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Lyapunov characterization of uniform exponential stability for

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Lyapunov characterization of uniform exponential stability for nonlinear infinite-dimensional systems
Ihab Haidar, yacine Chitour, Paolo Mason, Mario Sigalotti
To cite this version:
Ihab Haidar, yacine Chitour, Paolo Mason, Mario Sigalotti. Lyapunov characterization of uniform exponential stability for nonlinear infinite-dimensional systems. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2022, 67 (4), pp.1685-1697. ￿10.1109/TAC.2021.3080526￿. ￿hal-02479777￿

HAL Id: hal-02479777 https://hal.inria.fr/hal-02479777
Submitted on 14 Feb 2020

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Lyapunov characterization of uniform exponential stability for nonlinear infinite-dimensional systems
Ihab Haidar∗, Yacine Chitour†, Paolo Mason‡, and Mario Sigalotti§
February 17, 2020
Abstract
In this paper we deal with infinite-dimensional nonlinear forward complete dynamical systems which are subject to external disturbances. We first extend the well-known Datko lemma to the framework of the considered class of systems. Thanks to this generalization, we provide characterizations of the uniform (with respect to disturbances) local, semi-global, and global exponential stability, through the existence of coercive and non-coercive Lyapunov functionals. The importance of the obtained results is underlined through some applications concerning 1) exponential stability of nonlinear retarded systems with piecewise constant delays, 2) exponential stability preservation under sampling for semilinear control switching systems, and 3) the link between input-to-state stability and exponential stability of semilinear switching systems.
Keywords: Infinite-dimensional systems, Nonlinear systems, Switching systems, Converse Lyapunov theorems, Exponential stability.
1 Introduction
Various works have been recently devoted to the characterization of the stability of infinite-dimensional systems in Banach spaces through non-coercive and coercive Lyapunov functionals (see, e.g., [9, 12, 33, 35, 36]). By non-coercive Lyapunov functional, we mean a positive definite functional decaying along the trajectories of the system which satisfies
0 < V (x) ≤ α( x ), ∀ x ∈ X\{0},
where X is the ambient Banach space and α belongs to the class K∞ of continuous increasing bijections from R+ to R+. Such a function V would be coercive if there existed α0 ∈ K∞ such that V (x) ≥ α0( x ) for every x ∈ X. In [36] it has been proved that the existence of a coercive Lyapunov functional V represents a necessary and sufficient condition for the global asymptotic stability for a general class of infinite-dimensional forward complete dynamical systems. On the other hand, the existence of a non-coercive Lyapunov functional does not guarantee global asymptotic stability and some additional regularity assumption on the dynamics is needed (see, e.g., [12, 36]). Converse Lyapunov theorems can be helpful for many applications, such as stability analysis of interconnected systems [10] and for the characterization of input-to-state stability (see, e.g., [17, 42, 45]). Stability results based on non-coercive Lyapunov functionals may be more easily
∗Quartz EA 7393, ENSEA, Cergy-Pontoise, France, [email protected] †Laboratoire des Signaux et Syst`emes (L2S), Universit´e Paris-Saclay, CNRS, CentraleSup´elec, Universit´e Paris-Saclay, Gifsur-Yvette, France, [email protected] ‡CNRS & Laboratoire des Signaux et Syst`emes (L2S), Universit´e Paris-Saclay, CNRS, CentraleSup´elec, Gif-sur-Yvette, France, [email protected] §Laboratoire Jacques-Louis Lions (LJLL), Inria, Sorbonne Universit´e, Universit´e de Paris, CNRS, Paris, France, [email protected]
1

