Exponential stability for nonautonomous functional differential equations with state-dependent delay

The properties of stability of compact set $\mathcal{K}$ which is positively invariant for a semiflow $(\Omega\times W^{1,\infty}([-r,0],\mathbb{R}^n),\Pi,\mathbb{R}^+)$ determined by a family of nonautonomous FDEs with state-dependent delay taking values in $[0,r]$ are analyzed. The solutions of the variational equation through the orbits of $\mathcal{K}$ induce linear skew-product semiflows on the bundles $\mathcal{K}\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and $\mathcal{K}\times C([-r,0],\mathbb{R}^n)$. The coincidence of the upper-Lyapunov exponents for both semiflows is checked, and it is a fundamental tool to prove that the strictly negative character of this upper-Lyapunov exponent is equivalent to the exponential stability of $\mathcal{K}$ in $\Omega\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and also to the exponential stability of this minimal set when the supremum norm is taken in $W^{1,\infty}([-r,0],\mathbb{R}^n)$. In particular, the existence of a uniformly exponentially stable solution of a uniformly almost periodic FDE ensures the existence of exponentially stable almost periodic solutions.


Introduction
State-dependent delay differential equations (SDDEs for short) have been extensively investigated during the last years, due to the theoretical interest of the related problems and to the great number of potential applications in many areas of interest, as automatic control, mechanical engineering, neural networks, population dynamics and ecology. Among the extensive list of works devoted to this field, we can mention Hartung [7,8,9,10], Wu [29], Walther [27,28], Hartung et al. [11], Chen et al. [3], Hu and Wu [16], Mallet-Paret and Nussbaum [19], Hu et al. [15], Barbarossa and Walther [2], and He and de la Llave [12,13], and Krisztin and and Rezounenko [18], as well as the many references therein.
In this paper, we analyze the exponential stability properties of the solutions of a nonautonomous SDDE. The use of the skew-product formulation allows us to use techniques arising from the topological dynamics.
More precisely, let (Ω, σ, R) be a continuous flow on a compact metric space. We write ω·t := σ(t, ω), and consider the family of SDDEs with maximum delay r > 0, given byẏ (t) = F (ω·t, y(t), y(t − τ (ω·t, y t ))) , t > 0 (1.1) for ω ∈ Ω, where F : Ω × R n × R n → R n is continuous and admits continuous partial derivatives with respect to the vectorial components. Let C := C([−r, 0], R n ) be endowed with the supremum norm. The state-dependent delay is given by a continuous function τ : Ω × C → [0, r], which is supposed to be continuously differentiable with respect to its second argument and to satisfy some standard Lipschitz conditions. And, as usual, we represent y t (s) := y(t + s) for s ∈ [−r, 0] whenever y is a continuous function on [t − r, t].
Additional recurrence conditions can be assumed on f and τ in order to ensure that the flow (Ω, σ, R) is minimal. This is the situation in the particular cases for which the pair (f, τ ) is uniformly periodic, almost periodic or almost automorphic, properties which in fact ensure the same for the flow on the corresponding hull. However, our approach in this paper is more general: we assume neither that (1.1) comes from a single SDDE, nor the minimality of (Ω, σ, R). This last condition will be indeed required for some of the results, but it will be imposed in due time.
The compact metric space Ω is the base of the bundle which constitutes the phase space of a skew-product semiflow, whose fiber component is determined by the solutions of the family (1.1). In our setting, the fiber of the bundle will be the Banach space W 1,∞ ⊂ C of the Lipschitz-continuous functions endowed with the standard norm. The already mentioned conditions assumed on the vector field and on the delay are intended to ensure the existence, uniqueness and some regularity properties of the solutions. It is convenient to keep in mind the idea that they are more exigent than those ensuring similar properties in the study of fixed or time-dependent delay equations. Strongly based on previous results of [7], we have established in [20] the existence of a unique maximal solution y(t, ω, x) of the equation (1.1) given by ω ∈ Ω for every initial data x ∈ W 1,∞ (i.e., with y(s, ω, x) = x(s) for s ∈ [−r, 0]), which are defined on [−r, β ω,x ), with β ω,x ≤ ∞. Since, if t ∈ [0, β ω,x ), the map u(t, ω, x)(s) := y(t + s, ω, x) belongs to W 1,∞ , then (1.1) determines the local skew-product semiflow on Ω × W 1,∞ Π : U ⊆ R + × Ω × W 1,∞ → Ω × W 1,∞ , (t, ω, x) → (ω·t, u(t, ω, x)) .
In general, this semiflow is not continuous. But it satisfies strong continuity properties, described in Theorem 3.2 below. We will call it a pseudo-continuous semiflow. Its interest relies on the fact that, despite the lack of global continuity, it allows us to use the classical tools of topological dynamics in the analysis of the behaviour of its orbits, i.e., in the qualitative analysis of the solutions of (1.1). In particular, the restriction of Π to positively invariant compact subsets is continuous.
