One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates

This paper is concerned with the analysis of a one dimensional wave equation $z_{tt}-z_{xx}=0$ on $[0,1]$ with a Dirichlet condition at $x=0$ and a damping acting at $x=1$ which takes the form $(z_t(t,1),-z_x(t,1))\in\Sigma$ for every $t\geq 0$, where $\Sigma$ is a given subset of $\mathbb R^2$. The study is performed within an $L^p$ functional framework, $p\in [1, +\infty]$. We aim at determining conditions on $\Sigma$ ensuring existence and uniqueness of solutions of that wave equation as well as strong stability and uniform global asymptotic stability of its solutions. In the latter case, we also study the decay rates of the solutions and their optimality. We first establish a one-to-one correspondence between the solutions of that wave equation and the iterated sequences of a discrete-time dynamical system in terms of which we investigate the above mentioned issues. This enables us to provide a simple necessary and sufficient condition on $\Sigma$ ensuring existence and uniqueness of solutions of the wave equation as well as an efficient strategy for determining optimal decay rates when $\Sigma$ verifies a generalized sector condition. As an application, we solve two conjectures stated in the literature, the first one seeking a specific optimal decay rate and the second one associated with a saturation type of damping. In case the boundary damping is subject to perturbations, we derive sharp results regarding asymptotic perturbation rejection and input-to-state issues.

and hence the corresponding condition on z x (t, 1) and z t (t, 1) reduces to z x (t, 1) = −σ(z t (t, 1)), ∀t ≥ 0, (2) which can be interpreted as a feedback law prescribing z x (t, 1) in terms on z t (t, 1) at the boundary x = 1 for every t ≥ 0. Note that considering Σ more general that the mere graph of a function is a possible alternative to model the fact that the function σ is subject to uncertainties or discontinuities, as in the case, for instance, where Σ is equal to the graph of the sign set-valued map sgn : R ⇒ R defined by sgn(s) = {s/|s|} for nonzero s and sgn(0) = [−1, 1], cf. [27]. The use of a set Σ in (1) can also model switching boundary conditions, such as those considered in [3,8,12,16]. In this setting, the boundary condition is usually written as z x (t, 1) = −σ a(t) (z t (t, 1)), where a(·) is piecewise constant and takes values in a given (possibly infinite) index set I and σ i : R → R for i ∈ I. Defining Σ as the set of pairs (x, y) ∈ R 2 such that y ∈ σ i (x) for some i ∈ I, one obtains that any solution of the wave equation with the boundary condition (3) is a solution of (1). This construction is the analogue of the classical transformation of finite-dimensional switched systems into differential inclusions (see, e.g., [4,11,18]).

Existing results
The standard issues addressed for solutions of (1) (either with (2) or Σ equal to the graph of sgn) may be divided into three main questions: find conditions on σ so that (Q1) for every initial condition, there exists a global and possibly unique solution of (1), (Q2) in case solutions of (1) tend to zero as the time t tends to infinity, one can characterize their decay rates, and (Q3) one can try to establish optimality of these decay rates, where optimality is defined more precisely below as in [26]. Question (Q1) is usually addressed within a Hilbertian framework, i.e., (weak) solutions of (1) belong to X 2 = W 1,2 * (0, 1) × L 2 (0, 1) where W 1,2 * (0, 1) is the Hilbert space made of the absolutely continuous functions u : [0, 1] → R so that u ′ ∈ L 2 (0, 1) and u(0) = 0. Functional analysis arguments are then used, typically by considering appropriate unbounded operators and their associated C 0 semigroups. Most of the time, the function σ is assumed to be locally Lipschitz, nondecreasing and subject to the classical damping condition, i.e., sσ(s) ≥ 0 for every s ∈ R, which allows one to get a maximal monotone operator, and hence bringing a positive answer to (Q1).
Let us emphasize on the damping condition, since not only it helps to address (Q1) but it is also a first step to handle (Q2). Indeed, such a condition on σ makes the natural energy E(t) of (1) defined by E(t) = 1 0 |z x (t, x)| 2 + |z t (t, x)| 2 dx nonincreasing along trajectories of (1). This is why one usually refers to such a function σ as a damping function. Note that [25] (as other few earlier works mentioned in that reference) assumes σ to be a damping function not necessarily monotone. Addressing (Q1) in that case relies instead on the d'Alembert decomposition of solutions of (1).
Other functional frameworks have been considered recently [2,7,14], where the functional spaces are of L p -type, p ∈ [1, +∞], but these works consider 1D wave equations with localized distributed damping, i.e., z tt − z xx = −a(x)σ(z t ). Recall that the semigroup generated by the D'Alembertian z := z tt − ∆z with Dirichlet boundary conditions on an open bounded subset in R n , n ≥ 2, is not defined for any suitable extension of the Hilbertian framework to L p -type spaces for p = 2, as explained in [24]. This is why the study of issues of asymptotic behavior involving L p spaces with p = 2 makes sense only in the one-dimensional case.
Regarding more precisely results obtained for (Q2), there exist two main concepts of convergence of solutions of (1) to zero: the basic one referred to as strong stability, which says that the X 2 -norm of every solution of (1) tends to zero as t tends to infinity and a stronger notion, that of uniform globally asymptotic stability (UGAS for short), which says that there exists a KL-function β : R + × R + → R such that z(t) X 2 ≤ β( z(0) X 2 , t) for every t ≥ 0 and solution z(·) of (1). Recall that a KL-function (or a function of class KL) β is continuous with β(0, ·) ≡ 0, increasing with respect to its first argument and, for every s ≥ 0, t → β(s, t) is decreasing and tends to zero as t tends to infinity. One may interpret the function β as a generalized rate of convergence to zero of solutions of (1) and one may even ask what could be the "best" KL function β for which UGAS holds true. After [26] a reasonable definition that we will adopt in the paper goes as follows: we say that a KL function β is optimal for (1) if the latter is UGAS with rate β and there exists an initial condition in X 2 yielding a nontrivial solution z of (1) so that z(t) X 2 ≥ εβ( z(0) X 2 , t) for some positive constant ε and every t ≥ 0.
With the exception of [25], results on strong stability rely on a LaSalle argument and assume that σ is nondecreasing and locally Lipschitz: strong stability is first established for a dense and compactly embedded subset of X 2 made of regular solutions of (1) and then it is extended to the full X 2 by a density argument, cf. [1]. As regards UGAS, it can be shown that, under the damping assumption and a linear cone condition (i.e., there exist positive a, b such that as 2 ≤ sσ(s) ≤ bs 2 for every s ∈ R), the stability is exponential, i.e., one can choose β(s, t) = Cse −µt for some positive constants C, µ, cf. [26] for instance. If the linear cone condition only holds in a neighborhood of zero then exponential stability cannot hold in general, as shown in [26] where σ is chosen as a saturation function, i.e., such as σ(s) = arctan(s). Besides the linear cone condition, several results establishing UGAS have been obtained, cf. [17,19,20,27] where σ verifies a linear cone condition for large s and is either of polynomial type or weaker than any polynomial in a neighborhood of the origin, see also [26] for a extensive list of references. It has to be noticed that many of these studies deal with wave equations in dimension not necessarily equal to one and, for all of them, the estimates are obtained by refined arguments based on the multiplier method or highly nontrivial Lyapunov functionals.
Finally, for results handling (Q3), most of the existing results are gathered in [26] and [1] (cf. Theorems 1.7.12, 1.7.15 and 1.7.16 in the last reference) where several of the above mentioned upper estimates (of UGAS type) are shown to be optimal in the sense defined previously and more particularly in the case where the damping function σ in (4) is of class C 1 and verifies σ ′ (0) = 0. In particular, a list of examples for which optimality is shown is provided in [1,Theorem 1.7.12] while an example is also given (Example 5 in that list) for which only an upper estimate is given and it is stated in [1] that the general case (even under the condition σ ′ (0) = 0) is still open. In these references, the upper estimates are derived by delicate manipulations of Lyapunov functions relying on the multiplier method and lower estimates results are obtained with appropriate solutions of (1) with piecewise constant Riemann invariants, where the computations are actually similar in spirit to those of [25].

Discrete-time dynamical system
The approach proposed in this paper is inspired by [25], since, as in that reference, we focus on the Riemann invariants of a solution of (1). To describe our basic finding on the matter of interest, let us consider the discrete-time dynamical system S defined on the Hilbert space Y 2 = L 2 (−1, 1) which associates with every h ∈ Y 2 the subset S(h) of Y 2 made of the functions j so that S : (h(s), j(s)) ∈ RΣ, for a.e. s ∈ [−1, 1], where R is the planar rotation of angle −π/4. One should notice that a related discretetime dynamical system has been first characterized in [25] (see also Remark 9 below). We show that there exists an isometry of Hilbert spaces I : X 2 → Y 2 such that, for every (z 0 , z 1 ) ∈ X 2 , (1) admits a (global in time) weak solution z in X 2 starting at (z 0 , z 1 ) if and only if there exists a sequence (g n ) n∈N of elements in Y 2 with g 0 = I(z 0 , z 1 ) such that g n+1 ∈ S(g n ) for n ∈ N. The concatenation of the g n 's, which yields an element of L 2 loc (−1, +∞), is exactly the Riemann invariant zt−zx √ 2 associated with z. It is immediate that the previously described correspondence between (1) and (4) can be adapted for any p ∈ [1, +∞] after replacing Y 2 by Y p = L p (−1, 1) and X 2 by X p defined as X p := W 1,p * (0, 1) × L p (0, 1), where W 1,p * (0, 1) = {u ∈ L p (0, 1) | u ′ ∈ L p (0, 1) and u(0) = 0}, and equipped with the norms (u, v) p Xp := The norm · Xp , previously used, for instance, in [14], is equivalent to the standard norm in X p , but it has the advantage of being well-adapted to the analysis of wave equations, since it is expressed in terms of Riemann invariants and it is nonincreasing as soon as Σ satisfies a damping condition (see Proposition 27). In addition, with this choice of norm, the mapping (still denoted by) I : X p → Y p becomes an isometry between Banach spaces. One can reformulate appropriately the definition of S by considering the set-valued map S : R ⇒ R whose graph is RΣ ⊂ R 2 . For instance (4) reads j(s) ∈ S(h(s)) for a.e. s ∈ [−1, 1]. A first interesting application of this formalism concerns the case where Σ is equal to the graph of the sign set-valued map. Then the set-valued map S becomes single-valued and equal to the odd function defined on R + by S(s) = s for s ∈ [0, 1/ √ 2] and S(s) = √ 2−s for s ≥ 1/ √ 2. We also show how one can easily adapt the previously described approach to the case where the Dirichlet boundary condition at x = 0 in (1) is replaced by the more general boundary condition (z t (t, 0), z x (t, 0)) ∈ Σ ′ , t ≥ 0, where Σ ′ ⊂ R 2 .
By using the one-to-one correspondence between (global in time) solutions of (1) and iterated sequences of (4), we can translate the previously described issues of (Q1)-(Q3) associated with the original 1D wave equation (1) into the study of equivalent questions in terms of S. Question (Q1) of existence and uniqueness of solutions of (1) in X p for every initial condition is equivalently restated as follows: determine conditions on Σ so that, for every h ∈ Y p , there exists some (possibly unique) j ∈ Y p such that j ∈ S(h). Similarly, questions regarding the decay rates, i.e. (Q2) and (Q3), can be equivalently simply expressed as questions regarding the asymptotic behavior of iterated sequences (g n ) n∈N associated with S. the decay of solutions of (1) depending on the value of Q ′ (0) ∈ [0, 1]. First of all, we are able to recover all the cases listed in [1,Theorem 1.7.12], even for Example 5, which was left open, all of them corresponding to situations where Q ′ (0) = 1. Note that we obtain the previously known cases with simpler arguments and we also characterize the largest possible sets of initial conditions admitting optimal decay rates. If Q ′ (0) ∈ (0, 1), one has (local) exponential stability, a decay rate which is optimal. The handling of this case is rather elementary and is known in the literature, as least in the Hilbertian case p = 2. In the particular case where Q(s) = µs for every s ≥ 0 for some µ ∈ (0, 1), we provide alternative arguments for exponential stability and also give a necessary and sufficient and condition in terms of S only for exponential stability to hold true. Our results concerning the case Q ′ (0) = 0 seem to be new and exhibit convergence rates faster than any exponential ones.
We close the set of our findings on decay rates with the solution of a conjecture formulated in [26] asking whether arbitrary slow convergence is possible in case Σ is a damping set of saturation type, i.e., the values of |y| for (x, y) ∈ Σ and |x| large remain bounded by a given positive constant. We bring a positive answer for the possible occurrence of such an arbitrarily slow convergence in any space X p , p ∈ [1, +∞).
We also have sharp results when the wave equation (1) is subject to boundary perturbations, i.e., the condition (z t (t, 1), −z x (t, 1)) ∈ Σ for t ∈ R + is replaced by (z t (t, 1), −z x (t, 1)) ∈ Σ + d(t) for t ∈ R + , where d : R + → R 2 is a measurable function representing the perturbation. We provide two sets of results, the first one dealing with asymptotic perturbation rejection (i.e., conditions on d and Σ so that solutions of (1) converge to zero despite the presence of d) and another set of results proposing sufficient conditions on Σ ensuring input-to-state stability for the perturbed wave equation (cf. [21] for a definition of input-to state-stability).
Finally, we revisit the case where Σ is equal to the graph of the sign set-valued map sgn and extend all the results obtained in [27] regarding this question. In particular, we provide optimal results for existence and uniqueness of solutions of (1) in any X p , p ∈ [1, +∞] without relying on semigroup theory and we characterize the ω-limit set of every solution of (1) in an explicit manner in terms of the initial condition.

Structure of the paper
The paper is organized in six sections and four appendices. After the present introduction describing the contents of our paper and gathering the main notations, Section 2 is devoted to the description of the precise correspondence between the wave equation described by (1) and the discrete-time dynamical system given in (4), as well as the list of meaningful hypotheses one can assume on Σ and auxiliary results on the set-valued map S. Section 3 provides results on the existence and uniqueness of solutions of (1). Section 4 gathers results dealing with stability concepts, asymptotic behavior, decay rates, and their optimality for solutions of both iterated sequences of S and solutions of (1). Section 5 treats the case of boundary perturbations. Section 6 addresses the situation where Σ is equal to the graph of the sign set-valued map sgn. The four appendices collect lemmas and technical arguments used in the core of the text.