applied in practice, while the existence of a coercive Lyapunov functional may be exploited to infer additional information on a stable nonlinear system.
Here, we consider the same class of abstract forward complete dynamical systems, subject to a shiftinvariant set of disturbances, as in [36]. The novelty of our approach is that we focus on exponential (instead of asymptotic) stability. For the rest of the paper, the word uniform will refer to uniformity with respect to disturbances. We provide theorems characterizing different types of uniform local, semi-global, and global exponential stability, through the existence of non-coercive and coercive Lyapunov functionals. Using a standard converse Lyapunov approach, we prove that uniform semi-global exponential stability is characterized by the existence of a 1-parameter family of Lyapunov functionals, each of them decaying uniformly on a bounded set, while the union of all such bounded sets is equal to the entire Banach space X. Concerning the non-coercive case, we first give a generalization of the Datko lemma [5, 37]. Recall that the latter characterizes the exponential behavior of a linear C0-semigroup in a Banach space in terms of a uniform estimate of the Lp-norm of the solutions. This result has been extended in [14] to the framework of nonlinear semigroups. Here, we generalize the Datko lemma to the considered class of infinite-dimensional forward complete dynamical systems. Thanks to such a generalization, we prove that the existence of a noncoercive Lyapunov functional is sufficient, under a uniform growth estimate on the solutions of the system, for the uniform exponential stability. The importance of the obtained results is underlined through some applications as described in the sequel.
Retarded functional differential equations form an interesting class of infinite-dimensional systems that we cover by our approach. Converse Lyapunov theorems have been developed for systems described by retarded and neutral functional differential equations (see, e.g., [18, 41]). Such results have been recently extended in [9] to switching linear retarded systems through coercive and non-coercive Lyapunov characterizations. After representing a nonlinear retarded functional differential equation as an abstract forward complete dynamical system, all the characterizations of uniform exponential stability provided in the first part of the paper can be applied to this particular class of infinite-dimensional systems. In particular, we characterize the uniform global exponential stability of a retarded functional differential equation in terms of the existence of a non-coercive Lyapunov functional.
Another interesting problem when dealing with a continuous-time model is the practical implementation of a designed feedback control. Indeed, in practice, due to numerical and technological limitations (sensors, actuators, and digital interfaces), a continuous measurement of the output and a continuous implementation of a feedback control are impossible. This means that the implemented input is, for almost every time, different from the designed controller. Several methods have been developed in the literature of ordinary differential equations for sampled-data observer design under discrete-time measurements (see, e.g., [2, 19, 24, 29]), and for sampled-data control design guaranteeing a globally stable closed-loop system (see, e.g., [1, 13]). Apart from time-delays systems (see, e.g., [7, 20, 39] for sampled-data control and [29, 30] for sampled-data observer design), few results exist for infinite-dimensional systems. The difficulties come from the fact that the developed methods do not directly apply to the infinite-dimensional case, for which even the well-posedness of sampled-data control dynamics is not obvious (see, e.g., [21] for more details). Some interesting results have been obtained for infinite-dimensional linear systems [21, 23, 47]. In the nonlinear case no standard methods have been developed and the problem is treated case by case [22]. Here, we focus on the particular problem of feedback stabilization under sampled output measurements of an abstract semilinear infinite-dimensional system. In particular, we consider the dynamics

x˙ (t) = Ax(t) + fσ(t)(x(t), u(t)), t ≥ 0,

(1)

where x(t) ∈ X, U is a Banach space, u ∈ U is the input, A is the infinitesimal generator of a C0-semigroup of bounded linear operators (Tt)t≥0 on X, σ : R+ → Q is a piecewise constant switching function, and fq : X × U → X is a Lipschitz continuous nonlinear operator, uniformly with respect to q ∈ Q. Assume that only discrete output measurements are available

y(t) = x(tk), ∀ t ∈ [tk, tk+1), ∀ k ≥ 0,

(2)

where (tk)k≥0 denotes the increasing sequence of sampling times. It is well known that, in general, no feedback of the type u(t) = K(y(t)) stabilizes system (1). Moreover, suppose that system (1) in closed-loop

2

with

u(t) = K(x(t)), ∀ t ≥ 0,

(3)

where K : X → U is a globally Lipschitz function satisfying K(0) = 0, is uniformly semi-globally expo-

nentially stable. Using our converse Lyapunov theorem, we show that if the maximal sampling period is

small enough then, under some additional conditions, system (1) in closed-loop with the predictor-based

sampled-data control

u(t) = Tt−tk y(tk), ∀ t ∈ [tk, tk+1), ∀ k ≥ 0,

(4)