The results of Section 3 of [7] and Section 4 of [20] prove that, if (ω, x) ∈ C 0 and t ∈ [0, β ω,x ), then: there exists the linear map u x (t, ω, x) : W 1,∞ → W 1,∞ and is continuous; it determines the Fréchet derivative of u(t, ω, x) with respect to x; and (u x (t, ω, x) v)(s) = z(t + s, ω, x, v) where z(t, ω, x, v) is the solution of the variational equationż(t) = L(Π(t, ω, x)) z t with z(s) = v(s) for s ∈ [−r, 0]. In addition, K ⊂ C 0 , which allows us to consider the linear skew-product semiflow for w(t, ω, x, v)(s) = z(t + s, ω, x, v) (which is a new pseudo-continuous semiflow) in order to define the Lyapunov exponents and to derive the stability properties of K from the characteristics of these exponents. Note that this question is not trivial, since: C 0 has empty interior, and there are cases for which the map u x (t, ω, x) is not defined for all (ω, x) ∈ Ω×W 1,∞ , i.e., for which u(t, ω, x) does not admit directional derivatives in W 1,∞ (see [10]). Let us briefly explain the structure and main results of the paper. In Section 2, we introduce the concepts of topological dynamics required in the following pages. We also recall the definition of exponential stability, and the notion and basic properties of the upper Lyapunov exponent for a positively invariant compact set (in the terms of Sacker and Sell [24], Chow and Leiva [4,5], and Shen and Yi [25]).
In Section 3 we describe in detail the family of SDDEs and analyze some of its properties. In particular, we prove that every bounded and positively Π-invariant set contains a positively Π-invariant compact subset which is maximal for the property of existence of backward extension of its semiorbits. In the rest of the Introduction, K will be a positively Π-invariant compact subset such that all its elements admit backward extension in it. Such a set K is contained in C 0 , and so we can define the semiflow Π L on K × W 1,∞ by (1.2). In addition, the conditions assumed on the vector field and the standard theory of FDEs ensure that the solutions of the variational equation also define the continuous skew-product semiflow where w(t, ω, x, v) represents the same function as above. We show that, given (ω, x) ∈ K and T ≥ r, the map C → W 1,∞ , v → w(T, ω, x, v) is continuous, and that it is also compact if T ≥ 2r. This property is the main tool in the proof of a result which will be fundamental in the paper: if we follow the classical way to define the upper Lyapunov exponent of K with respect to the pseudo-continuous flow Π L , it agrees with the (classical) one with respect to Π L .
In Section 4 we strength slightly the Lipschitz conditions assumed on τ , and consider a set K as described above, with the additional property that it projects over the whole base. Let λ K be its upper Lyapunov exponent. We prove that the condition λ K < 0 is equivalent to the exponential stability of K for the usual Lipschitz norm, and also to the exponential stability of K expressed in terms of the supremum norm. This extends to our nonautonomous setting results previously proved by Hartung in [8] in the case of periodic SDDEs.
Section 5 considers again the initial conditions assumed on τ , and contains the adequate version of the characterization of the exponential stability. The results are very similar to that of Section 4: the only difference relies in the expression of the exponential stability in terms of the norm in C. In this less restrictive setting, we go further in the analysis. We prove that, if base Ω is minimal and λ K < 0, then K is an m-cover of the base flow (Ω, σ, R) admitting a flow extension. We also establish several properties on its domain of attraction. The paper is completed with the following nice extension: if P is a positively Π-invariant compact set such that λ M < 0 for every minimal set M ∈ P, then P only contains a finite number of minimal sets; and, in addition, the subsets of P determined by its intersection with the domains of attraction of its minimal subsets agree with the connected components of P. A conclusion of all the preceding results closes the paper: the existence of a uniformly exponentially stable solution of a single uniformly almost periodic SDDE ensures the existence of exponentially stable almost periodic solutions.
We close this introduction by pointing out that the conclusions of this paper provide the tools to develop appropriate versions for the context of nonautonomous SDDEs of some applied models described by Arino et al. [1], Smith [26], Wu [29], Hartung et al. [11], Novo et al. [21], Insperger and Stépán [17], and some of the references therein. In particular, the results of this paper are the key point in the extension to the case of nonautonomous SDDEs of the results about exponential stability for biological neural networks of [21], which will be developed elsewhere.

Some preliminaries
In this section we introduce the basic notions of topological dynamics which will be used throughout the paper. They can be found in Sacker and Sell [23,24], Chow and Leiva [4,5], Shen and Yi [25], and references therein.
Let Ω be a complete metric space. A (real, continuous) flow (Ω, σ, R) is defined by a continuous map σ : and its only nonempty compact σ-invariant subset is itself. Zorn's lemma ensures that every compact and σ-invariant set contains a minimal subset. Note that a compact σ-invariant subset is minimal if and only if each one of its orbits is dense. We say that the continuous flow (Ω, σ, R) is recurrent or minimal if Ω itself is minimal. The flow is local if the map σ is defined, continuous, and satisfies (f1) and (f2) (this last one whenever it makes sense) on an open subset O ⊆ R × Ω containing {0} × Ω. And, in the case of a compact base Ω, the flow (Ω, σ, R) is almost periodic if for every ε > 0 there exists δ = δ(ε) > 0 such that, if ω 1 , ω 2 ∈ Ω satisfy d Ω (ω 1 , ω 2 ) < δ (where d Ω is the distance on Ω), then d Ω (σ t (ω 1 ), σ t (ω 2 )) < ε for all t ∈ R.