Notations
The sets of integers, nonnegative integers, real numbers, nonnegative real numbers, and nonpositive real numbers are denoted in this paper respectively by Z, N, R, R + , and R − . For A ∈ {Z, N, R, R + , R − }, we use A * to denote the set A \ {0}.
For x ∈ R, we use ⌊x⌋ and ⌈x⌉ to denote, respectively, the greatest integer less than or equal to x and the smallest integer greater than or equal to x. The set R d , d ∈ N * , is assumed to be endowed with its usual Euclidean norm, denoted by |·|, and, given M > 0, B(0, M ) denotes the ball of R d centered at zero and of radius M . If A ⊂ R, we define A = sup a∈A |a|. All along the paper, we use the letter R to denote the matrix corresponding to the plane rotation of angle − π 4 , i.e., The identity function of R is denoted by Id and, for Σ ⊂ R 2 and d ∈ R 2 , we define the sum Σ + d as the set {x + d | x ∈ Σ}.
A set-valued map F : R ⇒ R is a function that, with each x ∈ R, associates some (possibly empty) F (x) ⊂ R, and its graph is the set i.e., the graph of F contains the graph of a function ϕ : R → R. It is said to be single-valued when F (x) is a singleton for every x ∈ R, i.e., the graph of F is the graph of a function ϕ : R → R. In that case, we usually make the slight abuse of notation of considering F = ϕ.
The composition of two set-valued maps S and T is the set-valued map S•T : R ⇒ R which, to each x ∈ R, associates the set of points z ∈ R such that there exists y ∈ R for which z ∈ S(y) and y ∈ T (x).
Consider a function f : R → R. Then, for n ∈ N, the n-th iterate of f , i.e., the composition of f with itself n times, is denoted by f [n] , with the convention that f [0] = Id. This notation is extended in a straightforward manner to set-valued functions F : R ⇒ R: y ∈ F [n] (x) if and only if there exists x 0 , . . . , x n ∈ R such that x 0 = x, x n = y, and x i+1 ∈ F (x i ) for every i ∈ {0, . . . , n − 1}.
Several notions of measurability are used in some parts of the paper (see Appendix A). Unless otherwise specified, the word "measurable" means "Lebesgue measurable". For an interval I ⊂ R, d ∈ N * , and p ∈ [1, +∞], the space L p (I, R d ) is endowed with the norm defined by The space R d is omitted from the notation when d = 1. We use Y p to denote the space L p (−1, 1) and we write its norm simply by · p . A function γ : R + → R + is said to be of class K if γ is continuous, increasing, and γ(0) = 0. If moreover lim x→+∞ γ(x) = +∞, we say that γ is of class K ∞ .
A function β : R + × R + → R + is said to be of class KL if it is continuous, β(·, t) is of class K for every t ∈ R + , and, for every x ∈ R + , β(x, ·) is decreasing and lim t→+∞ β(x, t) = 0.
2 Description of the model

Equivalent discrete-time dynamical system
In order to introduce a notion of weak solution of (1) adapted both to L p spaces and to the one-dimensional case, let us recall the following classical result on regular solutions to the one-dimensional wave equation, which corresponds to its d'Alembert decomposition into traveling waves (see, e.g., [10, Section 2.4.1.a]). 1] if and only if there exist functions f ∈ C 1 ([0, +∞)) and g ∈ C 1 ([−1, +∞)) such that Proof. Assume that z satisfies z xx = z tt in R + × [0, 1] and let u, v : one can easily check that f ∈ C 1 ([0, +∞)), g ∈ C 1 ([−1, +∞)), and u(t, x) = f (t + x) and v(t, x) = g(t − x) for every (t, x) ∈ R + × [0, 1]. In particular, it follows from (8) that Hence as required. Conversely, if z is given by (7), it is easy to see that z tt = z xx .
The functions f and g from (9) are called Riemann invariants in the classical literature of hyperbolic PDEs (see, for instance, [5]). Proposition 1 motivates the following definition of weak solution to (1). Definition 2. Let (z 0 , z 1 ) ∈ X p . We say that z : R + × [0, 1] → R is a weak global (in time) solution of (1) in X p with initial condition (z 0 , z 1 ) if there exist f ∈ L p loc (0, +∞) and g ∈ L p loc (−1, +∞) such that In that case, we use e p (z)(t) to denote the X p norm of the weak global solution z of (1) at time t ∈ R + , defined by Note that, if z is a weak global solution of (1) in X p , then (z(t, ·), z t (t, ·)) ∈ X p for every t ∈ R + and z tt = z xx is satisfied in R * + × (0, 1) in the sense of distributions. In the sequel, we refer to weak global solutions of (1) simply as solutions or trajectories of (1) and, by a slight abuse of expression, we refer to e p (z)(t) as the energy of z at time t.
By rewriting the boundary and initial conditions of (1) in terms of the functions f and g from Definition 2, one obtains at once the following characterization of solutions of (1), when they exist.
The main technique underlying all the results of our paper consists in establishing links between trajectories of (1) and trajectories of discrete-time dynamical systems which are defined next. Definition 4. Let S : R ⇒ R be a set-valued map. We refer to the inclusion as the discrete-time dynamical system associated with S on R and its corresponding trajectories (x n ) n∈N are called real iterated sequences for S. Similarly, we refer to the inclusion g n+1 (s) ∈ S(g n (s)), n ∈ N, a.e. s ∈ [−1, 1], as the discrete-time dynamical system associated with S on the space of real-valued measurable functions defined on [−1, 1] and its corresponding trajectories (g n ) n∈N are called iterated sequences for S. For p ∈ [1, +∞], the discrete-time dynamical system associated with S on Y p is defined as the restriction of the above dynamical system to sequences (g n ) n∈N in Y p .
In the sequel of the paper, we will connect solutions of (1) and solutions of (13) with S being the set-valued map whose graph is RΣ, in which case (13) is another way of writing (12c). For that purpose, we need to introduce some additional notations. In the following definition, we use Y N p to denote the set of sequences taking values in Y p .
(b) We use I : X p → Y p to denote the isometry which associates, with each (z 0 , z 1 ) ∈ X p , the element g 0 ∈ Y p defined by for a.e. s ∈ [0, 1].
As a consequence of Proposition 3 and the above definitions, one immediately obtains the following one-to-one correspondence between solutions of (1) in X p and trajectories of the dynamical system (13) in Y p . Proposition 6. Let p ∈ [1, +∞], Σ ⊂ R 2 , and S : R ⇒ R be the set-valued map whose graph is RΣ.
(a) Let z be a solution of (1) with initial condition (z 0 , z 1 ) ∈ X p , g ∈ L p loc (−1, +∞) be the corresponding function from Definition 2, and (g n ) n∈N = Seq(g). Then (g n ) n∈N is an iterated sequence for S on Y p starting at g 0 = I(z 0 , z 1 ).
(b) Conversely, let (g n ) n∈N be an iterated sequence for S on Y p starting at some g 0 ∈ Y p . Let g = Seq −1 ((g n ) n∈N ), f ∈ L p loc (0, +∞) be given by f (s) = −g(s) for a.e. s ≥ 0, and z be defined from f and g as in the first equation of (11). Then z is a solution of (1) in X p with initial condition (z 0 , z 1 ) = I −1 (g 0 ).
(c) Let z, g, and (g n ) n∈N be as in (a) or (b). Then and, in particular, e p (z)(2n) = g n p , for all n ∈ N.
Proof. Items (a) and (b) are reformulations of Proposition 3. As for (14), it follows from the definition of e p , (10), and (12b).
Saying that (g n ) n∈N is an iterated sequence for the set-valued map S given in the statement of Proposition 6 is equivalent to (g n (s), g n+1 (s)) ∈ RΣ, n ∈ N, a.e. s ∈ [−1, 1], which is nothing but (12c) rewritten in terms of the sequence (g n ) n∈N . It is now clear, at the light of what precedes, that addressing standard issues for solutions of (1) such as existence, uniqueness, and decay rates and their optimality is completely equivalent to addressing the same issues for sequences (g n ) n∈N in Y p verifying (16). This is the point of view that we will adopt all along the paper.
Remark 7. Thanks to the iterative nature of discrete-time dynamical systems in Y p , Proposition 6 reduces the issue of existence (resp. existence and uniqueness) of solutions of (1) in X p for every initial condition in X p to the following equivalent statement in terms of S: for every g ∈ Y p , there exists (resp. there exists a unique) h ∈ Y p such that h(s) ∈ S(g(s)) for a.e. s ∈ [−1, 1].
• Remark 8. For p < +∞, one deduces from (14) that, for every t ≥ 0, and hence t → e p p (z)(t) is absolutely continuous and • Remark 9. When Σ is the graph of a function σ : R → R, the set-valued map S : R ⇒ R whose graph is RΣ can be described as follows: for x ∈ R, S(x) is the set of solutions y ∈ R of the equation A similar equation has been given in [25] but, instead of working with a set-valued map S, the authors consider instead solutions y of the above equation of minimal absolute value. •

Hypotheses on Σ
To prepare for the sequel of the paper, we provide a list of assumptions on Σ that will be useful to characterize existence, uniqueness, or asymptotic behavior of (1). The results of this paper will require subsets of these assumptions, which are explicitly stated in each result. We stress the fact that we do not assume all these assumptions on Σ at the same time, since most of our results do not require all of them. Note that we will use the two notions of universally measurable function (whose definition is recalled in Appendix A) and function with linear growth, i.e., functions ϕ : R → R for which there exist a, b ∈ R + such that Hypotheses 10. The following hypotheses concern a set Σ ⊂ R 2 and the set-valued map S whose graph is RΣ.
(H2) ∞ RΣ contains the graph of a universally measurable function ϕ mapping bounded sets to bounded sets.
(H3) RΣ is equal to the graph of a universally measurable function ϕ with linear growth.
(H3) ∞ RΣ is equal to the graph of a universally measurable function ϕ mapping bounded sets to bounded sets.
Throughout the paper we will often assume Σ to be a damping set, whose definition is given next.
Definition 11 (Damping set). A set Σ ⊂ R 2 is called a damping set (or simply damping) if it satisfies (H1), (H2), and (H4). It is said to be strict when one requires (H5) to be satisfied instead of (H4).
By a slight abuse of notation, we will also refer to the set-valued function S whose graph is RΣ as a (resp. strict) damping when Σ is a (resp. strict) damping.
Assumption (H1) is used to guarantee that z ≡ 0 is a solution of (1). When Σ is the graph of a function σ, then (H1) reduces to σ(0) = 0.
In the case where Σ is the graph of a linear function σ(x) = αx, it is standard that a necessary and sufficient condition for the existence of solutions of (1) is α = −1, i.e., that RΣ is not the vertical axis x = 0. Hypotheses (H2)-(H3) ∞ prevent this phenomenon of nonexistence of solutions of (1) by imposing a positive distance between the vertical axis x = 0 and some points of RΣ outside of a neighborhood of 0. Moreover, we will show in Theorem 20 that (H2) (resp. (H2) ∞ ) is a sufficient condition for the existence of solutions of (1) in X p for p finite (resp. p = +∞) and the necessity of the linear growth condition (resp. the condition of mapping bounded sets to bounded sets).
Regarding uniqueness, we have a more precise result, namely that (H3) (resp. (H3) ∞ ) is necessary and sufficient for p finite (resp. for p = +∞), as shown in Theorem 21. Note that a standard assumption in the literature for obtaining uniqueness of solutions of (1) is that σ or Id +σ are monotone, cf. for instance [25,Proposition 1] and also Proposition 19 below. Either of these properties implies conditions (H3) and (H3) ∞ on Σ.
Condition (H4) is a generalization of the damping assumption on a function σ, which states that sσ(s) ≥ 0 for every s ∈ R, and which implies that the X 2 norm of solutions of (1) is nonincreasing. Similarly, (H5) is a strict version of (H4) and generalizes the condition of strict damping for a function σ, i.e., sσ(s) > 0 for every s ∈ R * . Hypothesis (H6) is used in the sequel to show that the stability concept of UGAS does not hold in general in X p for finite p and it can be restated, in the case where Σ is the graph of a continuous function σ : R → R, as In particular, Hypothesis (H6) is verified if σ is either a saturation function (see Figure 3 below) or has a superlinear growth at infinity. Hypotheses (H7) and (H8) are generalizations of linear sector conditions in neighborhoods of the origin and infinity respectively, which are classical in the case where Σ is the graph of a continuous function σ : R → R and which can be stated in that case respectively as We next translate these hypotheses in equivalent statements when one replaces Σ by RΣ and get the following proposition.
Proposition 12. Let Σ ⊂ R 2 and consider the list given in Hypotheses 10. Then Hypothesis (H1) is equivalent to the same statement when replacing Σ by RΣ. As for (H4), (H5), (H6), (H7), (H8), (H9), and (H10) they can be expressed in terms of RΣ (or, equivalent, of S) respectively as follows: For every x ∈ R and y ∈ S(x), one has |y| ≤ |x|, For every x ∈ R and y ∈ S(x) with (x, y) = (0, 0), one has |y| < |x|, Proof. This proposition follows after immediate computations, after noticing though that the equivalences between (H9) and (H9) and between (H10) and (H10) are obtained via the follow explicit relation between the functions q and Q: where indeed x → x + q(x) is an increasing bijection from R + to R + when (H9) or (H10) holds true.
Remark 13. Hypotheses (H9) and (H10) are borrowed from [26], where they are only considered in the case where Σ is the graph of a function. These hypotheses can be seen as nonlinear sector conditions, which can be clearly understood when written in terms of RΣ, see Figure 2 below. Indeed, these conditions naturally arise when one considers, instead of Σ, the set Σ −1 = {(y, x) | (x, y) ∈ Σ}, which generalizes the situation where Σ is the graph of an invertible function σ and in which case Σ −1 becomes the graph of σ −1 and is obtained from Σ by the symmetry with respect to the diagonal line y = x. Instead, RΣ −1 is obtained from RΣ by the symmetry with respect to the axis y = 0 which corresponds to a simple sign change for S. Hence, all the issues related to iterated sequences associated with S (such as existence, uniqueness and asymptotic behavior) remain unchanged when they are considered for −S. • We next provide figures illustrating the region RΣ for different choices of Σ. The first one, provided in Figure 1, is an example of a set Σ satisfying (H4) together with the corresponding set RΣ, which satisfies (H4). Figure 2 provides the regions where a set Σ must be included in order to satisfy either (H9) (in light blue) or (H10) (in light red). The figure also represents the corresponding regions for RΣ to satisfy (H9) or (H10).
We represent in Figure 3 a set Σ as a saturation-type sector, i.e., a region comprised between the graphs of two piecewise linear saturation functions. The latter are defined as functions f of the form f  for some positive constants λ and M . More generally, we call saturation function any continuous function whose graph is contained in a saturation-type sector. Finally, Figure 4 represents the set Σ M , M > 0, given by the graph of the sign set-valued map sgn M : R ⇒ R defined by i.e., Note that Σ M is not the graph of a single-valued function, but RΣ M is.