is uniformly locally exponentially stable in each ball around the origin. Furthermore, if the closed loop system (1)-(3) is uniformly globally exponentially stable, then the same property holds for the closed loop system (1)-(4), under sufficiently small sampling period. We give an example of a wave equation (see, e.g., [4, 26]) showing the applicability of our result.
In recent years, the problem of characterizing input-to-state stability (ISS) for infinite-dimensional systems has attracted a particular attention. Roughly speaking, the ISS property, introduced in [44] for ordinary differential equations, means that the trajectories of a perturbed system eventually approach a neighborhood of the origin whose size is proportional to the magnitude of the perturbation. This concept has been widely studied in the framework of complex systems such as switching systems (see, e.g., [25] and references therein), time-delay systems (see, e.g., [40, 46, 49] and references therein), and abstract infinite-dimensional systems (see, e.g., [32, 34]). For example, in [34] a converse Lyapunov theorem characterizing the input-to-state stability of a locally Lipschitz dynamics through the existence of a locally Lipschitz continuous coercive ISS Lyapunov functional is given. Recently in [16] it has been shown that, under regularity assumptions on the dynamics, the existence of non-coercive Lyapunov functionals implies input-to-state stability. Here, we provide a result of ISS type, proving that the input-to-state map has finite gain, under the assumption that the unforced system corresponding to (1) (i.e., with u ≡ 0) is uniformly globally exponentially stable.
The paper is organized as follows. Section 2 presents the problem statement with useful notations and definitions. In Section 3 we state our main results, namely three Datko-type theorems for uniform local, semiglobal, and global exponential stability, together with direct and converse Lyapunov theorems. In Section 4 we compare the proposed Lyapunov theorems with the current state of art. The applications are given in Section 5. In Section 6 we consider an example of a damped wave equation. The proofs are postponed to Section 7.

1.1 Notations

By (X, · ) we denote a Banach space with norm · and by BX (x, r) the closed ball in X of center x ∈ X

and radius r. By R we denote the set of real numbers and by | · | the Euclidean norm of a real vector.

We

use

R+

and

R⋆
+

to denote

the

sets

of non-negative and

positive real numbers respectively.

A

function

α : R+ → R+ is said to be of class K if it is continuous, increasing, and satisfies α(0) = 0; it is said to be of

class K∞ if it is of class K and unbounded. A continuous function κ : R+ × R+ → R+ is said to be of class

KL if κ(·, t) is of class K for each t ≥ 0 and, for each s ≥ 0, κ(s, ·) is nonincreasing and converges to zero as

t tends to +∞.

2 Problem statement
In this paper we consider a forward complete dynamical system evolving in a Banach space X. Let us recall the following definition, proposed in [36].
Definition 1. Let Q be a nonempty set. Denote by S a set of functions σ : R+ → Q satisfying the following conditions:
a) S is closed by time-shift, i.e., for all σ ∈ S and all τ ≥ 0, the τ -shifted function Tτ σ : s → σ(τ + s) belongs to S;

3

b) S is closed by concatenation, i.e., for all σ1, σ2 ∈ S and all τ > 0 the function σ defined by σ ≡ σ1 over [0, τ ] and by σ(τ + t) = σ2(t) for all t > 0, belongs to S.
Let φ : R+ × X × S → X be a map. The triple Σ = (X, S, φ) is said to be a forward complete dynamical system if the following properties hold:

i) ∀ (x, σ) ∈ X × S, it holds that φ(0, x, σ) = x;

ii) ∀ (x, σ) ∈ X × S, ∀ t ≥ 0, and ∀ σ˜ ∈ S such σ˜ = σ over [0, t], it holds that φ(t, x, σ˜) = φ(t, x, σ);

iii) ∀ (x, σ) ∈ X × S, the map t → φ(t, x, σ) is continuous;

iv) ∀t, τ ≥ 0, ∀ (x, σ) ∈ X × S, it holds that φ(τ, φ(t, x, σ), Ttσ) = φ(t + τ, x, σ). We will refer to φ as the transition map of Σ.

Observe that if Σ is a forward complete dynamical system and S contains a constant function σ then (φ(t, ·, σ))t≥0 is a strongly continuous nonlinear semigroup, whose definition is recalled below.
Definition 2. Let Tt : X → X, t ≥ 0, be a family of nonlinear maps. We say that (Tt)t≥0 is a strongly continuous nonlinear semigroup if the following properties hold:

i) ∀ x ∈ X, T0x = x;

ii) ∀ t1, t2 ≥ 0, Tt1 Tt2 x = Tt1+t2 x;

iii) ∀ x ∈ X, the map t → Ttx is continuous. An example of forward complete dynamical system is given next.