As usual, we represent R ± = {t ∈ R | ±t ≥ 0}. If σ : R + ×Ω → Ω, (t, ω) → σ(t, ω) is a continuous map which satisfies the properties (f1) and (f2) described above for all t, s ∈ R + , then (Ω, σ, R + ) is a (real, continuous) semiflow . The set {σ t (ω) | t ≥ 0} is the (positive) semiorbit of the point ω ∈ Ω. If this semiorbit is relatively compact, the omega-limit set O(ω) of the point ω ∈ Ω (or of its semiorbit) is the set of limits of sequences of the form . This is the case of all the omega-limit sets. A positively σ-invariant compact set M is minimal if it does not contain properly any positively σ-invariant compact set. If Ω is minimal, we say that the semiflow is minimal. The semiflow is local if the map σ is defined, continuous, and satisfies (f1) and (f2) on an open subset O ⊆ R + × Ω containing {0} × Ω. In this case, the definitions of positively invariant set and minimal set are the same as above. In particular, they are composed of globally defined positive semiorbits, so that the restriction of the semiflow to one of these sets is global. Note that, in the local case, we need to be sure that a semiorbit is (at least) globally defined in order to talk about its omega-limit set.
A continuous semiflow (Ω, σ, R + ) admits a continuous flow extension if there exists a continuous flow (Ω, σ, R) such that σ(t, ω) = σ(t, ω) for all t ∈ R + and ω ∈ Ω. Let M be a positively σ-invariant compact set. A point ω ∈ M admits a backward extension in M if there exists a continuous map θ ω : R − → M such that θ ω (0) = ω and σ(t, θ ω (s)) = θ ω (t + s) whenever s ≤ −t ≤ 0. We will use the words "admits at least a backward extension in M" to emphasize the fact that the extension may be non unique. The set M admits a continuous flow extension if the semiflow restricted to it admits one. It is known that, if the semiorbit of a point ω ∈ Ω is relatively compact, then any element of the omega-limit set O(ω) admits at least a backward extension in O(ω) (see Proposition II.2.1 of [25]); and that, in the case that Ω is locally compact, the existence of a continuous flow extension for M is equivalent to the existence and uniqueness of a backward extension for each of its points (see in Theorem II.2.3 of [25]).
A (local or global, continuous) semiflow is of skew-product type when it is defined on a vector bundle and has a triangular structure. More precisely, let (Ω, σ, R + ) be a global semiflow on a compact metric space Ω, and let X be a Banach space. We will represent ω·t = σ t (ω) = σ(t, ω). A local semiflow (Ω × X, Π, R + ) is a skew-product semiflow with base (Ω, σ, R) and fiber X if it takes the form (2.1) Property (f2) means that the map u satisfies the cocycle property u(t + s, ω, x) = u(t, ω·s, u(s, ω, x)) whenever the right-hand function is defined. It is frequently assumed that the base semiflow is in fact a flow. We will add explicitly this hypothesis when we use it. Now we state some definitions about stability. All of them refer to properties of the skew-product semiflow Π defined by (2.1). The norm on X and the corresponding distance are represented by · X and d X . A compact set K ⊂ Ω × X projects over the whole base if for any ω ∈ Ω there exists x ∈ X such that (ω, x) ∈ K. This is the type of sets on which the concept of stability make sense. Note that this is always the case if K is positively Π-invariant and Ω is minimal.
The next definitions and properties refer to the special case of a linear skewproduct semiflow. A global continuous skew-product semiflow Π is linear if it takes the form Π : 2) where φ(t, ω) is a bounded linear operator on X; in other words, if u(t, ω, x) is linear in x for each (t, ω) ∈ R + × Ω. In what follows, we assume that the base (Ω, σ, R) is a flow (not just a semiflow) on a compact metric space. This hypothesis will be weakened later: see Remark 2.5.
Definition 2.4. The upper Lyapunov exponent λ + s (ω) of ω ∈ Ω for the semiflow (Ω × X, Π, R + ) given by (2.2) is and the upper Lyapunov exponent of the set Ω for the semiflow (Ω × X, Π, R + ) is Proposition 2.1 of [5] proves that λ Ω < ∞, and Theorem 4.2 of [4] shows that In addition, Proposition II.4.1 and Corollary II.4.2 of [25] show that Remark 2.5. We will very often work with a linear skew-product semiflow for which the base (Ω, σ, R + ) is a global semiflow on a compact metric space, with the fundamental property that each one of its elements admits at least a backward extension in Ω. (Recall that this is the situation at least in the case that Ω is minimal, which we do not assume in what follows.) Our next purpose is to show that the previous definitions of Lyapunov exponents and the properties that we will require make sense also in this setting, in which the existence of a flow extension on Ω is not required. Part of the argument is taken from Section II.2.2 of [25] and from Theorem 10 of Chapter 4 of [23]. Let us define that is, the elements of Ω * are the global orbits provided by all the backward extensions of all the elements of Ω. Then Ω * is a compact subset of C(R, Ω) for the compact-open topology of C(R, Ω), which agrees with the topology given by the distance d Ω (ξ 1 (s), ξ 2 (s)) ; that is, Ω * is a compact metric space. Note that we have assumed that for every ω ∈ Ω there exists at least a point ξ ∈ Ω * with ξ(0) = ω. As said before, it is proved in Theorem II.2.3 of [25] that this correspondence is one-to-one if and only if the semiflow (Ω, σ, R + ) admits a continuous flow extension. In this more general setting, it is also possible to define a continuous flow on the set Ω * , called the lifting flow , which, roughly speaking, projects onto Ω. It is given by σ * : R × Ω * → Ω * , (t, ξ) → ξ·t, with (ξ·t)(s) = ξ(t + s). Hence, whenever ω = ξ(0) we have, for t ≥ 0, ω·t = σ(t, ω) = σ(t, ξ(0)) = ξ(t) = σ * (t, ξ)(0) = (ξ·t)(0) . Now we can define which is a continuous linear skew-product semiflow with base flow (Ω * , σ * ), and define the corresponding upper Lyapunov exponent (λ * ) + s (ξ) for ξ ∈ Ω * and λ * Ω * := sup ξ∈Ω * (λ + ) * s (ξ). It is clear that (λ * ) + s (ξ) only depends on ξ(0), which belongs to Ω. In other words, we can define λ + s (ω) and λ Ω directly from Π, as in Definition 2.4, and then we have (λ * ) + s (ξ) = λ + s (ω) for ω = ξ(0), and λ Ω = λ * Ω * . And it is clear that (2.3) and (2.4) are still valid.