Other models of wave equations with set-valued boundary damping
The flexibility of the viewpoint consisting of translating the study of solutions of (1) into that of iterated sequences of the set-valued map S is well illustrated when considering the wave equation where Σ 0 and Σ 1 are subsets of R 2 . We have taken this example from [26] where precise decay rates of solutions of (21), as time tends to infinity, have been given in the case where both Σ 0 and Σ 1 are graphs of functions sgn(x)|x| 1+α , α > 0.
Similarly to Proposition 6, we aim at characterizing a correspondence between the solutions of (21) and the iterated sequences of a discrete-time dynamical system. For that purpose, we first provide the appropriate counterpart to Definition 5.
(a) We use Seq 0 : L p loc (−1, +∞) → Z N p to denote the bijection which associates, with each h ∈ L p loc (−1, +∞), the sequence (h n ) n∈N in Z p defined by h n (s) = h(s + n) for n ∈ N and a.e. s ∈ [−1, 0].
(b) We use I 2 : X p → Z p × Z p to denote the isometry which associates, with each (z 0 , z 1 ) ∈ X p , the element (h 0 , g 0 ) ∈ Z p × Z p defined by for a.e. s ∈ [0, 1].
We also assume in the sequel of this subsection that Definition 2 is suitably modified in order to take into account the new boundary condition at x = 0 and the fact that z(0, 0) is not necessarily zero, and that the contents of Definition 4 are extended to set-valued maps defined on R 2 in order to deal with discrete-time dynamical systems on R 2 and Z p × Z p .
Proposition 15. Let p ∈ [1, +∞], Σ 0 , Σ 1 be two subsets of R 2 , and S i : R ⇒ R be the set-valued map whose graph is equal to RΣ i for i ∈ {0, 1}. Define the set-valued map S on R 2 which associates with every (x, y) ∈ R 2 the Cartesian product of the subsets (−S 1 )(y) and (−S 0 )(x) of the real line.
(b) Conversely, let (h n , g n ) n∈N be an iterated sequence for S on Z p × Z p starting at some (h 0 , g 0 ) ∈ Z p × Z p . Let f (· + 1) = Seq −1 0 ((h n ) n∈N ) and g = Seq −1 ((g n ) n∈N ). Then f ∈ L p loc (0, +∞), g ∈ L p loc (−1, +∞) and, if z is defined from f and g as in the first equation of (11), one gets that z is a solution of (21) in X p with initial condition (z 0 , z 1 ) = I −1 2 (h 0 , g 0 ). (c) Let z, f , g, and (h n , g n ) n∈N be as in (a) or (b). Then, for all t ∈ R + , and, in particular, for all n ∈ N, Proof. The proof for the proposition exactly follows the line of arguments that led to Proposition 6. The only difference appears when replacing the Dirichlet boundary condition at x = 0 in (1) by the boundary condition at x = 0 given by (z t (t, 0), z x (t, 0)) ∈ Σ 0 in (21). One then replaces by (12b) and (12c) by the equations − g(t) ∈ S 0 (f (t)), −f (t + 1) ∈ S 1 (g(t − 1)), for every t ≥ 0, respectively.
At the light of the above proposition, one can see that the issues of existence and uniqueness of solutions of (21) boil down to the study of the set-valued map S, the latter question being equivalent to the separate study of S 0 and S 1 . On the other hand, asymptotic stability of solutions of (21) and related issues are addressed through the study of iterated sequences associated with S in R 2 . Due to the structure of S, the latter question can be reduced to the study of real iterated sequences associated with the set valued maps (−S 1 ) • (−S 0 ) or (−S 0 ) • (−S 1 ) since, combining the two equations of (22) When S 0 is single-valued, the sequence (h n ) n∈N suffices to describe solutions z of (21), since the corresponding sequence (g n ) n∈N is uniquely determined by the first relation of (22) and the second relation in (22) can be expressed solely in terms of (h n ) n∈N . If moreover the single-valued map S 0 is invertible, we may alternatively describe solutions z of (21) in terms of (g n ) n∈N only, since (h n ) n∈N can be computed using the first relation of (22). This is precisely what is done for (1) in this paper, in which case S 0 = Id and we can use this simple expression of S 0 to further simplify the relation between e p (z)(·) and the norms of the elements of the sequence (g n ) n∈N . A similar remark applies when S 1 is single-valued.
Remark 16. Any boundary condition involving only z t and z x at the same endpoint can be recovered by our formalism. Indeed, an homogeneous Neumann condition reads Σ equal to the horizontal axis R × {0}, while an homogeneous Dirichlet condition can be seen as taking Σ equal to the vertical axis {0} × R. For instance, proceeding in such a way at the extremity x = 0 with an homogeneous Dirichlet boundary condition yields that S 0 = Id, from which we recover the set-valued map S of Proposition 6 as equal to S 1 . •

Additional auxiliary results
We now present some important auxiliary results providing properties of the solutions of (1) and of the set Σ that will be used several times in the paper. We start with the following result on a decomposition of solutions of (1), whose proof is immediate.
Proposition 17. Let p ∈ [1, +∞], Σ ⊂ R 2 , and S be the set-valued function whose graph is RΣ. Assume that S(0) = {0}. Let (f (k) ) k∈N be a sequence in Y p so that f (k) and f (k ′ ) have disjoint supports for every integers k = k ′ and assume moreover that f = k≥0 f (k) belongs to Y p . Hence, for every n ≥ 0, Moreover, if (f n ) n≥0 is any iterated sequence for S starting at f , then, for every k ≥ 0, there exists an iterated sequence (f n for every n ≥ 0. In particular, if p is finite, it holds and, if p = +∞, one has The next proposition gathers some elementary properties of set-valued dampings, as given in Definition 11. (d) If T is a damping, then S • T is also a damping, and it is strict if S or T is strict.
In particular, the iterates S [n] , n ∈ N * , are also dampings, and they are strict if S is strict.
(e) Assume that, for every real iterated sequence (x n ) n∈N in R for S which is not identically zero, there exists n ∈ N such that |x n | < |x 0 |. Then S [2] is a strict damping.
(f ) S [2] is a strict damping if and only if there does not exist x ∈ R * such that either Proof. By (H1), one has (0, 0) ∈ RΣ and thus 0 ∈ S(0) since the graph of S is RΣ.
One can deduce easily that (H4) implies S(0) = {0}. The second part of (a) follows immediately from (H5). The fact that RΣ contains the graph of a function, which is a consequence of (H2), also immediately implies (b). The closedness of S implies that, for every x ∈ R, ({x} × R) ∩ RΣ is closed, and (H4) yields that this set is also bounded, showing that it is compact. Then S(x) is compact as the image of ({x} × R) ∩ RΣ through the projection onto the second coordinate, proving (c).
To prove (d), notice that S • T trivially satisfies (H1), and (H4) for S • T follows immediately from the corresponding properties for S and T , with S • T satisfying (H5) as soon as one of S or T satisfies this assumption. Finally, if ϕ S : R → R and ϕ T : R → R are universally measurable functions with linear growth contained in the graphs of S and T , respectively, one immediately verifies that ϕ S • ϕ T is a function with linear growth contained in the graph of S • T . This function is also universally measurable as the composition of two universally measurable functions (see Proposition 77 in Appendix A), showing that S • T also satisfies (H2), as required.
Let us show (e) by contraposition. By (d), S [2] is a damping and, if it is not strict, then there exists x ∈ R * , y ∈ S(x), and z ∈ S(y) such that |z| = |y| = |x|. If x = y or y = z, then the sequence (x n ) n∈N defined by x n = y for every n ∈ N is a real iterated sequence for S with |x n | = |x 0 | for every n ∈ N. Otherwise, one has z = −y = x and we define the sequence (x n ) n∈N by x n = (−1) n x for n ∈ N. It is then clearly a real iterated sequence for S with |x n | = |x 0 | for every n ∈ N, leading thus to the proof of the desired result.
Finally, to prove (f), notice that, if the damping S [2] is not strict, then, letting x, y, z be as in the proof of (e), the previous argument shows that one has either y ∈ S(y) or z = −y = x, in which case −x ∈ S(x) and x ∈ S(−x), as required. Conversely, if there exists x ∈ R * such that x ∈ S(x), or such that −x ∈ S(x) and x ∈ S(−x), then, in both cases, x ∈ S [2] (x), showing that S [2] is not a strict damping.
In the case where Σ is the graph of a continuous function, items (a)-(c) have been already obtained in [25,Lemma 1].
We conclude this section with a technical result providing a necessary and sufficient condition on the set-valued map S to be the graph of a continuous function when Σ is the graph of a continuous function.
Proposition 19. Assume that Σ is the graph of a continuous function σ : R → R. Then the set-valued function S is single-valued and continuous if and only if Id +σ : R → R is a bijection. Moreover, if σ is a damping function, then S is single-valued and continuous if and only if Id +σ is strictly monotone.
If Id +σ is a bijection, then T is invertible and thus RΣ is the graph of the continuous function The last assertion of the proposition follows from the fact that, if σ is a damping function, then showing that Id +σ is surjective. Hence Id +σ is bijective if and only if it is strictly monotone.

Existence and uniqueness of solutions in X p
In this section, we will show that the hypotheses (H2)-(H3) ∞ introduced in Hypotheses 10 actually yield necessary and sufficient conditions for existence and uniqueness of solutions of (1) in X p , p ∈ [1, +∞]. We start with the following existence result.
(b) Assume that, for every (z 0 , z 1 ) ∈ X p , there exists a solution of (1) in X p with initial condition (z 0 , z 1 ). Then RΣ contains the graph of a Lebesgue measurable function. Moreover, RΣ also contains the graph of a function with linear growth if p < +∞ or the graph of a function mapping bounded sets to bounded sets if p = +∞.
Proof. According to Remark 7, the existence, for every initial condition (z 0 , z 1 ) ∈ X p , of a solution of (1) in X p with initial condition (z 0 , z 1 ), is equivalent to the following statement: for every g ∈ Y p , there exists h ∈ Y p such that h(s) ∈ S(g(s)) for a.e. s ∈ [−1, 1], where S denotes the set-valued map whose graph is RΣ. We start by showing Item (a). It is clear that (H2) (resp. (H2) ∞ ) is sufficient to get the statement for p finite (resp. for p = +∞) by taking h = ϕ • g, where ϕ is the function whose existence is asserted in (H2) (resp. (H2) ∞ ). Indeed, since ϕ is universally measurable and using Proposition 78 in Appendix A, one gets that h is measurable, and the linear growth assumption (resp. the assumption of mapping bounded sets to bounded sets) on ϕ guarantees that h ∈ Y p .
We next turn to an argument for Item (b). Assume that, for every g ∈ Y p , there exists h ∈ Y p such that h(s) ∈ S(g(s)) for a.e. s ∈ [−1, 1]. Notice first that, for every x ∈ R, S(x) is nonempty. Indeed, given x ∈ R, consider g identically equal to x and apply the working hypothesis on g. Then clearly h(s) ∈ S(x) for a.e. s ∈ [−1, 1], and hence S(x) is nonempty.
Let g : (−1, 1) → R be a diffeomorphism. Thanks to Lemma 80 in Appendix D, there exists a measurable function h : x ∈ R, and this inclusion can be made to hold everywhere up to modifying ϕ on set of measure zero.
We finally prove the last parts of the statement regarding linear growth for 1 ≤ p < +∞ and the condition of mapping bounded sets to bounded sets for p = +∞. Reasoning by contradiction yields the existence of a sequence (x n ) n∈N such that, for p < +∞, one has |y| > n(|x n | + 1) for every n ∈ N and y ∈ S(x n ), while, for p = +∞, (x n ) n∈N is bounded and |y| > n for every n ∈ N and y ∈ S(x n ).
Let {A n } n∈N be a family of disjoint measurable subsets of [−1, 1] of positive Lebesgue measure α n . In the case p < +∞, we further require that α n = 1 (|xn|+1) p (n+1) 2 , which is possible since ∞ n=0 α n ≤ 2. Consider the measurable function g = ∞ n=0 x n χ An . Since one has that Then, for p < +∞, one deduces by (26) that |h(s)| > n(|x n | + 1) for every n ∈ N and a.e. s ∈ A n and therefore which contradicts the fact that h ∈ Y p . Similarly, for p = +∞, one gets by (27) that |h(s)| > n for every n ∈ N and a.e. s ∈ A n and, since A n has positive measure for every n ∈ N, one has h ∞ = +∞, yielding the required contradiction also in that case.
We next provide our result on the uniqueness of solutions of (1). Theorem 21. Let Σ ⊂ R 2 and p ∈ [1, +∞]. Then, for every (z 0 , z 1 ) ∈ X p , there exists a unique solution of (1) in X p with initial condition (z 0 , z 1 ) if and only if either (H3) holds and p < +∞, or (H3) ∞ holds and p = +∞.
Proof. As in the proof of Theorem 20, we use Remark 7 to equivalently reformulate the existence and uniqueness, for every initial condition (z 0 , z 1 ) ∈ X p , of a solution of (1) in X p with initial condition (z 0 , z 1 ), as the following statement: for every g ∈ Y p , there exists a unique h ∈ Y p such that h(s) ∈ S(g(s)) for a.e. s ∈ [−1, 1].
Conversely, by Theorem 20(b), RΣ contains the graph of a measurable function ϕ : R → R. If RΣ were not equal to the graph of ϕ, there would exist x ∈ R such that S(x) contains more than one element, and thus, by considering the initial condition g constant equal to x, one would construct two different solutions to (1), contradicting the uniqueness assumption. Hence RΣ is equal to the graph of ϕ. Applying once again Theorem 20(b), one deduces that ϕ is necessarily a function with linear growth in the case p < +∞ or a function mapping bounded sets to bounded sets in the case p = +∞.
One is left to show that ϕ is universally measurable. This follows by first using Lemma 80 from Appendix D to conclude that ϕ preserves Lebesgue measurability by left composition and then Proposition 78 from Appendix A.