Example 3 (Piecewise constant switching system). We denote by PC the set of piecewise constant σ :
R+ → Q, and we consider here the case S = PC. Let σ ∈ PC be constantly equal to σk over [tk, tk+1), with 0 = t0 < t1 < · · · < tk < t < tk+1, for k ≥ 0. With each σk we associate the strongly continuous nonlinear semigroup (Tσk (t))t≥0 := (φ(t, ·, σk))t≥0. By concatenating the flows (Tσk (t))t≥0, one can associate with σ the family of nonlinear evolution operators

Tσ(t) := Tσk (t − tk)Tσk−1 (tk − tk−1) · · · Tσ1 (t1),

t ∈ [tk, tk+1). By consequence, system Σ can be identified with the piecewise constant switching system

x(t) = Tσ(t)x0, x0 ∈ X, σ ∈ PC.

(5)

Thanks to the representation given by (5), this paper extends to the nonlinear case some of the results obtained in [12] on the characterization of the exponential stability of switching linear systems in Banach spaces.
Various notions of uniform (with respect to the functions in S) exponential stability of system Σ are given by the following definition.

Definition 4. Consider the forward complete dynamical system Σ = (X, S, φ).

1. We say that Σ is uniformly globally exponentially stable at the origin (UGES, for short) if there exist M > 0 and λ > 0 such that the transition map φ satisfies the inequality

φ(t, x, σ) ≤ M e−λt x , ∀ t ≥ 0, ∀ x ∈ X, ∀ σ ∈ S.

2. We say that Σ is uniformly locally exponentially stable at the origin (ULES, for short) if there exist

R > 0, M > 0, and λ > 0 such that the transition map φ satisfies the inequality, for every t ≥ 0,

x ∈ BX (0, R), and σ ∈ S,

φ(t, x, σ) ≤ M e−λt x .

(6)

If inequality (6) holds true for a given R > 0 then we say that Σ is uniformly exponentially stable at the origin in BX (0, R) (UES in BX (0, R), for short).

4

3. We say that Σ is uniformly semi-globally exponentially stable at the origin (USGES, for short) if, for every r > 0 there exist M (r) > 0 and λ(r) > 0 such that the transition map φ satisfies the inequality

φ(t, x, σ) ≤ M (r)e−λ(r)t x ,

(7)

for every t ≥ 0, x ∈ BX (0, r), and σ ∈ S.
Remark 5. Up to modifying r → M (r), one can assume without loss of generality in the definition of USGES that r → λ(r) can be taken constant and r → M (r) nondecreasing. Indeed, let us fix M := M (1) and λ := λ(1). One has, by definition of (M, λ),
φ(t, x, σ) ≤ M e−λt x ,

for every t ≥ 0, x ∈ BX (0, 1), and σ ∈ S. For R > 1, by using (7) there exists tR such that, for every x ∈ BX (0, R) with x ≥ 1, one has for t ≥ tR,

M (R)e−λ(R)tR R = 1, φ(t, x, σ) ≤ x ≤ min{1, x }. R
This implies that, for every t ≥ tR, x ∈ BX (0, R), and σ ∈ S, φ(t, x, σ) ≤ M e−λ(t−tR) φ(tR, x, σ) ≤ M e−λ(t−tR) x .

By setting

M (R) =

M max{M (R), M (R)e|λ−λ(R)|tR , M eλtR }

R ≤ 1, R > 1,

one has that φ(t, x, σ) ≤ M (r)e−λt x , for every t ≥ 0, x ∈ BX (0, r), and σ ∈ S. Finally, we may replace r → M (r) with the nondecreasing function r → infρ≥r M (ρ).
The property of semi-global exponential stability introduced in Definition 4 turns out to be satisfied by some interesting class of infinite-dimensional systems, as described in the following two examples.

Example 6. For L > 0, let Ω = (0, L) and consider the controlled Korteweg–de Vries (KdV) equation

 ηt + ηx + ηxxx + ηηx + ρ(t, x, η) = 0 (x, t) ∈ Ω × R+,

η(t, 0) = η(t, L) = ηx(t, L) = 0

t ∈ R+,

(8)

η(0, x) = η0(x)

x ∈ Ω,

where ρ : R+ × Ω × R → R is a sufficiently regular nonlinear function. The case ρ ≡ 0 is a well known model describing waves on shallow water surfaces [15]. The controllability and stabilizability properties of (8) have
been extensively studied in the literature (see, e.g., [31, 43]). In the case where the feedback control is of the form ρ(t, x, η) = a(x)η, for some non-negative function a(·) having nonempty support in Ω, system (8) is globally exponentially stable in X = L2(0, L). In [27] the authors prove that, when a saturation is introduced
in the feedback control ρ, the system is only semi-globally exponentially stable in X.