Note finally that, if Ω is minimal, then Ω * is also minimal. In order to prove this assertion, we must take ξ 0 , ξ in Ω * , and find a sequence (t m ) in R such that ξ 0 ·t m converges to ξ uniformly on [−k, k] for all k > 0. Let us take ω = ξ(−k) and ω 0 = ξ 0 (−r), use the minimality of Ω to take a sequence (t m ) in R + with ω = lim m→∞ ω 0 ·t m , and deduce from the uniform continuity of σ on , which is the sought-for property.
We complete this section by fixing some notation which will be used throughout the paper. Given two Banach spaces (X, · X ) and (Y, · Y ), Lin(X, Y ) represents the set of bounded linear maps φ : X → Y equipped with the operator norm φ Lin(X,Y ) = sup x X =1 φ(x) Y . The maximum delay of the equations that we will consider is represented by r > 0. The set C represents the Banach space of continuous functions C([−r, 0], R n ) equipped with the norm ψ C := sup s∈[−r,0] |ψ(s)|, where | · | represents the Euclidean norm in R n . The subset C 1 ⊂ C is given by the functions which have continuous derivative on [−r, 0] (one-sided derivatives at the end points of the interval). The set L ∞ is the space of Lebesgue-measurable functions ψ : [−r, 0] → R n which are essentially bounded , which means that there exists k ≥ 0 such that the set {x ∈ [−r, 0] | |ψ(x)| > k} has zero measure. The norm on L ∞ , which is defined as the inferior of the set of real numbers k ≥ 0 with the previous property, is denoted by · L ∞ . The set W 1,∞ is the Banach space of Lipschitz-continuous functions ψ : [−r, 0] → R n equipped with the Lipschitz norm ψ W 1,∞ := max{ ψ C , ψ L ∞ }. Note that Arzelá-Ascoli theorem ensures that any bounded set of W 1,∞ is relatively compact in C. Finally, given a continuous

FDEs with state-dependent delay
Let (Ω, σ, R) be a continuous flow on a compact metric space. As in the previous section, we write ω·t = σ(t, ω) for t ∈ R and ω ∈ Ω. Given F : Ω × R n × R n → R n and τ : Ω × C → [0, r], we consider the family of nonautonomous SDDEṡ for ω ∈ Ω. All or part of the following conditions will be assumed on F and τ : H1 F : Ω × R n × R n → R n is continuous, and its partial derivatives w.r.t. its second and third arguments exist and are continuous on Ω × R n × R n . In particular, the functions D i F : Ω × R n × R n → Lin(R n , R n ) exist and are continuous for i = 2, 3. H2 (1) τ : Ω × C → [0, r] is continuous and differentiable w.r.t. its second argument, with D 2 τ : Ω × C → Lin(C, R) continuous. (2) D 2 τ is locally Lipschitz-continuous in the following sense: for every compact subset K ⊂ Ω × C there exists a constant L 2 = L 2 (K) > 0 such that for all (ω, x 1 ) and (ω, x 2 ) in K.
Remark 3.1. Note that H2(1) ensures the next property: H2 (3) τ is locally Lipschitz-continuous in this sense: for every compact subset for all (ω, x 1 ) and (ω, x 2 ) in K. In order to prove this assertion, we take a compact subset K ⊂ Ω × C and note that the setK = {(ω, s whenever ω ∈ Ω and x 1 , x 2 ∈ K, as asserted. Let us now summarize the most basic properties of the solutions of the equation (3.1) ensured by hypotheses H1 and H2(1). In the statement of the next theorem a fundamental role is played by the set of pairs "(equation, initial datum)" which satisfy the compatibility condition given by the vector field; namely The next result, strongly based on previous properties proved in [7], is proved in Theorem 3.3 and Corollary 3.4 of [20].
Then the restriction of Π to K defines a global continuous semiflow on K.
Note that point (i) states thaṫ where the derivative at t = 0 must be understood as the right-hand derivative. Remark 3.3. As anticipated in the Introduction, we will say that Π is a pseudocontinuous semiflow. The definitions of semiorbit, positively Π-invariant set and of minimal set are the same. Note that the positively Π-invariance of a set M ensures that R + × M ⊆ U. If K is a positively Π-invariant compact set K, then also the definition of existence of backward extension of its element (ω, x) in K is the same. In addition, if a point (ω, x) has bounded Π-semiorbit (which ensures that β ω,x = ∞ and that {(ω·t, u(t, ω, x)) | t ∈ [r, ∞)} ⊂ Ω × W 1,∞ is relatively compact), we can define its omega-limit set as in Section 2: Theorem 3.6(ii) will show that this causes no confusion. Finally, also Definitions 2.1, 2.2 and 2.3 can be directly adapted to Π.