Remark 22.
It is clear from Theorems 20 and 21 that existence and uniqueness of solutions of (1) in X p solely depends on the fact that the set RΣ contains (or is equal to) the graph of a function with appropriate properties. Thanks to this fact, it is possible to obtain existence and uniqueness of solutions of (1) for a large variety of sets Σ. For instance, this allows us to consider Σ as graphs of discontinuous functions completed by vertical segments at jump discontinuities without facing the usual issues addressed in [11,27]. An example of the above discussion is the sign function, illustrated in Figure 4 and treated in more details in Section 6.
• Remark 23. Hypothesis (H2) falls short of being necessary for existence of solutions of (1) in X p for p finite. Indeed, it would be the case if not only the measurable function and that of linear growth provided by Theorem 20(b) would be equal to the same function ϕ, but also if this function ϕ were universally measurable. Note that the further assumption of uniqueness of solutions of (1), together with Proposition 78, implies such properties on ϕ and yields Theorem 21. We conjecture that (H2) is actually necessary for the existence of solutions of (1) in X p for p finite. Similar comments can be made on (H2) ∞ in the case p = +∞. • Remark 24. The notion of solution of (1) introduced in Definition 2 covers only the case of solutions which are global in time. One can easily adapt Definition 2 to allow for solutions of (1) defined locally in time: given T > 0, it suffices to require that f ∈ L p loc (0, T + 1) and g ∈ L p loc (0, T ) and that (11) is satisfied for t ∈ [0, T ) instead of for t ∈ R + . The local formulation allows one to also consider solutions of (1) that blow up in finite time, but this topic is outside the scope of the present paper, which focuses instead on the convergence to zero of solutions as time tends to infinity.
Theorem 20 concerns the existence of global solutions of (1), but, since one does not necessarily have uniqueness of solutions, there may also exist local solutions blowing up in finite time under the assumptions of Theorem 20(a). For finite p, one may prevent the existence of such local solutions by requiring the linear growth assumption on S instead of only on the universally measurable function ϕ from (H2), i.e., by requiring that there exist a, b ∈ R + such that S(x) ≤ a|x| + b for every x ∈ R. Under this assumption, any local solution of (1) in X p can be extended to a global solution. A similar remark holds in the case p = +∞ by requiring x∈A S(x) to be bounded for every bounded set A ⊂ R (which is in particular satisfied when S has linear growth).
Notice that the linear growth condition on S is satisfied under (H3) (and, similarly, in the case p = +∞, the condition of S being bounded on bounded sets is satisfied under (H3) ∞ ), meaning that, given an initial condition, the unique solution of (1) from Theorem 21 is unique not only on the class of global solutions, but also on the class of local solutions. In the sequel of the paper, we will mostly often work with sets Σ ⊂ R 2 which are dampings, in which case (H3) or (H3) ∞ are not necessarily satisfied, but the linear growth assumption trivially holds due to (H4).
• Remark 25. Since (H3) is independent of p, it follows from Theorem 21 that existence and uniqueness of solutions of (1) in X p for any initial condition in X p for some finite p is equivalent to having the same property for every finite p. However, existence and uniqueness in the case p = +∞ are equivalent to the weaker assumption (H3) ∞ . A way to interpret this fact is to remark that the linear growth assumption from (H3) is designed to avoid blow-up in finite time of solutions of (1) in L p norm for finite p, but, for bounded initial conditions, this blow-up phenomenon can be prevented with the weaker assumption from (H3) ∞ . • Remark 26. If g 0 is a simple function (i.e., a function whose range is a finite set), one may wonder whether there exists a solution of (1) with initial condition I −1 (g 0 ). It turns out that a necessary and sufficient condition for the existence of such a solution for every simple function g 0 is that S is a multi-valued function (i.e., S(x) = ∅ for every x ∈ R). Indeed, in this case, starting from a simple function g 0 , one builds at once a sequence (g n ) n∈N satisfying (16) where, for every n ≥ 0, g n is a simple function which is constant on every set on which g 0 is constant. The same question may be asked in the more general case where g 0 ∈ Y p has countable range. The necessary and sufficient condition for the existence of solutions of (1) starting at I −1 (g 0 ) for every such g 0 is simply now that RΣ contains the graph of a function with linear growth in the case p < +∞ or that of a function mapping bounded sets to bounded sets in the case p = +∞.
Both statements follow immediately from the arguments provided in the proof of Theorem 20.
• a necessary and sufficient condition for the energy of a solution to be nonincreasing and providing suitable definitions of stability. We then provide necessary and sufficient conditions for these notions of stability in Subsection 4.2 in terms of the behavior of real iterated sequences for the set-valued map S whose graph is RΣ, and relate them to properties of S [2] in Subsection 4.3. A detailed study of the decay rates of solutions of (1) and their optimality is then presented in Subsection 4.4, and the section is concluded in Subsection 4.5 by an example showing that the decay of solutions can be arbitrarily slow when Σ is of saturation type, answering a conjecture of [26].

Basic results and definitions
We start with the following basic property of a damping set Σ as regards the behavior of e p along solutions of (1). When Σ is the graph of a function, this result is classical for p = 2 and has been essentially given for p ∈ [1, +∞] in the case of internal distributed damping in [14].
, S be the set-valued map whose graph is RΣ, and assume that S is a multi-valued function. Then t → e p (z)(t) is nonincreasing for every solution z of (1) in X p if and only if (H4) holds.
Proof. Assume that (H4) holds and recall that it is equivalent to (H4). Let z be a solution of (1) in X p and g be as in Definition 2. By (H4) and (12c), one deduces that |g(t + 1)| ≤ |g(t − 1)| for every t ≥ 0, and the result follows immediately in the case p < +∞ by (17). If p = +∞, notice that e ∞ (z)(t) = lim q→+∞ e q (z)(t) for every t ≥ 0. Since e q (z)(·) is nonincreasing for every q ∈ [1, +∞), the same holds true for e ∞ (z)(·). Conversely, reasoning by contraposition, assume that (H4) does not hold. Therefore, there exist (x, y) ∈ RΣ with |y| > |x|. Fix p ∈ [1, +∞] and let g 0 , g 1 in Y p be the constant functions equal to x and y, respectively. Let (g n ) n≥1 be the sequence in Y p defined in Proposition 6 corresponding to a solution of (1) in X p with initial condition I −1 (g 1 ), which exists thanks to Remark 26 since g 1 is a simple function. Then the sequence (g n ) n≥0 corresponds, in the sense of Proposition 6, to a solution z of (1) in X p with initial condition I −1 (g 0 ). Using (15), one has where 2 1 p = 1 for p = +∞. This shows that the function t → e p (z)(t) is not nonincreasing, concluding then the proof of our result.
For the rest of the section, Σ will be assumed to be a damping set (recall Definition 11) and we aim at understanding the asymptotic behavior of solutions of (1) in X p for p ∈ [1, +∞]. Taking into account the previous proposition, we next aim at providing necessary and sufficient conditions on Σ so that all solutions of (1) in X p converges to zero as t → +∞. Since we are in an infinite-dimensional setting, there are several meaningful definitions of convergence to zero, and we state in the next definition the ones that will be of interest in this paper.
Strong stability The wave equation defined by (1) in X p is said to be strongly stable if, for every (z 0 , z 1 ) ∈ X p and any solution z of (1) starting at (z 0 , z 1 ), one has lim t→+∞ e p (z)(t) = 0.
UGAS The wave equation defined by (1) in X p is said to be uniformly globally asymptotically stable (UGAS) if there exists β ∈ KL such that, for every (z 0 , z 1 ) ∈ X p and any solution z of (1) starting at (z 0 , z 1 ), one has GES The wave equation defined by (1) in X p is said to be globally exponentially stable The definition of strong stability is classical in the context of stabilization of PDEs (see, e.g., the survey [1]), while that of UGAS stems from control theory (see, e.g., [22]).
Note that, thanks to Proposition 6, these stability concepts in X p admit equivalent statements in terms of the sequences (g n ) n∈N in Y p corresponding to solutions of (1). The function β ∈ KL from the definition of UGAS can be interpreted as a rate of decrease for solutions of (1) in X p . In the sequel, we will hence say that (1) is UGAS in X p with rate β.
As a first step towards the characterization of the asymptotic behavior of (1), it is useful to consider real iterated sequences for S since they provide particular solutions of (1) whose corresponding sequence (g n ) n∈N in the sense of Proposition 6 is made of constant functions. Thanks to (H4) from Proposition 12, all real iterated sequences are nonincreasing in absolute value as soon as Σ is a damping, and we seek extra conditions on Σ so that these real iterated sequences converge to 0. One might think that a natural sufficient condition for that purpose would be that Σ is a strict damping set, i.e., it satisfies (H5), but this is not the case, as shown by the following example.
Example 29. Let Σ 0 be a strict damping set, p ∈ [1, +∞], and (a n ) n∈N be a decreasing sequence in R + converging to a limit a * > 0. Let Σ be the set such that i.e., the sequence (a n ) n∈N becomes a real iterated sequence for the set-valued map whose graph is RΣ. Clearly, Σ is still a strict damping set since, for every (x, y) ∈ RΣ with (x, y) = (0, 0), one has either (x, y) ∈ RΣ 0 , in which case |y| < |x| since Σ 0 is a strict damping, or (x, y) = (a n , a n+1 ) for some n ∈ N, in which case |y| = a n+1 < a n = |x| since (a n ) n∈N is decreasing.
Consider the sequence (g n ) n∈N in Y p such that, for every n ∈ N, g n is constant and equal to a n . Since this sequence clearly satisfies (16), Proposition 6 ensures that this sequence corresponds to a solution z of (1) in X p . The sequence (g n ) n∈N converges in Y p to the constant function equal to a * > 0, and thus, in particular, z does not converge to 0 in X p . • The previous example furnishes an instance of a set Σ which is a strict damping but for which there exists a real iterated sequence not converging to 0. Conversely, we next provide an example of a set Σ which is a nonstrict damping set for which every real iterated sequence converges to 0.
Example 30. Let Σ ⊂ R 2 be the graph of the function σ : R → R given by σ(x) = min(0, x) for x ∈ R. Then Σ is a nonstrict damping set with the corresponding setvalued map S given by S(x) = {− max(x, 0)} for x ∈ R. It is immediate to check that every iterated sequence converges to 0 since S [2]

Real iterated sequences and stability
In order to present our results, we introduce the following convergence properties for real iterated sequences.
Definition 31. Let S : R ⇒ R be a set-valued map.
(a) We say that real iterated sequences for S converge simply to zero if every real iterated sequence converges to zero.
(b) We say that real iterated sequences for S converge to zero uniformly (on compact sets) if, for every r ≥ 0 and ε > 0, there exists N ∈ N such that, for every x ∈ R satisfying |x| ≤ r and every real iterated sequence (x n ) n∈N for S starting from x, one has |x n | < ε for every n ≥ N .
Our first result provides necessary and sufficient conditions for the strong stability of (1) in X p in terms of convergence properties of real iterated sequences.
(a) If p < +∞, then the wave equation (1) is strongly stable in X p if and only if real iterated sequences for S converge simply to zero.
(b) For p = +∞, the following statements are equivalent: Real iterated sequences for S converge uniformly to zero.
Proof. To prove (a), notice first that real iterated sequences for S provide particular solutions of (1) for which the corresponding iterated sequence (g n ) n∈N from Proposition 6 is made of constant functions (cf. also Remark 26). Hence, convergence of all such sequences to zero is a necessary condition for the strong stability of (1). Conversely, assume that real iterated sequences for S converge simply to zero. Let z be a solution of (1) in X p and consider the corresponding sequence (g n ) n∈N in Y p from Proposition 6. Then (g n (s)) n∈N is a real iterated sequence for S for a.e. s ∈ [−1, 1] and hence (g n ) n∈N converges pointwise to zero almost everywhere. Since Σ is a damping set, one has |g n (s)| ≤ |g 0 (s)| for every n ∈ N and a.e. s ∈ [−1, 1], and one concludes by the dominated convergence theorem that g n → 0 in Y p as n tends to infinity.
Let us now prove (b). The implication (b-i) =⇒ (b-ii) is trivial by definition. We prove that (b-ii) =⇒ (b-iii) reasoning by contraposition. Hence, assume that real iterated sequences for S do not converge uniformly to zero. This implies that there exists r > 0 and ε > 0 such that, for every k ∈ N, there exists x (k) ∈ [−r, r] and a real iterated sequence (x Without loss of generality, one can assume N k = k by possibly replacing the initial value x (k) by x x (k) n χ A k .
Let z be a solution of (1) in X ∞ and consider the corresponding sequence (g n ) n∈N in Y ∞ from Proposition 6. For a.e. s ∈ [−1, 1], (g n (s)) n∈N is a real iterated sequence for S starting from g 0 (s), and thus for a.e. s ∈ [−1, 1] and every n ∈ N.
Using the fact that β 0 is nondecreasing with respect to its first argument, we deduce that, for every n ∈ N, g n ∞ ≤ β 0 ( g 0 ∞ , n).

From Propositions 6 and 27, one has
One concludes the proof by applying Lemma 81 from Appendix D to the function (r, t) → β 0 (r, ⌊t/2⌋).
Remark 33. Since the notion of real iterated sequences of S converging simply to zero is independent of p, it follows from Proposition 32(a) that the convergence of solutions of (1) to 0 for p finite is independent of p: if solutions of (1) converge to 0 in some X p with p finite, then solutions will converge to 0 in every X p with p finite. The situation is different for p = +∞, in which case one must require a uniformity property on the convergence to zero of real iterated sequences for S in order to obtain convergence in the strong topology of X ∞ . By adapting the proof of Proposition 32(a), one can show that simple convergence to zero of real iterated sequences for S is also a necessary and sufficient condition for the convergence to 0 of solutions of (1) in X ∞ in the weak- * topology of X ∞ . • with the convention that 1/p = 0 for p = +∞. In case the wave equation (1) is UGAS in X ∞ , one deduces at once a UGAS-like estimate of e p (z)(t) for every finite p, involving however, as regards the dependence with the respect to the initial condition, its X ∞ norm e ∞ (z)(0) only. In the next result, we refine the above mentioned trivial estimate of e p (z)(t), with another one involving the X p norm of the initial condition.
Proposition 34. Let Σ ⊂ R 2 be a damping set and p ∈ [1, +∞). Assume that the wave equation (1) associated with Σ is UGAS in X ∞ with rate β. Then, for every solution z of (1) in X ∞ , one has, for every t ≥ 0, Proof. Let z be a solution of (1) in X ∞ and consider the corresponding sequence (g n ) n∈N in Y ∞ in the sense of Proposition 6. Let us denote Z q = e q (z)(0) for q ∈ {p, ∞}. Consider the partition of [−1, 1] in the disjoint subsets Let χ i and α i , i ∈ {1, 2}, be the characteristic functions and Lebesgue measures associated with E i , respectively. One clearly has, for every n ∈ N, g n = g n χ 1 + g n χ 2 , For i = 1, since β(·, t) is increasing for all t ≥ 0, one deduces that and, for i = 2, one gets By Chebyshev's inequality, it follows that α 2 ≤ Z p/2 p . Putting together (30), (31) and the above estimate of α 2 , one obtains Since p ≥ 1, one deduces that g n p ≤ 2 1/p β(Z 1/2 p , 2n) + Z 1/2 p β(max(Z ∞ , Z 1/2 p ), 2n).
We conclude using the fact that t → e p (z)(t) is nonincreasing.
At the light of the equivalence between (b-i) and (b-ii) from Proposition 32, one may wonder if such an equivalence holds in X p with p finite. The answer is negative, which is a consequence of the next proposition.
Proof. We prove that (1) is not UGAS in X p reasoning by contradiction. Assume then that (1) is UGAS in X p with rate β, i.e., there exists some β ∈ KL such that, for every solution z of (1), one has for every t ≥ 0.
Consider now the iterated sequence of functions (g n ) n∈N for S in Y p such that g n = x n χ A , where A ⊂ [−1, 1] is an interval of length 1/|x 0 | p . Then clearly g n p = xn x 0 . Letting z be the solution of (1) corresponding to (g n ) n∈N in the sense of Proposition 6, one deduces from (15) that This is a contradiction since, from (33), one also has that This contradiction establishes the desired result.
Remark 36. A particular instance of the above proposition has been established in [26,Theorem 4.1.1] where it is shown that (1) is not GES in the case p = 2 and Σ is the graph of a saturation function. • Proposition 35 raises the question of whether one can provide conditions on Σ under which (1) is UGAS for finite p. Our next result identifies the assumption (H8) as such a condition.
Theorem 37. Let Σ ⊂ R 2 be a damping satisfying (H8) and p ∈ [1, +∞). Assume moreover that the wave equation defined in (1) is UGAS in X ∞ with rate β. Then it is also UGAS in X p .
Proof. Let M > 0 and µ ∈ (0, 1) be given by (H8) and set η = 1/µ. Let z be a solution of (1) in X p and denote by (g n ) n∈N the corresponding sequence in Y p in the sense of Proposition 6. For simplicity of notation, we set g = g 0 , Z p = g p , Z = max(M, Z 1/2 p ). Consider the partition of [−1, 1] in disjoints subsets defined by Let χ E and α E , χ F and α F , χ k and α k , k ≥ 0, be the characteristic functions and Lebesgue measures associated with E, F and E k , k ≥ 0, respectively. One clearly has Moreover, for every n ∈ N, one has Note that, for every set S ∈ {E, F, E 0 , E 1 , . . . }, the sequence (g n χ S ) n∈N corresponds, in the sense of Proposition 6, to a solution z S of (1) with initial condition I −1 (gχ S ). In particular, since gχ S ∈ Y ∞ , z S is a solution of (1) in X ∞ and one has, for every n ∈ N, for a.e. s ∈ [−1, 1].
since (1) is UGAS in X ∞ with rate β.
One deduces for S = E that and, for By Chebyshev's inequality, we have that α F ≤ p . Then, we obtain We next estimate the sum appearing in the right-hand side of (35). For k ≥ 0 and s ∈ E k , by applying (H8), it holds that S(g(s)) ≤ η k Z and, more generally, an immediate inductive argument using (H8) shows that S [n] (g(s)) ≤ η max(0,k+1−n) Z for every n ≥ 0. Then, for every k ∈ N and n ∈ N, one has g n χ k p p ≤ α k η p max(0,k+1−n) Z p ≤ α k ξ p k η p(k−max(0,k+1−n)) .
Note that Proposition 34 and Theorem 37 both assume that (1) is UGAS in X ∞ , this property being characterized in terms of real iterated sequences for S in Proposition 32. Proposition 38(e) in the next subsection provides an easy to check sufficient condition to have the latter property.