Example 7. Consider the 1D wave equation with boundary damping



ψtt − ∆ψ = 0

(x, t) ∈ (0, 1) × R+,

ψ(0, t) = 0

t ∈ R+,

(9)

ψx(1, t) = −σ(t, ψt(1, t)) t ∈ R+,

ψ(0) = ψ0, ψt(0) = ψ1 x ∈ (0, 1),

where σ : R+ × R → R is continuous. This system is of special interest when the damping term σ(t, ψt(1, t)) represents a nonlinear feedback control. Once again, different types of stability can be established (for global and semi-global exponential stability, see e.g., [28, 3]). In particular, if σ is a nonlinearity of saturation type, only semi-global exponential stability holds true in
X = {(ψ0, ψ1) | ψ0(0) = 0, ψ0′ and ψ1 ∈ L∞(0, 1)}.

5

The systems considered in Examples 6 and 7 do not depend on σ ∈ S as in Definition 1. In Section 6 we introduce and study variants of such systems in a switching framework.

3 Main results

3.1 Datko-type theorems

In this section we give Datko-type theorems [5] for an abstract forward complete dynamical system Σ. The uniform (local, semi-global, and global) exponential stability is characterized in terms of the Lp-norm of the trajectories of the system. This provides a generalization of the results obtained in [14] for nonlinear
semigroups. The following theorem characterizes the local exponential stability of system Σ.

Theorem 8. Consider a forward complete dynamical system Σ = (X, S, φ). Let t1, G0 > 0, and β be a

function

of

class

K∞

such

that

lim supr↓0

β(r) r

is

finite

and

φ(t, x, σ) ≤ G0β( x ), ∀ t ∈ [0, t1], ∀ x ∈ X, ∀ σ ∈ S.

(10)

The following statements are equivalent

i) System Σ is ULES;

ii) for every p > 0 there exist a nondecreasing function k : R+ → R+ and R > 0 such that

+∞

φ(t, x, σ) pdt ≤ k( x )p x p,

(11)

0

for every x ∈ BX (0, R) and σ ∈ S;

iii) there exist p > 0, k : R+ → R+ nondecreasing, and R > 0 such that (11) holds true.

Remark 9. Observe that hypothesis (10) in Theorem 8 is global over X. Indeed, we do not know if the stability at 0 may be deduced from inequality (11) if one restricts (10) to a ball BX (0, r).

The following theorem characterizes the semi-global exponential stability of system Σ.

Theorem 10. Consider a forward complete dynamical system Σ = (X, S, φ). Let t1, G0 > 0, and β be a

function

of

class

K∞

such

that

lim supr↓0

β(r) r

is

finite

and

φ(t, x, σ) ≤ G0β( x ), ∀ t ∈ [0, t1], ∀ x ∈ X, ∀ σ ∈ S.

(12)

The following statements are equivalent

i) System Σ is USGES;

ii) for every p > 0 there exists a nondecreasing function k : R+ → R+ such that, for every x ∈ X and σ ∈ S

+∞

φ(t, x, σ) pdt ≤ k( x )p x p;

(13)

0

iii) there exist p > 0 and k : R+ → R+ nondecreasing such that (13) holds true.
The particular case of uniformly globally exponentially stable systems is considered in the following theorem.

6

Theorem 11. Consider a forward complete dynamical system Σ = (X, S, φ). Let t1 > 0 and G0 > 0 be

such that

φ(t, x, σ) ≤ G0 x , ∀ t ∈ [0, t1], ∀ x ∈ X, ∀ σ ∈ S.

(14)

The following statements are equivalent

i) System Σ is UGES;

ii) for every p > 0 there exists k > 0 such that

+∞

φ(t, x, σ) pdt ≤ kp x p,

(15)

0

for every x ∈ X and σ ∈ S;

iii) there exist p, k > 0 such that (15) holds true.

