\beta\left(\frac{1}{L}, t\right) \|x - y\| $$

for all $t \ge 0$. For some $\gamma \in (0, 1)$, let $T > 0$ be such that

$$ KL\beta\left(\frac{1}{L}, T\right) \le \gamma. $$

This implies that

$$ \begin{aligned} & \|\phi(t, x) - \phi(t, y)\| \le KL\beta\left(\frac{1}{L}, T\right) \|x - y\| \\ & \le \gamma \|x - y\|. \end{aligned} $$
If

$$ \|\phi(t, x) - \phi(t, y)\| \geq \frac{1}{L(K-1)}, $$

then

$$ \begin{aligned} \|\phi(2T, x) - \phi(2T, y)\| &= \|\phi(T, \phi(T, x)) - \phi(T, \phi(T, y))\| \\ &\leq \gamma \|\phi(t, x) - \phi(t, y)\| \\ &\leq \gamma^2 \|x - y\|. \end{aligned} $$

Let $N \in \mathbb{N}_0$ be the minimal number such that

$$ \|\phi((N+1)T, x) - \phi((N+1)T, y)\| < \frac{1}{L(K-1)}. $$

By repeating the above procedure, we obtain

$$ \|\phi((p+1)T, x) - \phi((p+1)T, y)\| \leq \gamma^{p+1} \|x - y\| $$

for all $p \in \{0, 1, \dots, N\}$. Also, for each $p \in \{0, 1, \dots, N\}$ and each $t \in [pT, (p+1)T]$, we have

$$ \begin{aligned} \|\phi(t, x) - \phi(t, y)\| &= \|\phi(t - pT, \phi(pT, x)) - \phi(t - pT, \phi(pT, y))\| \\ &\leq KL\beta \left(\frac{1}{L}, t - pT\right) \|\phi(pT, x) - \phi(pT, y)\| \\ &\leq KL\beta \left(\frac{1}{L}, 0\right) \gamma^p \|x - y\|. \end{aligned} $$

By noting that

$$ \gamma^p = \frac{1}{\gamma} e^{\ln \gamma^{p+1}} \leq \frac{1}{\gamma} e^{\frac{\ln \gamma}{T} t} $$

for all $t \in [pT, (p+1)T]$ and all $p \in \{0, 1, \dots, N\}$, we get

$$ \|\phi(t, x) - \phi(t, y)\| \leq \frac{KL\beta(1/L, 0)}{\gamma} e^{\frac{\ln \gamma}{T} t} \|x - y\| $$

for all $t \in [0, (N+1)T]$. On the other hand, we have

$$ \begin{aligned} \|\phi(t, x) - \phi(t, y)\| &\leq \beta \left( \frac{1}{L(K-1)}, t \right) \\ &\leq \beta \left( \frac{1}{L(K-1)}, 0 \right) \\ &\leq R \end{aligned} $$

for all $t \geq (N+1)T$. Therefore, we have shown that

$$ \|\phi(t, x) - \phi(t, y)\| \leq \max \left\{ \frac{KL\beta(1/L, 0)}{\gamma} e^{\frac{\ln \gamma}{T} t} \|x - y\|, R \right\} $$

for all $t \geq 0$ and all $x, y \in \mathbb{R}^n$.

## Appendix B. TECHNICAL LEMMA

Suppose that there exists a class-KL function $\beta$ such that

$$ \|\phi(t, x) - \phi(t, y)\| \leq \beta(\|x - y\|, t) $$

for all $t \geq 0$ and all $x, y \in \mathbb{R}^n$. Given $x$ and $y$, we define

$$ v_k := \frac{x - y}{k}, \quad k \in \mathbb{N}. $$

By the triangle inequality, we have

$$ \begin{aligned} \|\phi(t, x) - \phi(t, y)\| &\leq \sum_{j=1}^{k} \|\phi(t, x - (j-1)v_k) - \phi(t, x - jv_k)\| \\ &\leq k\beta(\|v_k\|, t) \end{aligned} $$

for all $t \geq 0$ and all $k \in \mathbb{N}$. Now, we define a function $\tilde{\beta}: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ by

$$ \tilde{\beta}(s, t) := \inf_{k \in \mathbb{N}} k\beta \left( \frac{s}{k}, t \right). $$

Then, we obtain

$$ \|\phi(t, x) - \phi(t, y)\| \leq \tilde{\beta}(\|x - y\|, t) $$

for all $t \geq 0$ and all $x, y \in \mathbb{R}^n$.

Based on this new bound, we prove the following result.

**Lemma 17.** Suppose that the family of all solutions to (1) is $\delta$GAS. Let $K > 1$ and $L > 0$ be given. If

$$ \|x - y\| \geq \frac{1}{L(K-1)}, $$

then

$$ \|\phi(t, x) - \phi(t, y)\| \leq KL\beta \left(\frac{1}{L}, t\right) \|x - y\| $$

for all $t \geq 0$.

**Proof.** Let $c$ be a constant such that

$$ c \geq \frac{1}{K-1}. $$

Recall that

$$ \tilde{\beta}(c\|v\|, t) \leq k\beta \left( \frac{c\|v\|}{k}, t \right) $$

for all $t \geq 0$, all $k \in \mathbb{N}$, and all $v \in \mathbb{R}^n$. Because $[c]$ is a positive integer, it follows that

$$ \begin{aligned} \tilde{\beta}(c\|v\|, t) &\leq [c]\beta \left( \frac{c\|v\|}{[c]}, t \right) \\ &\leq [c]\beta(\|v\|, t) \end{aligned} $$

for all $t \geq 0$ and all $v \in \mathbb{R}^n$. Since $c+1 \leq Kc$, it holds true that

$$ \begin{aligned} \tilde{\beta}(c\|v\|, t) &\leq (c+1)\beta(\|v\|, t) \\ &\leq Kc\beta(\|v\|, t) \end{aligned} $$

for all $t \geq 0$ and all $v \in \mathbb{R}^n$. Given $x$ and $y$, we set

$$ v = \frac{1}{L} \frac{x - y}{\|x - y\|}, \quad c = L \|x - y\|, $$

Thus, if

$$ \|x - y\| \geq \frac{1}{L(K-1)}, $$

then

$$ \begin{aligned} \|\phi(t, x) - \phi(t, y)\| &\leq \tilde{\beta}(\|x - y\|, t) \\ &= \tilde{\beta}(c\|v\|, t) \\ &\leq Kc\beta(\|v\|, t) \end{aligned} $$

for all $t \geq 0$. The proof is complete.