Stability properties based on properties of S [2]
Proposition 32 provides necessary and sufficient conditions for the strong stability of (1) in X p in terms of convergence to zero of real iterated sequences for the multi-valued map S : R ⇒ R. These necessary and sufficient conditions are not yet satisfactory because they are difficult to verify. Instead, we provide in the sequel necessary or sufficient conditions on S that are simpler to check. For that purpose, we introduce the function ρ : R + → R + defined next which will be useful for several results in the sequel of the paper.
Proposition 38. Let Σ ⊂ R 2 be a damping set. Let ρ : R + → R + be the function defined by ρ(r) = lim Then the following properties hold.
(c) Assume that S [2] is closed. Then ρ(r) < r for every r > 0 if and only if S [2] is a strict damping.
(d) For every n ∈ N and r ∈ R + , one has In particular, for every real iterated sequence (x n ) n∈N for S, one has (e) If ρ(r) < r for every r > 0, then real iterated sequences for S converge uniformly to zero.
Let us now prove (c). The only nontrivial implication is that ρ(r) < r for every r > 0 as soon as S [2] is closed and a strict damping. Fix r > 0. By definition of ρ(r), there exist sequences (x n ) n∈N and (y n ) n∈N with y n ∈ S [2] (x n ) for every n ∈ N and such that, up to extracting subsequences, (x n ) n∈N converges to some x * ∈ [−r, r] and (y n ) n∈N converges to some y * ∈ R with ρ(r) = |y * |. Since S [2] is closed, one has y * ∈ S [2] (x * ), which implies, using the fact that S [2] is a strict damping, that (x * , y * ) = (0, 0) or |y * | < |x * |. In both cases, one deduces that ρ(r) < r.
To prove (d), notice that it suffices to prove (42) for even integers since S(x) ≤ |x| for every x ∈ R, and that (42) is trivially true for n = 0 and n = 2. The argument goes on by induction: let n ≥ 4 be an even integer so that (42) holds for even integers m < n and set k = n 2 . For every r ≥ 0, y ∈ [−r, r], and z ∈ S [2(k−1)] (y), one has where we used the definition of ρ in the first inequality, the fact that ρ is nondecreasing in the second, third, and fourth inequalities, and the induction hypothesis in the third inequality. Hence (42) follows at once. It is clear that (43) is an immediate consequence of (42).
Remark 39. Notice that, if S is closed, then S [2] is closed, and therefore the conclusion of item (c) holds true. This is the case, in particular, if S is the graph of a continuous function. • We can now state necessary and sufficient conditions on S only (and not relying on real iterated sequences) for strong stability of (1) in X p .
Conversely, assume that S [2] is a strict damping and that ρ defined in Proposition 38 satisfies ρ(r) < r for every r > 0. Then (1) is strongly stable in X p .
Proof. The first part of the statement follows immediately from Propositions 18(e) and 32, while the second part is an immediate consequence of Propositions 38(e) and 32.
Remark 41. In the case where the damping set Σ is the graph of a function σ : R → R, the second part of Theorem 40 has been essentially already obtained in [25] in the case p = 2 under the assumption that either σ(s) > 0 for all s > 0 or σ(s) < 0 for all s < 0 (referred to in that reference as a unilateral condition), which is stronger than requiring S [2] to be a strict damping.
• Remark 42. Recall that, for p = +∞, Proposition 32 ensures that strong stability and UGAS are equivalent and (43) immediately implies that, for every solution z of (1) in X ∞ , the corresponding sequence (g n ) n∈N in Y ∞ from Proposition 6 satisfies, for every n ≥ 0, which yields If now ρ verifies that ρ(r) < r for every r > 0, we have UGAS for solutions of (1) in X ∞ and one can build a corresponding KL function β by applying Lemma 81 to the function (r, t) → ρ [⌊t/4⌋] (r). • Remark 43. One could have replaced S [2] by S [n] for n ≥ 1 in Proposition 38 to define functions ρ n similar to ρ (= ρ 2 ) satisfying the same properties. Note that (ρ n ) n∈N * is a nonincreasing sequence of functions. We focus on the case n = 2 due to Proposition 18(e) as well as to the fact that, from Theorem 40, S [2] being a strict damping is a necessary condition for the strong stability of (1), which is not the case for S due to Example 30. An ultimate justification for sticking to n = 2 is the fact that we were not able to come up with a result interesting enough to justify the use of ρ n with n > 2, even though, for n ≥ 3, the condition ρ n (r) < r for every r > 0 is strictly weaker than the corresponding condition with n = 2. • As an immediate consequence of Proposition 38(c) and Theorem 40, one deduces the following result.
Corollary 44. Let Σ ⊂ R 2 be a damping set, p ∈ [1, +∞], and assume that S [2] is closed. Then (1) is strongly stable in X p if and only if S [2] is a strict damping.

Decay rates and their optimality
In the previous sections, we have considered the convergence to zero of solutions of (1) and the speed of convergence to zero (or decay rate) was in some cases upper bounded by a KL function β arising from the stability concept of UGAS. In this section, we intend to be more explicit on the dependence in time of β and also to provide lower bounds for the decay rate in some cases. More precisely, we say that a KL function β is optimal for (1) if the latter is UGAS with rate β and there exists a nonzero initial condition (z 0 , z 1 ) ∈ X p such that In the sequel, we will essentially work with Hypotheses (H9) or (H10). Our results in this section are decomposed in two parts: first, in Section 4.4.1, we present some preliminary results concerning the decay rates of real iterated sequences, and then, in Section 4.4.2, we apply these results to solutions of (1).

Decay rates for real iterated sequences
The main results of this section are the next two propositions dealing with real iterated sequences for S. They will be crucial in order to establish our subsequent results for decay rates of solutions of (1) and their optimality in X p , p ∈ [1, +∞]. The first proposition translates (H9) and (H10) in terms of upper and lower bounds, respectively, on real iterated sequences for S, relying on iterates of the function Q from (H9) and (H10). Its proof is an immediate consequence of the formulation of (H9) and (H10).
Proposition 45. Let Σ ⊂ R 2 be a damping set and S be the corresponding set-valued map whose graph is equal to RΣ.
(a) Assume that (H9) holds and let Q be as in (H9). Then, for every real iterated sequence (x n ) n∈N for S with |x 0 | ≤ M/ √ 2, one has (b) Assume that (H10) holds and let Q be as in (H10). Then, for every real iterated sequence (x n ) n∈N for S with |x 0 | ≤ M/ √ 2, one has |x n | ≥ Q [n] (|x 0 |), for every n ∈ N.
The next proposition provides an explicit asymptotic behavior for the sequence (Q [n] (|x 0 |)) n∈N , which serves as either an upper or a lower bound in the previous proposition.
Let us now show that, under (49), one has the stronger conclusion that x n ∼ F −1 (n) as n → +∞. Notice that x n = F −1 (t n ) = F −1 (n) − (t n − n)q(F −1 (ξ n )) for some ξ n between n and t n . Set z n = F −1 (ξ n ), which tends to zero as n → +∞. Then, for n ∈ N, In addition, one has that t n − n ξ n = max(t n , n) min(t n , n) − 1 min(t n , n) ξ n ≤ max(t n , n) min(t n , n) − 1 .
We next turn to an argument for Item (b). Notice first that λ = − ln Q ′ (0). For n ∈ N, one has x n+1 = Q ′ (0)x n (1 + Q 1 (x n ))), where Q 1 (x) = Q(x) xQ ′ (0) − 1 for x > 0. Notice that Q 1 (x) tends to zero as x tends to zero since Q is differentiable at zero. Moreover, there exists n 0 ∈ N so that |Q 1 (x n )| < 1 for n ≥ n 0 . One deduces that, for n ≥ n 0 , x n e λn = C(x 0 , n 0 ) n−1 k=n 0 where C(x 0 , n 0 ) > 0 only depends on x 0 and n 0 . Hence, Since Q 1 (x n ) → 0 as n tends to infinity, the first part of Item (b) follows at once. In particular, it implies that, for n large enough, one has the estimate Assume moreover that (50) holds true. Notice that, for x small enough, one has where y = (Id +q) −1 ( √ 2x) ≤ √ 2x. Indeed, for x > 0, one has and (54) follows since (H9) or (H10) are supposed to hold true. The second part of Item (b) is equivalent to the fact that ln(x n ) + λn remains bounded as n tends to infinity. By (52), it is then enough to prove that the series of general term |Q 1 (x n )| is convergent. By (53) and (54), for n large enough, one has where we also use the fact that ψ is nondecreasing. The conclusion follows from (50). We finally prove Item (c). Note that Q ′ (0) = 0 and hence x n+1 xn tends to zero as n tends to infinity. Fix C > 0. For every n ≥ 0, one has e Cn x n = x 0 n−1 k=0 y k , y n := e C x n+1 x n .
The first part of Item (c) immediately follows since lim n→+∞ y n = 0. For the second part, we start by noticing that, for every x > 0, one has Q(x) = 1 √ 2 (y − q(y)) with y = (Id +q) −1 ( √ 2x). For x ≤ x * √ 2 and since one has y ≤ √ 2x ≤ x * , one deduces by assumption that Let n 2 ∈ N be such that x n ≤ x * √ 2 for n ≥ n 2 . Then one has, for n ≥ n 2 , x n+1 ≤ C * x 1+α n .
By setting z n = ln x n C 1/α * , an elementary computation yields that z n+1 ≤ (1 + α)z n , hence, for n ≥ n 2 , one has z n ≤ (1 + α) n−n 2 z n 2 and one gets the desired conclusion after setting This is how F −1 is introduced in [26]. We postpone to Remark 55 a comparison of our results from Proposition 46(a) with the corresponding ones of [26]. Condition (49) is satisfied in several cases considered in the literature (see, e.g., [1, Theorem 1.7.12, Examples 1 to 4] and [26]), such as polynomial feedbacks of the form q(s) = s|s| p−1 for p > 1, or when q is an odd function given for s > 0 by q(s) = e −β(s) with β(s) = 1/s p for p > 0 or β(s) = e 1/s . Concerning (50), it holds true if for instance there exist positive C, α such that |q ′ (x) − q ′ (0)| ≤ C|x| α in a neighborhood of zero. Moreover one has quasi-optimality of (50) in the following sense: if the series of general term Q 1 (x n ) is unbounded, then lim sup n→+∞ e λn Q Proposition 49. Let q ∈ C 1 (R + , R + ) be as in the statement of Proposition 46(a), i.e., q(0) = q ′ (0) = 0, 0 < q(x) < x, and |q ′ (x)| < 1 for every x > 0. Let x 0 ∈ R * + and Q be defined from q as in (18). Proof. To show (a), note that, for n ∈ N, where ϕ is defined as in the proof of Proposition 46(a). Since Q [n+j] (x 0 ) → 0 as n → +∞ for every j ∈ {0, · · · , n 0 − 1} and ϕ(0) = ϕ ′ (0) = 0, one gets the conclusion. In order to prove (b), assume, with no loss of generality, that y 0 ≤ x 0 . Then there , and one gets the conclusion from (a).
A quick look at the argument in Proposition 46(c) for the decrease of (x n ) n∈N faster than any exponential in the case q ′ (0) = 1 may let one think that more precise decay rates can be obtained. However, it is not really the case, as explained in the following proposition, whose proof is given in Appendix C.