In most of this section, we will be working with a subset K of Ω×W 1,∞ satisfying the following conditions (see Section 2 and Remark 3.3): Hypotheses 3.4. Conditions H1 and H2(1) hold, and K ⊂ Ω×W 1,∞ is a positively Π-invariant compact set such that each one of its elements admits a backward extension in K.
Remark 3.5. If Hypotheses 3.4 hold, then the semiflow (K, Π, R + ) is globally defined and continuous: see Theorem 3.2(ix), and note that we denote with the same symbol Π the restriction Π| K . In addition, the existence of backward extension in K of its elements ensures that K ⊂ C 0 , where C 0 is defined by (3.2).
Such a set K will be fixed once we have proved the next theorem. It shows that any positively Π-invariant bounded set determines a positively Π-invariant compact set; and it explains that each positively Π-invariant compact set contains a maximal subset K satisfying the conditions of Hypotheses 3.4.
is a nonempty positively Π-invariant compact set, and is the maximal subset of K 2 with these properties.
(ii) Theorem 3.2(iii) shows that the classical definition of omega-limit set O(ω, x) of a point (ω, x) with bounded Π-semiorbit makes sense. Since it agrees with the omega-limit set of the point Π(2r, ω, x), we can adapt the proof of (i) to show that O(ω, x) is compact. Its positively Π-invariance follows from Theorem 3.2(vii). Theorem 3.2(ix) ensures that the restricted semiflow (O, Π, R + ) is continuous, and hence Proposition II.2.1 of [25] proves the last assertion in (ii).
(iii) It is clear that the set K 3 is a positively Π-invariant subset of K 2 . Since K 2 contains at least a minimal subset, point (ii) ensures that the set K 3 is nonempty. Therefore, since K 2 is compact, the goal is to check that K 3 is closed. Let us fix a point (ω, x) ∈ closure Ω×W 1,∞ K 3 . We will follow an iterative procedure. The first step is to find a point (ω·(−1), x −1 ) ∈ closure Ω×W 1,∞ K 3 and a continuous map To this end, we take a sequence ((ω m , x m )) in K 3 with limit (ω, x). For each m ∈ N we choose a backward orbit of (ω m , x m ) in K 2 , which we write as {(ω m ·s, θ ωm,xm (s)) | s ≤ 0}. It is clear that (ω·s, θ ωm,xm (s)) ∈ K 3 for any s ≤ 0: its backward orbit is in fact provided by the same map. In addition, θ ωm,xm (0) = x m , and Π(t, ω m ·s, θ ωm,xm (s)) = (ω m ·(t + s), θ ωm,xm (s + t)) whenever s ≤ −t ≤ 0 .
The compactness of K 2 provides a subsequence (ω k , x k ) of (ω m , x m ) such that there exists lim k→∞ (ω k ·(−1), θ ω k ,x k (−1)). We call this limit (ω·(−1), x −1 ) and note that (ω·(−1), x −1 ) ∈ closure Ω×W 1,∞ K 3 . We define , which satisfies the required conditions: the positively Π-invariance of K 2 ensures that it is well defined; it is obvious that . This completes the first step. Now we iterate the process in order to obtain a sequence ((ω·(−j), x −j )) of points in closure Ω×W 1,∞ K 3 and a sequence of continuous functions It is not hard to deduce from these facts that the continuous map θ ω,x : R − → K 2 obtained by concatenating the previous maps is a backward extension of (ω, x) in K 2 . This completes the proof of the compactness of K 3 . The last assertion of (iii) is obvious.
As said before, in the rest of the section we fix a set K satisfying Hypotheses 3.4 (see also Remark 3.5). Let us define L : and associate to (3.1) the family of linear variational equationṡ for (ω,x) ∈ K. Let us summarize the strategy of the remaining part of this section. The solutions of this family of linear FDEs (of time-dependent delay type) will allow us to define two semiflows on two different bundles with base (K, Π, R + ). More precisely, on K × W 1,∞ and on K × C. Corollary 4.3 of [20] states that the first one is pseudo-continuous and the second one continuous. The assumptions made on K ensure that the construction made in Remark 2.5 applies to both semiflows, despite the lack of global continuity of the first one. In particular, it makes sense to talk about the upper Lyapunov exponents of these two linear skew-product semiflows for which (K, Π, R + ) is the base. It is also proved in [20] (see Theorem 3.7 below) that the first semiflow is that usually called the linearized semiflow of Π. This means that the corresponding upper Lyapunov exponent (which can be defined despite the possible noncontinuity of the semiflow) is that which responds to the classical concept. But it also turns out that the second upper Lyapunov exponent is often "easier to handle". Theorem 3.10 solves the disjunctive: it shows that in fact these two quantities agree. We will now describe the two mentioned semiflows. First, for each (ω,x) ∈ K and v ∈ W 1,∞ , we denote by z(t, ω,x, v) the solution of (3.8) with initial condition v (that is, with z(s, ω,x, v) = v(s) for each s ∈ [−r, 0]), which is defined for all t ≥ −r and is linear with respect to v. In addition, the map K → Lin(W 1,∞ , R n ), (ω,x) → L(ω,x) is continuous. These properties allow us to define a global linear skewproduct semiflow on the set K × W 1,∞ by where w(t, ω,x, v)(s) = z(t + s, ω,x, v) for all s ∈ [−r, 0] and t ≥ 0. As said before, Corollary 4.3 of [20] proves that this semiflow is pseudo-continuous. In particular, for all (t, ω,x) ∈ R + × K, the linear map is continuous.