Remark 12. By the shift-invariance properties given by items a) and iv) of Definition 1, it is easy to see

that (14) implies

φ(t, x, σ) ≤ M eλt x , ∀ t ≥ 0, ∀ x ∈ X, ∀ σ ∈ S,

(16)

where M = G0 and λ = max{0, log( Gt10 )}. Notice that inequality (16) is a nontrivial requirement on system Σ. Even in the linear case, and even if (16) is satisfied for each constant σ ≡ σc, uniformly with respect to σc, it does not follow that a similar exponential bound holds for the corresponding system Σ (see [12, Example 1]).

3.2 Lyapunov characterization of exponential stability

In this section we characterize the exponential stability of a forward complete dynamical system through the existence of a Lyapunov functional. First, let us recall the definition of Dini derivative of a functional V : X → R+.

Definition 13. Consider a forward complete dynamical system Σ = (X, S, φ). The upper and lower Dini

derivatives DσV : X → R ∪ {±∞} and DσV : X → R ∪ {±∞} of a functional V : X → R+ are defined,

respectively, as

1

DσV

(x)

=

lim sup
h↓0

h

(V

(φ(h,

x,

σ))



V

(x))

,

and 1

DσV

(x)

=

lim inf
h↓0

h

(V

(φ(h, x, σ))



V

(x))

,

where x ∈ X and σ ∈ S.

Remark 14. When S contains PC, we can associate with every q ∈ Q the upper and lower Dini derivatives DqV and DqV corresponding to σ ≡ q. Notice that for every σ ∈ PC and sufficiently small h > 0, we have σ|(0,h) ≡ q, for some q ∈ Q. By consequence, we have DσV (ϕ) = DqV (ϕ) and DσV (ϕ) = DqV (ϕ).
The regularity of a Lyapunov functional associated with an exponentially stable forward complete dynamical system Σ = (X, S, φ) is recovered, in our results, from the regularity of the transition map φ. The S-uniform continuity of the transition map φ with respect to the initial condition is defined as follows.

Definition 15. We say that the transition map φ of Σ = (X, S, φ) is S-uniformly continuous if, for any t¯ > 0, x ∈ X, and ε > 0, there exists η > 0 such that

φ(t, x, σ) − φ(t, y, σ) ≤ ε,

for every t ∈ [0, ¯t], y ∈ BX (x, η), and σ ∈ S.

7

Similarly, the notion of S-uniform Lipschitz continuity of the transition map is given by the following definition.

Definition 16. We say that the transition map φ of Σ = (X, S, φ) is S-uniformly Lipschitz continuous (respectively, S-uniformly Lipschitz continuous on bounded sets) if, for any t¯ > 0 (respectively, t¯ > 0 and R > 0), there exists l(t¯) > 0 (respectively, l(t¯, R) > 0) such that

φ(t, x, σ) − φ(t, y, σ) ≤ l(t¯) x − y ,

for every t ∈ [0, ¯t], x, y ∈ X, and σ ∈ S (respectively,

φ(t, x, σ) − φ(t, y, σ) ≤ l(t¯, R) x − y ,

for every t ∈ [0, ¯t], x, y ∈ BX (0, R), and σ ∈ S).

The following theorem shows that the existence of a non-coercive Lyapunov functional is sufficient for proving the uniform exponential stability of the forward complete dynamical system Σ, provided that inequality (10) holds true.

Theorem 17. Consider a forward complete dynamical system Σ = (X, S, φ). Let t1, G0 > 0, and β be a

function

of

class

K∞

such

that

lim supr↓0

β(r) r

is

finite

and

φ(t, x, σ) ≤ G0β( x ), ∀ t ∈ [0, t1], ∀ x ∈ X, ∀ σ ∈ S.