Decay rates for solutions
We now use Section 4.4.1 to derive results regarding decay rates for solutions of (1). We start with the following consequence of Proposition 45.
(a) Suppose that Σ satisfies (H9). Let M 0 > 0 and Q be the constant and the function whose existences are asserted in (H9).
Then, for every solution z of (1) in X ∞ , one has where t z ≥ 0 is the first nonnegative time so that (a-iii) Assume that (H8) holds and that the wave equation defined in (1) is UGAS in X ∞ . Then, for every p ∈ [1, +∞] and every solution z of (1) in X p , one has, for t large enough, where Z p = e p (z)(0), Z = max(M, Z The conclusion follows at once since Q [n] is increasing for every integer n. As regards the second part of Item (a-i), we simply follow the argument of Proposition 34 replacing the bounds using the KL function β(·, t) by Q [⌊t/2⌋] (·) and the conclusion follows from (32). Since e ∞ (z)(t z ) ≤ M 0 √ 2 , Item (a-ii) follows immediately by applying Item (a-i) to g 0 = I(z(t z , ·), z t (t z , ·)).
As regards Item (a-iii), the argument consists in reproducing the proof of Theorem 37 with modifications taking into account Item (a-ii). In the sequel, we use the notations of the proof of Theorem 37. Thanks to Item (a-ii), one can replace the estimate (37) by where t 1 ≥ 0 only depends on Z p . We use again Item (a-ii) to replace equations (38) and (40) by where t 2 ≥ 0 only depends on Z. Putting together (39) and the previous inequalities, and up to increasing t 1 and t 2 , we deduce (59). We finally provide an argument for Item (b). Denote by (g n ) n∈N the sequence in Y p corresponding to the solution z in the sense of Proposition 6. For every n ∈ N, define the measurable set F n ⊂ [−1, 1] by Assume first that F n is of zero Lebesgue measure for every n ∈ N. In particular, Since z is nontrivial, the Lebesgue measure α E of E is positive. Moreover, our assumption on the sets F n and the fact that Σ ∩ ∆ = {(0, 0)} yield that, for almost every s ∈ E and every n ∈ N, one has |g n (s)| ≥ M √ 2 . In particular, g n p ≥ α 1/p E M √ 2 , and the conclusion follows with C 1 = α 1/p E M √ 2 , C 2 = 1, and by using Propositions 6 and 27 and the fact that Q(x) ≤ x for every x ≥ 0.
If now there exists n 0 ∈ N so that F n 0 has positive Lebesgue measure, then there exists C z > 0 and a subset G z of F n 0 of positive measure α z such that C z ≤ |g n 0 (s)| ≤ M √ 2 for s ∈ G z . By Proposition 45(b) and by using the fact that Q is increasing, it follows, for n ≥ n 0 , that |g n (s)| ≥ Q [n−n 0 ] (|g n 0 (s)|) ≥ Q [n] (C z ) for s ∈ G z . Thus g n p ≥ α 1/p z Q [n] (C z ) and, from Propositions 6 and 27, one deduces that e p (z)(t) ≥ g ⌊t/2⌋+1 p ≥ α Remark 52. If we assume in (a-iii) that (H8) holds with a constant M > 0 equal to the constant M 0 > 0 from (H9), then necessarily (1) is UGAS in X ∞ . Indeed, in that case, by combining the bounds from (H8) and (H9), we deduce that ρ(r) < r for every r > 0, where ρ is given by (41). The fact that (1) is UGAS in X ∞ then follows from Propositions 32(b) and 38(e).
• Remark 53. It is useful to notice that, for p = +∞, we have deduced the estimate (56) immediately from (61). One may wonder whether it is possible to deduce from (61) a similar estimate replacing e ∞ by e p for finite p. This is indeed the case under the extra assumption that Q is concave (or, equivalently, that q is convex): one deduces from Lemma 82 in Appendix D that as soon as e ∞ (z)(0) ≤ M 0 √ 2 . Note that (63) holds true for every solution of (1) in X p as soon as one assumes that (H9) holds globally, i.e., M = +∞ in its definition. In that case, the wave equation in (1) is UGAS in X p . This is in accordance with Theorem 37 since (H8) holds if Q is concave in R + and (H9) holds globally. • Theorem 51 together with Propositions 46 and 49 provide accurate estimates for the behavior of trajectories of (1) as time tends to +∞. Moreover, the optimality of such estimates can be addressed using Theorem 51(b). Even though this can be done in full generality in the case where Σ is the graph of a function q or q −1 , with q ∈ C 1 (R + , R + ) satisfying the statements in (H9) or (H10), we only focus in the sequel on the case where q ′ (0) = 0 since it is the one usually addressed in the literature (see, e.g., [1,26]). More precisely, we have the following result.
Corollary 54. Let Σ ⊂ R 2 be a damping set satisfying (H8) and M > 0 be the constant from (H8). Assume moreover that Then, for every p ∈ [1, +∞] and every nontrivial solution z of (1) in X p , there exist positive constants C 1 , C 2 such that, for every t ≥ 0, where Q is the function defined from q in (18).
Proof. Note that assumption (b) implies that both (H9) and (H10) are satisfied with the same function q and the same M > 0. Moreover, as noticed in Remark 52, (b) and (H8) imply that (1) is UGAS in X ∞ . Then, by Theorem 51(a-iii) and (b), one deduces that (59) and (60) hold, and the conclusion follows by Proposition 49.
Remark 55. Our results given in Proposition 46(a), Theorem 51 (equations (56), (58), and (60) with p = +∞), and Corollary 54 are directly inspired by a string of results of [26], namely Theorem 2.1 and all the results from Section 3 of that reference. Because of the flexibility of our approach, we provide simpler proofs and we are able to relax some assumptions on the function q and the set of initial conditions for which the appropriate estimates hold true. Note that Theorem 51(a-iii), together with the assumption that the function q from (H9) satisfies q ′ (0) = 0 and (49), yields the estimate e p (z)(t) ≤ 3(1 + e p (z)(0) 1/2 )F −1 (t/2) for every p ∈ [1, +∞], every solution z of (1) in X p , and t large enough, where F is as in Proposition 46(a). Indeed, this follows from (59), Proposition 46(a), and manipulations similar to those in the argument of Corollary 54. This estimate can be compared to that of [28, Theorem 2.1(b)], which obtains a similar estimate for wave equations in space dimension up to 3 but with stronger assumptions on q.
• An interesting instance of the preceding results is the classical linear case, which corresponds to Σ satisfying (H7). As a consequence of the above results and remarks, we have the following.
(b) If there exist positive constants a, b such that a|x| ≤ |y| ≤ b|x| for every (x, y) ∈ Σ, then (1) is GES in X p for every p ∈ [1, +∞]. More precisely, there exist C > 0 and λ > 0 such that, for every p ∈ [1, +∞] and every initial condition (z 0 , z 1 ) ∈ X p , every solution z of (1) starting at (z 0 , z 1 ) satisfies Proof. Item (a) can be deduced by combining the previous results and remarks and noticing that (H7) is a particular case of (H9) with a linear function q. We choose however to provide the following direct ad hoc argument. Let M > 0 and µ ∈ (0, 1) be as in (H7) and β be the KL function provided by the UGAS assumption in X ∞ . Let z be a solution of (1) in X ∞ and consider the corresponding sequence (g n ) n∈N in Y ∞ from Proposition 6. Set R = g 0 ∞ , let t R > 0 be such that β(R, t R ) ≤ M √ 2 , and define n R = ⌈t R /2⌉. For n ≥ n R and a.e. s ∈ [−1, 1], one has It follows from (H7) that, for n ≥ n R , one has The previous inequality still holds for n ∈ {0, . . . , n R −1} since (g n ) n∈N is nonincreasing. One immediately gets the conclusion with λ = − ln µ 2 using Propositions 6 and 27. Part (b) follows immediately since, from its assumptions, one deduces that (66) holds for every n ∈ N with n R = 0.
It is immediate to see that, under (H7) * , real iterated sequences for S converge uniformly to zero and hence (1) is UGAS in X ∞ , therefore (H7) * is stronger than the assumptions of Corollary 56(a). Nevertheless, we can prove Corollary 56(a) under (H7) * for p finite with an argument having its own interest, which is given next. Consider the Lyapunov function V p , based on a variant of e p , and defined, for a solution z of (1) in X p , by where f and g are the functions associated with z from Definition 2, ν is a positive constant to be chosen later, and F (s) = |s| p for s ∈ R. Note that V p coincides with e p p (z) if ν = 0. In the case p = 2, the corresponding Lyapunov function V 2 has been extensively used in the literature of hyperbolic systems of conservation laws, as detailed in [5].
Remark 57. Note that the previous argument stills works without considering the parameter ν in the definition of V p in (67), in which case V p = e p p (z). Equation (69) becomes, after taking into account (H7) * , dV p dt (t) ≤ −(1 − µ p )F (g(t − 1)), for a.e. t ≥ 0.
Integrating between t − 1 and t + 1 for t ≥ 1, one gets that e p (z)(t + 1) ≤ µe p (z)(t − 1), which yields exponential decay. More generally, at the light of Remark 8 and without assuming necessarily (H7) * , one can follow a reasoning relying on a Lyapunov function of the form (67) with any positive definite function F with no additional regularity assumption on the solution z ∈ X p of (1), as soon as a damping and sector conditions are satisfied. • We close this section by providing a necessary and sufficient condition of GES.
Corollary 58. Let Σ ⊂ R 2 be a damping set, p ∈ [1, +∞], and S be the set-valued map whose graph is RΣ. Then (1) is GES in X p if and only if there exists µ ∈ (0, 1) and n 0 ∈ N * such that, for every x ∈ R, one has S [n 0 ] (x) ≤ µ|x|.
Proof. Assume that there exists µ ∈ (0, 1) and n 0 ∈ N * such that, for every x ∈ R, one has S [n 0 ] (x) ≤ µ|x|. Let z be a solution of (1) in X p and consider the sequence (g n ) n∈N in Y p corresponding to z according to Proposition 6. Then, for every k ∈ N, one has g kn 0 p ≤ µ k g 0 p , and, by using Propositions 6 and 27, one gets that (1) is GES. Conversely, assume that (1) is GES and let C, λ be positive constants such that e p (z)(t) ≤ Ce −λt e p (z)(0) for every t ≥ 0 and every solution z of (1) in X p . Let n 0 ∈ N * be such that Ce −2λn 0 < 1. By considering constant initial conditions, one gets the conclusion with µ = Ce −2λn 0 .

Arbitrary slow convergence
In this subsection, we positively answer a conjecture posed in [26, Theorem 4.1, Remark 2] regarding the worst possible decay rate of solutions of (1) when Σ is of saturation type.
A graphical representation of the region R C and the rotated region RR C is provided in Figure 5. The darker shade represents the intersection between R C and the damping region {(x, y) ∈ R 2 | xy ≥ 0}.
Proof. Let S be the set-valued map whose graph is RΣ. The assumptions on Σ, namely the fact that Σ is a damping set and that Σ ⊂ R C , imply that, for every x ∈ R and y ∈ S(x), one has |x| − C ≤ |y| ≤ |x|.
From Propositions 6 and 27, it is enough to construct g 0 ∈ Y p such that, for every iterated sequence (g n ) n∈N in Y p for S starting at g 0 , one has g n p ≥ ϕ(2(n − 1)), ∀n ∈ N * .
Up to dividing the sequence (g n ) n∈N and the function ϕ by C, we assume with no loss of generality that C = 1.
R C R RR C Figure 5: Regions R C and RR C .
Let C p = 1 3 p−1 and φ : R + → R + be a decreasing function such that φ(t) = 1 Cp ϕ p (2( t 1/p 3 − 1)) for every t ≥ 3 p . Note that C p φ(3 p n p ) = ϕ p (2(n − 1)) for every n ∈ N * . Let (a n ) n∈N and (b n ) n∈N be the sequences obtained by applying Lemma 85 in Appendix D to φ. Define g 0 : and consider an iterated sequence (g n ) n∈N for S starting at g 0 . Using the fact that the sequence (b n ) n∈N converges to 0 and the relationship between the sequences (a n ) n∈N and (b n ) n∈N , it is easy to see that and thus g 0 ∈ Y p , which implies that g n ∈ Y p for every n ∈ N. Moreover, Proposition 17 shows that, for every n ≥ 1, one has Notice that, for every n ∈ N and k ≥ n p , one has |α k,n | ≥ k 1/p − n. Then, for every n ∈ N, one has We next prove that, for every n ∈ N * and k ≥ 3 p n p , one has To see that, we rewrite the left-hand side of (73) as k(A p − B p ) with Using that k ≥ 3 p n p , one deduces at once that 1 3 one concludes that (73) holds. It now follows from (72), (73), and the expression of the sequence (a k ) k∈N with respect to (b k ) k∈N that as required.
Remark 60. Note that Theorem 59 does not hold for p = +∞. Indeed, assume that Σ ⊂ R 2 is the graph of the piecewise linear saturation function σ defined by σ(x) = x for |x| ≤ 1 and σ(x) = x |x| for |x| ≥ 1. Then RΣ is the graph of the function S given by S(x) = 0 for |x| ≤ √ 2 and S(x) = −x + √ 2 x |x| for |x| ≥ √ 2. A straightforward computation using Proposition 6 shows that, for every (z 0 , z 1 ) ∈ X ∞ , we have e ∞ (z)(t) = 0 for every t ≥ 2 e ∞ (z)(0)/ √ 2 , and thus an estimate such as that of Theorem 59 cannot hold. •