There is a strong relation between the semiflows Π and Π L , as the next result shows. It is proved in Theorem 4.4 of [20], in turn based on Theorems 2 and 4 of [7].  H1 and H2(1) hold. Let us fix (ω, x) The definition of the second semiflow is now given. For each (ω,x) ∈ K and v ∈ C, let z(t, ω,x, v) denote the solution of (3.8) with initial condition v. Corollary 4.3 of [20] proves that the solutions of (3.8) induce a global continuous linear skewproduct semiflow on the set K × C, defined by which is a linear continuous map for all (t, ω,x) ∈ R + × K. Note that 14) It is easy to deduce from the fact that z(t, ω,x, v) solves (3.8), from Hypotheses 3.4 and from the expression of L(Π(t, ω,x)) obtained from (3.7) that, if (ω,x) ∈ K and v ∈ C, then w(t, ω,x, v) ∈ W 1,∞ for t ≥ r. This means that the map π L (t, ω,x), defined on C by (3.13), takes values in W 1,∞ for t ≥ r. The next goal is to check that this map is continuous when it is defined from C to W 1,∞ . This property will be used in the proof of Theorem 3.10.
The last assertion of the proposition is an immediate consequence of the previous one and of the fact that any bounded sequence in W 1,∞ determines a bounded sequence in C. This completes the proof.
For further purposes, we point out that Proposition 3.8 allows us to assert that is finite, where π L is defined by (3.15). In fact, it follows from (3.17) that C r ≤ C r (1 + C 0 ), with C 0 and C r given by (3.18). Despite the possible lack of continuity of the semiflow Π L defined by (3.9), Definition 2.4 provides two well-defined values, which we call and denote upper Lyapunov exponent λ + s (ω,x) of (ω,x) ∈ K for (K, Π L , R + ) and upper Lyapunov exponent λ K of the semiflow (K, Π L , R + ). We also denote by λ + s (ω,x) the upper Lyapunov exponent of Π L (given by (3.12)) for (ω,x) ∈ K; and by λ K the upper Lyapunov exponent of (K, Π L , R + ). Remark 3.9. Despite the lack of classic continuity of the semiflow Π L , we can repeat the arguments of Theorem 4.2 of [4] in order to check that, if (ω,x) ∈ K, then λ + s (ω,x) = lim sup The next theorem shows that the upper Lyapunov exponents of both semiflows coincide.

Exponential stability of invariant compact sets
The structure of Sections 4 and 5 is similar: we establish conditions characterizing the exponential stability of the positively invariant compact subsets of the pseudo-continuous semiflow Π defined on Ω × W 1,∞ by (3.3) in terms of the corresponding upper Lyapunov exponents. The hypotheses assumed in this section are more restrictive than those of the next one, and they allow us to obtain stronger conclusions: compare the statements of Theorems 4.2 and 5.2. Although part of the hypotheses are common, we write down now the whole list for the reader's convenience. Recall that (Ω, σ, R) is a continuous flow on a compact metric space, with ω·t = σ(t, ω). H1 F : Ω× R n × R n → R n is continuous, and its partial derivatives with respect to the second and third arguments exist and are continuous on Ω×R n ×R n . In particular, the functions D i F : Ω × R n × R n → Lin(R n , R n ) exist and are continuous for i = 2, 3. H2 * (1) τ : Ω × C → [0, r] is continuous and differentiable in the second argument, with D 2 τ : Ω × C → Lin(C, R) continuous. (2) τ is locally Lipschitz-continuous in this sense: for every bounded and closed subset B ⊂ C there exists a constant L 1 = L 1 (B) > 0 such that for all ω ∈ Ω and x 1 , x 2 ∈ B. (3) D 2 τ is locally Lipschitz-continuous in the following sense: for every bounded and closed subset B ⊂ C there exists a constant L 2 = L 2 (B) > 0 such that for all ω ∈ Ω and x 1 , x 2 ∈ B.
Let Π be defined by (3.3) from the family (3.1) of FDEs. Throughout this section, we will work under Hypotheses 4.1. Conditions H1 and H2 * hold, and K ⊂ Ω × W 1,∞ is a positively Π-invariant compact set projecting over the whole base and such that each one of its elements admits at least a backward extension in K.
Recall that the semiflow (K, Π, R + ) is global and continuous, and that K ⊂ C 0 : see Remark 3.5. The goal of this section is to prove that the exponential stability of K can be characterized in terms of its upper Lyapunov exponent by the condition λ K < 0. We will also show that the property of the exponential stability can be formulated either in terms of the W 1,∞ -norm or of the C-norm. These two results are stated in the following theorem, whose proof requires three preliminary technical lemmas. Recall Definition 3.11 of λ K , and that Theorem 3.10 shows that it is the upper Lyapunov exponent both form the linearized semiflow given by (3.9) on K × W 1,∞ and by (3.12) on K × C; in fact, both definitions of λ K will be used in the proof. It is interesting to remark that part of this result and the corresponding proof could be somehow standard if the semiflow Π were C 1 on an open neighborhood of K. But the assumptions of this paper do not allow us to deduce this condition. (1) λ K < 0.