(17)

Then,

i) if there exist R > 0, V : BX (0, R) → R+, and p, c > 0 such that, for every x ∈ BX (0, R) and σ ∈ S,

V (x) ≤ c x p,

(18)

DσV (x) ≤ − x p,

(19)

and V (φ(·, x, σ)) is continuous from the left at every t > 0 such that φ(t, x, σ) ∈ BX (0, R), then system Σ is ULES;

ii) if i) holds true for every R > 0 with V = VR, c = cR and p = pR and

1

lim sup β−1 R min 1, t1 pR

R→+∞

G0

cR

= +∞,

(20)

then system Σ is USGES;

iii) if β in (17) is equal to the identity function and there exist p, c > 0 and a functional V : X → R+ such that, for every x ∈ X and σ ∈ S, the map t → V (φ(t, x, σ)) is continuous from the left, and V satisfies inequalities (18)-(19) in X, then system Σ is UGES.
The following theorem states that the existence of a coercive Lyapunov functional is necessary for the uniform exponential stability of a forward complete dynamical system.
Theorem 18. Consider a forward complete dynamical system Σ = (X, S, φ) and assume that the transition map φ is S-uniformly continuous. If Σ is USGES then, for every r > 0 there exist cr, cr > 0 and a continuous functional Vr : X → R+, such that

cr x ≤ Vr(x) ≤ cr x , ∀ x ∈ BX (0, r),

(21)

DσVr(x) ≤ − x , ∀ x ∈ BX (0, r), ∀ σ ∈ S,

(22)

Vr = VR on X, ∀R > 0 such that λ(r) = λ(R)

and M (r) = M (R),

(23)

where λ(·), M (·) are as in (7). Moreover, in the case where the transition map φ is S-uniformly Lipschitz continuous (respectively, S-uniformly Lipschitz continuous on bounded sets), Vr can be taken Lipschitz continuous (respectively, Lipschitz continuous on bounded sets).

8

To conclude this section, we state the following corollary which characterizes the uniform global exponential stability of a forward complete dynamical system, completing Item iii) of Theorem 17.
Corollary 19. Consider a forward complete dynamical system Σ = (X, S, φ). Assume that the transition map φ is S-uniformly continuous. If there exist t1 > 0 and G0 > 0 such that

φ(t, x, σ) ≤ G0 x , ∀ t ∈ [0, t1], ∀ x ∈ X, ∀ σ ∈ S,

(24)

then the following statements are equivalent: i) System Σ is UGES;

ii) there exists a continuous functional V : X → R+ and positive reals p, c, and c such that c x p ≤ V (x) ≤ c x p, ∀ x ∈ X,

and

DσV (x) ≤ − x p, ∀ x ∈ X, ∀ σ ∈ S;

(25)

iii) there exist a functional V : X → R+ and positive reals p and c such that, for every x ∈ X and σ ∈ S, the map t → V (φ(t, x, σ)) is continuous from the left, inequality (25) is satisfied and the following
inequality holds V (x) ≤ c x p, ∀ x ∈ X.

Proof. The fact that item i) implies ii) is a straightforward consequence of Theorem 18, using, in particular, (23). Moreover ii) clearly implies iii). Finally, iii) implies i), as follows from Theorem 17.

4 Discussion: comparison with the current state of the art

We compare here the results stated in Section 3.2 with some interesting similar results, obtained recently in [35], concerning the Lyapunov characterization of the uniform global asymptotic stability of a forward complete dynamical system. In order to make this comparison, we briefly recall some definitions and assumptions from [35].
Definition 20. We say that a forward complete dynamical system Σ = (X, S, φ) is uniformly globally asymptotically stable at the origin (UGAS, for short) if there exists a function κ of class KL such that the transition map φ satisfies the inequality

φ(t, x, σ) ≤ κ( x , t), ∀ t ≥ 0, ∀ x ∈ X, ∀ σ ∈ S.

The notion of robust forward completeness of system Σ is given by the following.
Definition 21. The forward complete dynamical system Σ = (X, S, φ) is said to be robustly forward complete (RFC, for short) if for any C > 0 and any τ > 0 it holds that

sup

φ(t, x, σ) < ∞.

x ≤C,t∈[0,τ ],σ∈S

Notice that RFC property of Σ is equivalent to inequality (17), although it does not necessarily imply

that

lim supr↓0

β(r) r

is

finite.

The notion of robust equilibrium point, which may be seen as a form of weak stability at the origin, is

given as follows.

Definition 22. We say that 0 ∈ X is a robust equilibrium point (REP, for short) of the forward complete dynamical system Σ = (X, S, φ) if for every ε, h > 0, there exists δ = δ(ε, h) > 0, so that

x ≤ δ =⇒ φ(t, x, σ) ≤ ε, ∀t ∈ [0, h], ∀σ ∈ S.

9
StabilitySystemsUniformExistenceOrigin