Boundary perturbations
In this section, we show that the framework developed previously to address the stability of (1) can also be applied to handle wave equations with disturbances in the boundary condition (z t (t, 1), −z x (t, 1)) ∈ Σ. More precisely, the disturbed version of (1) we consider in this section is where d : R + → R 2 is a measurable function representing the disturbance. Given p ∈ [1, +∞] and a disturbance d as above, solutions of (74) in X p can be defined with an obvious modification of Definition 2, consisting in replacing the set Σ in the boundary condition at x = 1 in (11) by Σ + d(t). Proposition 3 still holds for (74) after replacing (12c) by (g(s − 2), g(s)) ∈ RΣ + Rd(s − 1), for a.e. s ≥ 1.
As regards existence and uniqueness of solutions of (74), similarly to Theorems 20 and 21, we deduce in a straightforward manner the following result.
(b) For every (z 0 , z 1 ) ∈ X p and every d ∈ L p loc (R + , R 2 ), there exists a unique solution of (74) in X p with initial condition (z 0 , z 1 ) if and only if either (H3) holds and p < +∞, or (H3) ∞ holds and p = +∞.
We now turn to the issue of asymptotic behavior of solutions of (74). Before stating our results, we need to introduce, for every damping set Σ ⊂ R 2 , the function µ : where we recall that S is the set-valued map whose graph is RΣ. Similarly to the function ρ introduced in (41), the function µ satisfies the properties stated in Proposition 38, with S [2] replaced by S in (c) and ρ [⌊n/2⌋] replaced by µ [n] in (d). We also need to introduce the space of disturbances D p , p ∈ [1, +∞), given by Note that D p ⊂ L p (R + , R 2 ) with equality if and only if p = 1.
We shall use the next definition of input-to-state stability for (74), which can can be seen as a generalization of the UGAS property in presence of disturbances (see, for instance, [22]).
Definition 62. Let p ∈ [1, +∞]. We say that the dynamical system (74) is inputto-state stable (ISS) with respect to the state space X p and the disturbance space L p (R + , R 2 ) if there exist a KL function β and a K function γ such that, for every trajectory z of (74) associated with an initial condition (z 0 , z 1 ) ∈ X p and a disturbance d ∈ L p (R + , R 2 ), one has The main result of this section is the next theorem, which provides conditions on µ ensuring that (74) is either strongly stable or ISS.
(a) Let d ∈ L p loc (R + , R 2 ) and z be a solution of (74) with disturbance d. Then e p (z)(t) → 0 as t → +∞ if one of the following conditions holds.
(b) The dynamical system (74) is ISS with respect to X p and L p (R + , R 2 ) if one of the following conditions holds.
We first prove (a) under the assumption (a-i). One deduces from (81) that, for n ∈ N and a.e. s ∈ [−1, 1], where D is the function in Y p associated with d ∈ D p in the sense of (79). Hence (h n ) n∈N is a sequence in Y p and it is dominated by the function |h 0 | + 2D, which also belongs to Y p . In addition, δ n (s) tends to 0 as n → +∞ for a.e. s ∈ [−1, 1] since D ∈ Y p . Then, in order to obtain the conclusion of (a), it suffices to show that (h n ) n∈N tends to zero almost everywhere. Let s ∈ [−1, 1] be such that the series of general term |δ n+1,1 (s) − δ n,2 (s)| is convergent. The sequence (|h n (s)|) n∈N being bounded thanks to (82), it is enough to prove that its only limit point is zero. Reasoning by contradiction, assume that there exists a subsequence (|h n k (s)|) k∈N converging to some r * > 0. Then, for every ε > 0, there exists k 0 ∈ N such that |h n k (s)| − r * < ε for every k ≥ k 0 and ∞ n=n k 0 |δ n+1,1 (s) − δ n,2 (s)| < ε. Using (81) and the fact that µ is increasing, one deduces that Hence r * ≤ µ(r * + ε) + 2ε and, since µ is upper semi-continuous and ε > 0 is arbitrary, we deduce that µ(r * ) ≥ r * , yielding the desired contradiction.
First of all, one obtains from Lemma 84 given in Appendix D that there exists a K ∞ function ϕ lower bounding the function r → r − µ(r) and such that Id −ϕ is nondecreasing. In particular, one then has that ϕ −1 : R + → R + is K ∞ . Moreover, if (H8) holds true, one has in addition that Id −ϕ is concave. From (76), one deduces that, for every n ∈ N and s ∈ [−1, 1], one has |g n+1 (s)| ≤ (Id −ϕ)(|g n (s)| + |δ n,1 (s)|) + |δ n,2 (s)|, and thus |g n+1 (s)| ≤ (Id −ϕ)(|g n (s)|) + |δ n,1 (s)| + |δ n,2 (s)|, since, by an immediate computation using the fact that ϕ is nondecreasing, for every a, b ∈ R + , one has (Id −ϕ)(a + b) ≤ (Id −ϕ)(a) + b. Since Id −ϕ is nondecreasing and it is concave when p is finite, one deduces, applying Lemma 82 in Appendix D in the case p < +∞, that, for every n ∈ N, g n+1 p ≤ 2 1/p (Id −ϕ)(2 −1/p g n p ) + 2 δ n p , with the convention that 1/p = 0 for p = +∞. It follows that g n p ≤ k n , where (k n ) n∈N is the trajectory of the one-dimensional discrete-time control system defined by with initial condition and control given respectively by To obtain the conclusion, it is enough to prove that the control system (83) is ISS according to the standard definition of ISS for finite-dimensional control systems as given in [15]. Indeed, in this case, one will deduce that there exist a KL function β and a K function γ such that g n p ≤ k n ≤ β(k 0 , n) + γ sup n∈N u n = β( g 0 p , n) + γ 2 sup n∈N δ n p and the conclusion follows by noticing that sup n∈N δ n p ≤ d L p (R + ,R 2 ) .
Thanks to [15,Theorem 1], the proof of ISS for (83) is reduced to establishing the existence of three K functions γ 0 , σ 1 , σ 2 such that, for every trajectory (k n ) n∈N of (83) starting at k 0 ∈ R and corresponding to a control (u n ) n∈N , one has lim sup n→+∞ |k n | ≤ γ 0 lim sup Finally, (85) follows by proving that, for every n ∈ N, where U = sup n∈N |u n |. Indeed, using an inductive argument, the above inequality holds trivially for n = 0, while the induction step follows by considering separately the cases where |k n | < 2 1/p ϕ −1 (2 −1/p U ) or not.
Remark 64. Consider a damping set Σ verifying the following generalized sector condition, which is a global version of (H9): there exist a positive constant M and a function q ∈ C 1 (R + , R + ) with q(0) = 0, 0 < q(x) < x, and |q ′ (x)| < 1 for every x > 0 such that q(|x|) ≤ |y| and q(|y|) ≤ |x| for every (x, y) ∈ Σ. In that case, the function µ defined in (78) satisfies µ ≤ Q, where the function Q is defined from q as in (18). Moreover, conditions on µ expressed in (a-ii) and (b-i) are satisfied if lim inf x→+∞ q(x) > 0 and lim x→+∞ q(x) = +∞, respectively. • Remark 65. One can weaken the assumption in (b-i) to p = +∞ and lim inf r→+∞ r − µ(r) =: ℓ > 0 and obtain an ISS-type result under additional assumptions on the L ∞ norm of the disturbance d. More precisely, in that case, one can adapt the proof of Lemma 84 in Appendix D to provide a K function ϕ lower bounding Id −µ such that Id −ϕ is nondecreasing, but whose range is [0, ℓ) instead of R + . By following the same lines of the arguments of Item (b-i), one deduces an ISS-type estimate for trajectories of (74) associated with disturbances d in L ∞ (R + , R 2 ) with d L ∞ < ℓ/2, i.e., one has an estimate of the form (80) but the function γ is defined only on [0, ℓ/2). • Remark 66. To the best of our knowledge, few results on (74) have been obtained and the most precise ones can be found in [27], which considers the case where Σ is the graph of a saturation function and essentially deals with disturbance rejection. Theorem 63(a) improves the results of that reference since the disturbance there is assumed to be matching and regular (both d(·) and d ′ (·) belong to D 1 ) and Remark 65 shows that we can achieve ISS-type of results in X ∞ . •

Application to the case of the sign function
In this section, we apply the framework introduced in the paper to the particular choice of Σ as the graph of the sign multi-valued function defined in (19), which is not the graph of a function σ : R → R and which is not a strict damping. We will illustrate how our techniques easily handle this case, obtaining in particular existence and uniqueness of solutions of (1) and the precise characterization of their asymptotic behavior. This boundary condition has been previously considered in the literature, for instance in [27]. We then consider that Σ = Σ M , where Σ M is defined in (20), and, for sake of simplicity, we assume in the sequel that M = √ 2. All the results that we will present in this section in that case readily extend to the general case of Σ M for any M > 0 by simply remarking that z is a solution of (1) associated with the set Σ M if and only if √ 2z/M is a solution of (1) associated with the set Σ √ 2 . A straightforward computation shows that RΣ is the graph of a function (which we denote by S in a slight abuse of notation) given by Note that this set satisfies both (H3) and (H3) ∞ and thus, as an immediate consequence of Theorem 21, we obtain the following.
Remark 68. In [27], the sign function has been introduced as the limit of linear saturated feedbacks, and the existence and uniqueness result therein, [27,Lemma 3], is shown in the Hilbertian framework of X 2 by proving that the generator of the corresponding equation in X 2 is a maximal monotone nonlinear operator. With respect to that framework, our techniques allow one to consider solutions of (1) in X p for any p ∈ [1, +∞]. • We now turn to the asymptotic behavior of trajectories of (1) in X p for p ∈ [1, +∞] with Σ given by (20) and M = √ 2. It is immediate to see that Σ is a damping set and hence, by Proposition 27, the energy e p (z)(·) is nonincreasing along trajectories z. On the other hand, S is continuous, and hence its graph is closed, as well as the graphs of its iterates S [n] for n ∈ N. Since S(x) = x for x ∈ [−1, 1], S [2] is not a strict damping, and thus Corollary 44 immediately implies that (1) is not strongly stable in X p for any p ∈ [1, +∞]. A more direct way to see that, which is similar to the argument provided in [27,Lemma 4], consists of considering a constant function g 0 in Y p whose constant value belongs to [−1, 1]. The corresponding solution z of (1) with initial condition I −1 (g 0 ) corresponds to the sequence (g n ) n∈N which is constant and equal to g 0 .
We next characterize the limits of solutions of (1). We start with the following preliminary result dealing with limits of real iterated sequences for S, which is obtained by straightforward computations.
(a) For every n ∈ N, one has S [n] (0) = 0 and, for x = 0, Let (x n ) n∈N be a real iterated sequence for S starting at x 0 = 0 and denote Then (x n ) n≥k 0 is constant and For sake of simplicity and using an abuse of notation, we will write the right-hand side of (87) also when x = 0, and it should be considered as being equal to zero in that case.
As an immediate consequence of Lemma 69, we have the following. Then S • g ∞ = g ∞ , g n converges to g ∞ a.e. on [−1, 1], and |g n | ≤ |g 0 |. In particular, if g 0 ∈ Y p for some p ∈ [1, +∞), then g n converges to g ∞ in Y p . In addition, if g 0 ∈ Y ∞ , then g n = g ∞ for every n ≥ K ∞ .
Translating the above result in terms of solutions of (1), one immediately gets the following.
Theorem 71. Let p ∈ [1, +∞], (z 0 , z 1 ) ∈ X p , and consider the solution z of (1) with Σ given by (20) with M = √ 2 and with initial condition (z 0 , z 1 ). Let g 0 = I(z 0 , z 1 ) ∈ Y p , g ∞ be defined from g 0 as in Proposition 70, and z ∞ be the solution of (1) starting at I −1 (g ∞ ). Then z ∞ is 2-periodic and z(t) − z ∞ (t) converges to 0 in X p as t → +∞. Moreover, if p = +∞, the convergence takes place in finite time less than or equal to T = 2 Proof. We only provide an argument for the 2-periodicity of z ∞ . Indeed, recall that, by Proposition 6(b), the Riemannian invariants of z ∞ are built after the S-iterates of g ∞ . The latter being a fixed point of S, the Riemannian invariants of z ∞ are then 2-periodic. One derives the 2-periodicity of z ∞ by using the first equation of (11).
Remark 72. Theorem 71 improves the result [27,Theorem 6] in the following directions: firstly, Theorem 71 applies to solutions of (1) with initial conditions in X p for some p ∈ [1, +∞], whereas the main convergence result in [27,Theorem 6] only consider regular solutions of (1) in the Hilbertian setting. Secondly, thanks to Theorem 71, we provide an expression for the limit z ∞ which is more explicit than the one based on Fourier series provided in [27,Theorem 6]. • Remark 73. At the light of the finite-time convergence in X ∞ in Theorem 71, one may wonder whether uniform bounds on the convergence rate can also be obtained for finite p. Unfortunately, the answer turns out to be negative, since, by Theorem 59, the energy of solutions of (1) in X p for finite p may decrease arbitrarily slow. •

A Universally measurable functions
Hypotheses (H2)-(H3) ∞ used throughout this paper use the notion of universally measurable functions. This appendix provides the definition of this class of functions together with their main properties used in this paper. Interested readers may find further properties of universally measurable sets and functions and their applications in analysis in [6,9,23]. For sake of simplicity, we only consider here real-valued universally measurable functions defined on a (possibly unbounded) interval I ⊂ R, since this particular setting is the only one used in the paper. We denote by B and L the σ-algebras of Borel and Lebesgue measurable subsets of R, respectively.
Definition 74. Let F be a σ-algebra on R.
(a) Let µ be a nonnegative measure on (R, F).
(b) A set M ⊂ R is said to be universally measurable with respect to F if, for every probability measure µ on (R, F), M is µ-measurable. The family of all universally measurable subsets of R with respect to F is denoted by F * . When F is the σ-algebra B of Borel subsets of R, we define U = B * and we say that the elements of U are the universally measurable subsets of R.
(c) Let I ⊂ R be an interval and f : I → R. We say that f is an universally measurable function if, for every A ∈ B, one has f −1 (A) ∈ U.
For every σ-algebra F of R, the set F * is also a σ-algebra and one has further that (F * ) * = F * . In particular, U * = U. The σ-algebra U is called the universal σ-algebra of R. Using the fact that the Lebesgue measure is complete with respect to the σ-algebra L, one also immediately checks that L * = L.
The above definition implies that B ⊂ U ⊂ L, and classical counterexamples presented in [6,9,23] show that these inclusions are strict. One deduces from these inclusions that every Borel measurable function is universally measurable and that every universally measurable function is Lebesgue measurable. Note that, as stated, e.g., in [23], the requirement on µ to be a probability measure in Definition 74(b) may be replaced with the requirement on µ being finite or also σ-finite with no change in the definition.
A classical result on universal measurability is the following property, whose proof can be found, for instance, in [9,Lemma 8.4.6].
Proposition 75. Let F and G be σ-algebras on R, I ⊂ R be an interval, and f : As an immediate consequence of Proposition 75, one obtains the following alternative characterizations of Lebesgue and universally measurable functions.  Corollary 76(b) implies in particular that universal measurability of functions is preserved under composition, as we state next.
Proposition 77. Let f : R → R and g : R → R be universally measurable functions. Then f • g is also universally measurable.
The major result on universally measurable functions that we need in this paper is the following, which characterizes the set of universally measurable functions as those which preserve Lebesgue measurability by left composition. The statement and the proof presented below were communicated to the authors 1 by Mateusz Kwaśnicki.
Proposition 78. Let f : R → R. Then f is universally measurable if and only if, for every Lebesgue measurable function g : (−1, 1) → R, f • g is Lebesgue measurable.
Proof. Assuming first that f is universally measurable, one immediately obtains from the characterizations in Corollary 76 that f • g is Lebesgue measurable for every Lebesgue measurable function g : (−1, 1) → R.
Let us now assume that f is not universally measurable. We will construct a continuous function g : (−1, 1) → R such that f • g is not Lebesgue measurable.
Since f is not universally measurable, there exists B ∈ B such that A = f −1 (B) / ∈ U. Thus, there exists a probability measure µ on (R, B) such that A is not µmeasurable. Let λ be the standard Gaussian probability measure on R, i.e., dλ(x) = 1 √ 2π e −x 2 /2 dx, and consider the probability measure ν 0 = 1 2 µ + 1 2 λ on (R, B). Clearly, A is not ν 0 -measurable either. Let ν be the probability measure obtained from ν 0 by removing its atoms and renormalizing the resulting measure, and notice that A is not ν-measurable.
We claim that h(A) is not Lebesgue measurable. Indeed, if it were not the case, there would exist two Borel subsets F 1 , F 2 of (0, 1) such that F 1 ⊂ h(A) ⊂ F 2 and m(F 2 \ F 1 ) = 0. Then, letting E i = h −1 (F i ) for i ∈ {1, 2}, we would have that E 1 and E 2 are Borel sets (since h is continuous) with E 1 ⊂ A ⊂ E 2 and ν(E 2 \ E 1 ) = m(h(E 2 \ E 1 )) = m(F 2 \ F 1 ) = 0, implying that A is ν-measurable, a contradiction.