Before stating and proving the mentioned lemmas, we fix some real parameters and a set which will play an important role in what follows. We define and represent by L 0 1 and L 0 2 the Lipschitz constants of the functions τ and D 2 τ on Ω × B 0 , respectively provided by conditions H2 * (2) and H2 * (3). We also denote It follows from H2 * (3) that D 2 τ 0 ≤ L 0 2 + sup{ D 2 τ (ω,x) Lin(C,R) | (ω,x) ∈ K}, so that it is finite. Condition H1 ensures the same property for |F | 0 , D 2 F 0 and D 3 F 0 . We assume without restriction that the six constants L 0 1 , L 0 2 , |F | 0 , D 2 F 0 , D 3 F 0 and D 2 τ 0 are strictly positive.

Thus,
so that the assertion of the lemma is equivalent to the property The chain rule implies that p t is continuously differentiable, and thaṫ .
The notation (4.3) is used in this statement.
We can finally prove the main theorem of this section.

Proof of Theorem 4.2.
(1)⇒(2) We consider the linear systeṁ where L is defined by (3.7). Let U (t, ω,x) be the fundamental solution of (4.13) in the terms given in Chapter 1 of [6]; i.e., for each (ω,x) ∈ K the n × n matrix-valued map t → U (t, ω,x) is a solution ofU (t) = L(Π(t, ω,x))U t for t ≥ 0, and it satisfies for all (ω,x) ∈ K. Here I n and 0 n are the n × n identity and zero matrices. We assume that λ K < 0, fix any β ∈ (0, −λ K ), and choose α with β < α < −λ K . Theorem 3.10 together with the expression (3.13) of the flow on K × C and relation (2.4) ensures the existence of a constant k 0 ≥ 1 such that w(t, ω,x, v) C = π L (t, ω, x) v C ≤ k 0 e −αt v C and |U (t, ω,x) c| ≤ k 0 e −α t |c| for every (ω,x) ∈ K, v ∈ C, c ∈ R n and t ≥ 0.
In order to check that last assertion of the theorem it is enough to have a look to the choice of β in the proof of (1)⇒ (2), and observe that the value of β in (3) is the same one as in (2).

Weakening the hypotheses
Let Π be the semiflow defined on Ω × W 1,∞ by (3.3) from the family (3.1) of FDEs. In this section we work under the following assumptions, which are less restrictive than those of the preceding one: Hypotheses 5.1. Conditions H1 and H2 hold, and K ⊂ Ω × W 1,∞ is a positively Π-invariant compact set projecting over the whole base and such that each one of its elements admits at least a backward extension in K.
As in the preceding sections, the set K will be fixed throughout most of this one. The first purpose now is to adapt to this less restrictive setting the characterization of the exponential stability of K in terms of its upper Lyapunov exponent. The difference with respect to Theorem 4.2 relies on the second equivalent condition, which characterizes the exponential stability in terms of · C instead of · W 1,∞ . To formulate it, we call (5.1) Theorem 5.2. Suppose that Hypotheses 5.1 hold. and let λ K and ρ 0 be respectively given by Definition 3.11 and (5.1). The following statements are equivalent: (2) There exists β > 0 satisfying the following property: if we fix ρ > ρ 0 , there exist constants and so that u(t, ω, x) − u(t, ω,x) C ≤ k 1 e βr e −βt x −x C for all t ≥ 0 .
The proof of this theorem reproduces basically that of Theorem 4.2. It is also based on three lemmas (Lemmas 5.4, 5.5 and 5.6), whose statements are very similar to those of Section 4 and whose proofs are almost identical. Just a little of previous work is required in order to adapt everything to the less restrictive hypotheses we are considering now. Given any γ > 0, we denote which is a compact subset of C, and represent by L γ 1 and L γ 2 the Lipschitz constants of the functions τ and D 2 τ on Ω × B γ , respectively provided by conditions H2(3) and H2 (2). As in Section 4, we take r 0 := 1 + sup{ x C | (ω,x) ∈ K} , and define Now we fix ρ > ρ 0 and define ρ * := r 0 + |F | 0 + ρ 0 + ρ , To check that D 2 τ 0 < ∞, we note that it agrees with the supremum of D 2 τ on a relatively compact subset of Ω × C, which is finite by condition H2(2). We assume without restriction that |F | 0 , D 2 F 0 , D 3 F 0 , and D 2 τ 0 are strictly positive. Lemma 5.3. Suppose that Hypotheses 5.1 hold, and fix ρ > ρ 0 . We fix (ω,x) ∈ K and (ω, x) ∈ Ω × B ρ . Then, for a time T ∈ (0, β ω,x ), then (5.7) The notation (4.3) is used in this statement.
Lemma 5.5. Suppose that Hypotheses 5.1 hold, and fix ρ > ρ 0 . Then, for every ε > 0 there exists The notation (4.3) is used in this statement.

The notation (4.3) is used in this statement.
This completes the summary of ideas regarding the proof of Theorem 5.2.
The following consequence of Theorem 5.2 in the case of minimal base flow will play a fundamental role in the rest of the paper. Recall that the set K is a k-cover of (Ω, σ, R) if each fiber K ω := {x ∈ W 1,∞ | (ω, x) ∈ K} contains exactly k elements.
Corollary 5.7. Suppose that the base flow (Ω, σ, R) is minimal, that Hypotheses 5.1 hold, and that λ K < 0. Then, there exists k ∈ N such that K is a k-cover of (Ω, σ, R), and the semiflow (K, Π, R + ) admits a flow extension. In addition, (i) for each ω ∈ Ω there exist a neighborhood U ω ⊂ Ω of ω and k continuous maps x 1 , . . . , x k : U ω → W 1,∞ such that for all ω ∈ U ω . (ii) The set K is the disjoint union of a finite number of minimal sets M 1 , . . . , M l , where M j is an exponentially stable m j -cover of the base for j = 1, . . . , l.
Proof. Theorem 5.2 ensures that K is exponentially stable, so that it is uniformly asymptotically stable. Theorem 3.5 of Novo et al. [22], which is based on previous results of Sacker and Sell [23], proves that K is a k-cover of the base for a k ∈ N. The fact that (K, Π, R + ) admits a flow extension follows for instance from Theorem 3.4 of [22].
(i) This assertion can be easily proved by combining two facts: first, the closed character of K ensures the continuity of the map ω → K ω in the Hausdorff topology of the set of compact subsets of W 1,∞ (see Theorem 3.3 of [22]); and second, K ω always contains k elements.
(ii) Let M ⊆ K be a minimal set. It is obvious that λ M < 0, and hence, as we have already proved, M is an exponentially stable m-cover of the base with m ≤ k. It is easy to deduce from the existence of flow extensions on K and M that K − M is also positively Π-invariant. Let us now take a sequence (ω k ) with limit ω. Theorem of [22] ensures that (K ω k ) and (M ω k ) respectively converge to K ω and M ω in the Hausdorff topology of the set of compact subsets of W 1,∞ , and it is not hard to deduce from here that ((K − M) ω k ) converges to (K − M) ω , and hence that K − M is a positively Π-invariant compact set. Obviously, λ K−M < 0. Altogether, we see that the set K − M satisfies the same conditions as K, so that it is a (k − m)-cover of the base. Repeating the process a finite number of times (at most k − 1) leads us to the desired conclusion.
In order to prove that D(M) is connected, write D(M) ⊆ V 1 ∪ V 2 for two disjoint open subsets V 1 and V 2 of Ω × W 1,∞ . Since M is connected (as any minimal set), then it is contained in one of these sets, say M ⊂ V 1 . But any point (ω, x) ∈ D(M) ∩ V 2 is connected with V 1 by a positive semiorbit, which together with the positively Π-invariance of D(M) shows that D(M) ∩ V 2 is empty. The conclusion is that D(M) is connected, which completes the proof of (i).
It follows easily from Corollary 5.7 that D 0 (M) is an open subset of D(M). Note also that there exists t 0 > 0 such that Π(t, D 0 (M)) ⊆ D 0 (M) for all t ≥ t 0 , as easily deduced from Theorem 5.2 and the definition of δ 2 .
(ii) Suppose that P contains two different minimal subsets M 1 and M 2 . It is obvious that D(M 1 ) and D(M 2 ) are disjoint. On the other hand, it follows from (i) that P is contained in the union of the domains of attraction of all its minimal subsets, each one of which is an open set with nonempty intersection with P. Hence, (ii) follows from compactness of P.
(iii) It follows from (i) and (ii) that P = l j=1 P ∩ D(M j ) . Our first goal is to prove that the positively Π-invariant set P ∩ D(M j ) is closed (and hence compact) for j = 1, . . . , l. Let us fix j ∈ {1, . . . , l} and take a sequence ((ω m , x m )) ∈ P ∩ D(M j ) with limit ( ω, x) ∈ P. We look for k ∈ {1, . . . , l} such that ( ω, x) ∈ P ∩ D(M k ). Since this set is open in P, there exists m 0 such that (ω m0 , x m0 ) ∈ P ∩ D(M k ), and since D(M j ) ∩ D(M k ) is empty if k = j, then k = j. That is, P ∩ D(M j ) is closed, as asserted.
In order to prove that each set P ∩ D(M j ) is connected, we assume by contradiction that for an index j ∈ {1, . . . , l} we can write P ∩ D(M j ) ⊂ V 1 ∪ V 2 for two disjoint open subsets V 1 and V 2 of Ω × W 1,∞ . Since M j is a connected subset of P ∩D(M j ), we can assume without restriction that M j ⊂ V 1 . And since P ∩D(M j ) is a positively Π-invariant subset of D(M j ), we conclude that P ∩ D(M j ) ∩ V 2 is empty. This completes the proof of (iii), The last assertion of the theorem follows trivially from (iii) together with the open character of D(M j ) ensured by Proposition 5.8(i).
Remark 5.10. It is important to emphasize the fact that, if the flow (Ω, σ, R) is almost periodic and M is a minimal m-cover of the base admitting a flow extension, then the flow (M, Π, R) is also almost periodic: see [23], Theorem 6 of Chapter 3. Therefore, Corollary 5.7 ensures the following property. Assume that our family (3.1) is constructed by the usual hull procedure (summarized in the Introduction) from a single FDEẏ(t) = f (t, y(t), y(t − τ (t, y t ))) given by a uniformly almost periodic pair (f, τ ). Then the existence of a minimal set M ⊂ Ω × W 1,∞ with λ M < 0 ensures the existence of exponentially stable almost-periodic solutions of the initial system. Note also that the existence of such a set M is ensured by the existence of a bounded and uniformly exponentially stable solution on [0, ∞) of the initial system: it is easy to check that the omega-limit set of such a solution for the flow Π associated to the family of FDE defined on the corresponding hull Ω is a minimal subset of Ω × W 1,∞ (just repeat the proof of Theorem 5.9(i)), which in addition is exponentially stable.