B An optimal decay rate
This appendix proves the following result, which identifies the optimal decay rate of Q [n] (x 0 ) when q(x) = x (− ln x) p for x > 0 small enough and Q is defined from q as in (18) (cf. Remark 48).
Set N = 1 2p . Then there exist N + 1 real numbers α k , k ∈ {0, . . . , M }, with α 0 = (2(p + 1)) 1 p+1 , such that, as n → +∞, one has Proof. Notice first that q ′ (0) = 0. Thanks to the assumptions on q, (x n ) n∈N is a decreasing sequence of positive real numbers with x n → 0 as n → +∞ and we assume that x n ∈ (0, M √ 2 ) for every n ≥ 0 with no loss of generality (cf. Proposition 49(a)). Let F be the diffeomorphism defined in Proposition 46(a). One computes that, for z ∈ (0, x 0 ], it holds where C is a positive constant. In particular, q does not satisfy (49). We start the argument for the theorem by setting some notations for the subsequent computations It follows at once that all the sequences defined above are positive, decreasing, and tend to zero as n tends to infinity. By manipulating their definitions and using also (18), the explicit expression of q, and the fact that x n+1 = Q(x n ) for n ∈ N, we deduce that, for n ∈ N and a ∈ R, and ξ a n = µ a n 1 − µ p+1 For the rest of the argument, we also use the standard symbols ∼, O(·) and o(·) as n tends to infinity without writing the latter fact. By (91), one has that ξ n ∼ µ n and it follows from (89) and Proposition 46(a) that Moreover, since 0 < µ n < 1 for every n ∈ N, one has On the other hand, one deduces from (91), (92), and the fact that ξ n ∼ µ n , that µ n ∼ 1 α 0 n 1/(p+1) , µ a n = ξ a n 1 + O n −1 , and k≥0 µ 2pk The above equation yields, together with (95), that (94) can be written, after taking its (p + 1)-th power, as By summing up the above equations between 1 and n, one deduces that which implies that By taking the exponential of the above relation, the theorem is proved in the case p > 1 2 . We next suppose that p ∈ (0, 1 2 ] and in that case N = 1 2p ≥ 1. By using (95), one rewrites (96) as k≥0 µ 2pk One then rewrites (97) as We next prove that there exists N real numbers γ k for k ∈ {1, . . . , N } such that To see that, we set γ 0 = α p+1 0 and we will prove by induction on j ∈ {0, . . . , N } the following property: there exist N + 1 real numbers γ k for k ∈ {0, . . . , N } so that, for every j ∈ {0, . . . , N }, setting Note that f j (n) ∼ f 0 (n) = α p+1 0 n and the property is clearly true for j = 0 by (98). For the inductive step, assume the property holds for some j ∈ {0, . . . , N − 1} and let us establish it for j + 1. It amounts to prove that there exists a real number γ j+1 such that Using the induction assumption (102) and the definition of F j (n), we have and, since j ≤ N −1, we have 2p(j+1) p+1 ≤ 1 p+1 < 1, showing that, for every k ∈ {1, . . . , N }, Moreover, we also have the estimate We inject the expression 1 ξ p+1 n = f j (n) + F j (n) into (100) and, after computations using (104) and (105), one obtains that Denoting the second sum by T j (n), one has that and write its Taylor expansion around Z = 0 as ϕ j,k (Z) = 1 + ℓ≥1 r j,k,ℓ Z ℓ . Then, letting Z = n − 2p p+1 in the previous expression, one gets for suitable coefficientsr j,ℓ , ℓ ∈ {1, . . . , j + 1}. The key point is to notice that, for ℓ ∈ {1, . . . , j}, the coefficients r j,k,ℓ only depend on γ 0 , . . . , γ ℓ and not on γ s for s > ℓ nor on j. As a consequence, the coefficientsr j,ℓ only depend on γ 0 , . . . , γ ℓ−1 and not on γ j nor j. Hence, γ ℓ , for ℓ ∈ {0, . . . , N }, is chosen according to the relation , which is possible since the right-hand side only involves γ 0 , . . . , γ ℓ−1 and does not depend on j > l. In other words, γ ℓ is determined exactly at step ℓ of the induction.
, one gets (103) after summation of the previous equation between one and n large. The induction step has been established, which concludes the proof of (101). One deduces from (101) that Since the term can be seen as a polynomial in the indeterminate p+1 with zero constant term, it is clear that there exist N real numbers α k with k ∈ {1, . . . , N } such that .
Plugging the above equation in (108), one gets that , for n large enough. Taking the exponential yields (88).

C Proof of Proposition 50
Proof. If (55) holds true for some ϕ as in the statement, then it still holds true for any function satisfying the same assumptions and which is larger than ϕ on any interval [x 0 , +∞), x 0 ≥ 0. By using Lemma 83, it is therefore enough to prove the proposition for ϕ which, in addition to the above mentioned hypotheses, is also C 2 , with ϕ ′ > 0, ϕ ′′ ≤ 0, ϕ(0) > 0, and such that 0 ≤ xϕ ′ (x) ϕ(x) ≤ 1 and 0 ≤ − xϕ ′′ (x) ϕ(x) ≤ 2 for every x ≥ 0, which we assume in the sequel.
We first note that, for every C > 0, one has since, for x ≥ 0, one has which tends to zero as x tends to infinity, yielding (109). Notice that it suffices to construct the functions q and Q in a neighborhood of zero (in R + ) and to prove (55) for x 0 > 0 in a neighborhood of zero (in R + ). Indeed, if that is done, one can immediately extend q and Q to R + in such a way that the assumptions from the statement are satisfied and, in this case, for any x 0 > 0, the sequence (Q [n] (x 0 )) n∈N is decreasing and converging to zero, showing that Q [n] (x 0 ) is in a neighborhood of zero for every n large enough. Hence, in the sequel, we only construct q and Q in a neighborhood of zero and we only show (55) for x 0 ∈ (0, 1) belonging to that neighborhood.
If q and Q are defined in a neighborhood of zero as in the statement, x 0 ∈ (0, 1) belongs to that neighborhood, and we let y n = − 1 ln(Q [n] (x 0 )) for n ≥ 0, then one verifies from straightforward computations that the sequence (y n ) n∈N satisfies the recurrence relation y n+1 = y n − U (y n ), n ∈ N, where U is defined in a neighborhood of zero by U (0) = 0 and U (y) = − y 2 ln ψ(e −1/y ) 1 − y ln ψ(e −1/y ) , y > 0, and ψ is defined in a neighborhood of zero by ψ(0) = 0 and ψ(x) = Q(x) x for x > 0. Conversely, given a function U , defining ψ in a neighborhood of zero in such a way that (111) holds, setting Q(x) = xψ(x) and defining q from Q using (18), any sequence (y n ) n∈N starting in a neighborhood of zero and satisfying (110) is of the form y n = − 1 ln(Q [n] (x 0 )) for some suitable x 0 ∈ (0, 1) in a neighborhood of zero. Moreover, in terms of the sequence (y n ) n∈N , (55) reads Hence, constructing q and Q as in the statement is equivalent to constructing a function U such that the functions q and Q defined from it as above satisfy the properties of the statement and such that any sequence (y n ) n∈N satisfying (110) and starting in a neighborhood of zero verifies (112). Define Ψ(x) = 2 xϕ(x) for x > 0. Then Ψ realizes a C 2 diffeomorphism from R * + to R * + , mapping a neighborhood of +∞ to a neighborhood of 0. Moreover, for x ≥ 0, one has and In particular, one has Ψ ′ < 0 and Ψ ′′ > 0 on R * + . Straightforward computations also show that 0 < − xΨ ′′ (x) Ψ ′ (x) ≤ 4 for every x > 0 and yΨ −1 (y) = 2Ψ −1 (y) Ψ −1 (y)ϕ(Ψ −1 (y)) → 0 as y → 0 + . Using the above bound on xΨ ′′ (x) Ψ ′ (x) and reasoning as in the argument to obtain (109), one deduces that, for every C > 0, We claim that the function U defined by U (0) = 0 and U = − 1 2 Ψ ′ •Ψ −1 in R * + meets all the requirements. Indeed, U is of class C 1 in R * + and, since U (y) = y with Z = Ψ −1 (y), one deduces that U is continuous at 0, 0 < U (y) < y for every y > 0, U ′ (0) = lim y→0 + U (y) y = 0, and, using that yΨ −1 (y) → 0 as y → 0 + , we also deduce that lim y→0 + U (y) Let ψ be defined by ψ(0) = 0 and for x in a neighborhood (0, x * ) of 0 with x * ∈ (0, 1), in such a way that (111) holds for y in a neighborhood of 0. Using the above properties on U , one deduces that ψ is of class C 1 in (0, x * ), continuous at 0, and ψ(x) ∈ (0, 1) for x ∈ (0, x * ). We claim that, up to reducing x * , one has xψ ′ (x) > 0 for x ∈ (0, x * ) and xψ ′ (x) → 0 as x → 0 + . Indeed, notice that ψ can be written for x ∈ (0, x * ) as A straightforward computation yields that , and the above properties of U show that xψ ′ (x) > 0 for x small enough and xψ ′ (x) → 0 as x → 0 + .
We are now left to prove (112) for every sequence (y n ) n∈N satisfying (110) and with y 0 > 0. Fix such a sequence (y n ) n∈N and notice that, since U is continuous and 0 < U (y) < y for every y ∈ R * + , (y n ) n∈N is a decreasing sequence of positive numbers converging to 0. We claim that there exists n 0 ∈ N such that, for every n ∈ N, one has y n ≥ Ψ(n + n 0 ).

D Technical lemmas
This appendix provides a series of technical results used in the paper. The first one is useful for establishing existence and uniqueness results for solutions of (1) in Section 3.
Proof. Let g : [−1, 1] → R be measurable. Let A 0 = g −1 ([−1, 1]) and, for n ∈ N * , let A n = g −1 ([−n − 1, −n) ∪ (n, n + 1]). Then clearly A n is measurable for every n ∈ N and the sequence (A n ) n∈N is a partition of [−1, 1]. For each n ∈ N, define g n : [−1, 1] → N by g n = gχ An , where χ An denotes the characteristic function of A n . Then g n is measurable and bounded, and hence g n ∈ Y p . Hence, there exists a sequence (h n ) n∈N in Y p such that h n (s) ∈ S(g n (s)) for every n ∈ N and a.e. s ∈ [−1, 1]. Let h = ∞ n=0 h n χ An , which is measurable as the countable sum of measurable functions. For every n ∈ N and a.e. s ∈ A n , one has h(s) = h n (s) ∈ S(g n (s)) = S(g(s)), and thus h(s) ∈ S(g(s)) for a.e. s ∈ [−1, 1], as required.
The definition of uniform global asymptotic stability of (1) requires (28) to be satisfied for some KL function β. Our next lemma provides sufficient conditions under which a function can be upper bounded by a KL function, and it is thus useful in several proofs of UGAS results.
Lemma 81. Let f : R + × R + → R + be a function so that (a) f (0, ·) ≡ 0; (b) for every t ≥ 0, x → f (x, t) is nondecreasing and tends to zero as x tends to 0; (c) for every x ≥ 0, t → f (x, t) is nonincreasing and tends to zero as t tends to infinity.
The next result, used in Remark 53 and in the proof of Theorem 63, is a generalization of Jensen's inequality to L p norms. Its proof follows closely the classical proof of Jensen's inequality and is provided here for sake of completeness.
The next two lemmas provide suitable constructions of functions and are used, respectively, in the proofs of Proposition 50 and Theorem 63.
Proof. It suffices to prove the result with the additional assumptions that ϕ is also continuous and piecewise affine with positive constant derivative on every interval of the form (n, n+1). Indeed, when this is not the case, one can easily construct a function φ using a procedure similar to that of Lemma 81 in such a way that φ(x) ≤ ϕ(x) for x large enough and φ satisfies the assumptions of the theorem as well as the previous additional assumptions. We then assume these additional assumptions in the sequel. For n ∈ N, let ϕ n = ϕ(n) and denote byφ n > 0 the constant value of the derivative of ϕ in the interval (n, n + 1). One has that ϕ n = ϕ 0 + n−1 k=0φ k , which is an increasing sequence tending to infinity. Also note that we can assume with no loss of generality that ϕ 0 andφ 0 are both positive.
We first construct a continuous increasing piecewise affine function F : R + → R + with F ≤ ψ, lim x→+∞ F (x) = +∞, and such that its derivative f = F ′ is nonincreasing and constant at every interval of the form (n, n + 1). For that purpose, it is sufficient to construct the sequence (F n ) n∈N of the values of F (x) at the points x = n and the sequence (f n ) n∈N of the constant values of f on the intervals (n, n + 1).
It remains to prove that F n tends to infinity as n tends to infinity. Arguing by contradiction yields that both sequences (F n ) n∈N and (f n ) n∈N are bounded. Hence there exists an integer n 0 so that F n + f n−1 ≤ ϕ n+1 for every n ≥ n 0 , since (ϕ n ) n∈N tends to infinity. Therefore, by definition, (f n ) n≥n 0 is constant and equal to some f > 0, yielding that F n = F n 0 + f (n − n 0 ) for n ≥ n 0 , which contradicts the fact that (F n ) n∈N is bounded, establishing this the result.
To obtain the required function ψ, we define ψ(x) = F (x) for x ∈ [0, 1 2 ] and we regularize the function F in a neighborhood of each positive integer as follows: for n ∈ N * and s ∈ [0, 1], we set ψ n − 1 2 + s = f n − f n−1 2 s 2 + f n−1 s + F n − f n−1 2 .
Note that ψ and ψ ′ coincide with F and f , respectively, at all points of the form n + 1 2 for n ∈ N, and that ψ ′ is positive, continuous, and nonincreasing. In particular, from the latter fact, we get which implies that 0 ≤ xψ ′ (x) ψ(x) ≤ 1. Moreover, for every x ≥ 1 2 , there exist n ∈ N * and s ∈ [0, 1] such that x = n − 1 2 + s, and one has which concludes the proof.
We are only left to prove that ϕ(x) → +∞ as x → +∞ or, equivalently, that F n → +∞ as n → +∞. Reasoning by contradiction yields that, since (F n ) n∈Z is increasing, there exists F * > 0 such that F n → F * as n → +∞ and F n < F * for every n ∈ Z. Let n 0 ∈ N be such that ϕ 0 (x n ) ≥ F * for every n ≥ n 0 , which exists since ϕ 0 (r) → +∞ as r → +∞. Hence, for every n > n 0 , we have F n < ϕ 0 (x n−1 ), and the inductive definition of F n implies that F n = F n−1 + δ n−1 . Thus F n = F n 0 + n−1 k=n 0 δ k = F n 0 + x n − x n 0 and, as n → +∞, one has x n → +∞, implying that F n → +∞ and yielding the required contradiction.
Finally, we turn to the second part of the statement, namely that, under the extra assumption of the existence of a ∈ (0, 1) and M ≥ 0 such that µ(r) ≤ ar for every r ≥ M , one may construct ϕ to be convex. With no loss of generality, we assume that M > 0. Let ϕ be constructed from µ as above, λ = min 1, (1−a)M
Our final technical result in this appendix is the following lemma, used in the proof of Theorem 59.
Proof. Notice first that b n > 0 for every n ∈ N since b n ≥ φ(n) > 0.
It now follows that a n > 0 for every n ∈ N, since a 0 = 1 and, for n ≥ 1, a n = b n−1 − b n and (b n ) n∈N is decreasing.
Let us now show that (a n ) n∈N is nonincreasing. One has b 1 ≥ b 0 − 1, and thus a 1 = b 0 − b 1 ≤ 1 = a 0 . For n ≥ 2, one has b n ≥ 2b n−1 − b n−2 , which implies that b n − b n−1 ≥ b n−1 − b n−2 , and hence a n ≤ a n−1 , as required.
Since (b n ) n∈N is decreasing, this sequence admits a limit b * ∈ [0, b 0 ). Then lim n→+∞ a n = lim Assume, to obtain a contradiction, that b * > 0. Using also the fact that φ(t) → 0 as t → +∞, one deduces that there exists N ≥ 2 such that, for every n ≥ N , one has b n > b * 2 and φ(n) < b * 2 . Hence, one has necessarily b n = 2b n−1 − b n−2 for every n ≥ N , which implies that b n−1 − b n = b n−2 − b n−1 , and thus a n = a n−1 for every n ≥ N . Since a n → 0 as n → +∞, this implies that a n = 0 for every n ≥ N , which contradicts the fact that a n > 0 for every n ∈ N. This contradiction establishes that b * = 0, as required.