Hedy Attouch
email: hedy.attouch@univ-montp2.fr

Alexandre Cabot
email: alexandre.cabot@u-bourgogne.fr

CONVERGENCE OF DAMPED INERTIAL DYNAMICS GOVERNED BY REGULARIZED MAXIMALLY MONOTONE OPERATORS

Keywords: asymptotic stabilization, damped inertial dynamics, Lyapunov analysis, maximally monotone operators, time-dependent viscosity, Yosida regularization AMS subject classification. 37N40, 46N10, 49M30, 65K05, 65K10, 90C25

. In this last paper, the authors considered the case γ(t) = α t , which is naturally linked to Nesterov's accelerated method. We unify, and often improve the results already present in the literature.

Introduction

Throughout this paper, H is a real Hilbert space endowed with the scalar product ., . and the corresponding norm . . Let A : H → 2 H be a maximally monotone operator. Given continuous functions γ : [t 0 , +∞[→ R + and λ : [t 0 , +∞[→ R * + where t 0 is a fixed real number, we consider the second-order evolution equation (RIMS) γ,λ ẍ(t) + γ(t) ẋ(t) + A λ(t) (x(t)) = 0, t ≥ t 0 ,

where

A λ = 1 λ I -(I + λA) -1
is the Yosida regularization of A of index λ > 0 (see Appendix A.1 for its main properties). The terminology (RIMS) γ,λ is a shorthand for "Regularized Inertial Monotone System" with parameters γ, λ.

Thanks to the Lipschitz continuity properties of the Yosida approximation, this system falls within the framework of the Cauchy-Lipschitz theorem, which makes it a well-posed system for arbitrary Cauchy data. The above system involves two time-dependent positive parameters: the damping parameter γ(t), and the Yosida regularization parameter λ(t). We shall see that, under a suitable tuning of the parameters γ(t) and λ(t), the trajectories of (RIMS) γ,λ converge to solutions of the monotone inclusion 0 ∈ A(x).

Indeed, the design of rapidly convergent dynamics and algorithms to solve monotone inclusions is a difficult problem of fundamental importance in many domains: optimization, equilibrium theory, economics and game theory, partial differential equations, statistics, among other subjects. Trajectories of Date: February 19, 2018.

(RIMS) γ,λ do so in a robust manner. Indeed, when A is the subdifferential of a closed convex proper function Φ : H → R ∪ {+∞}, we will obtain rates of convergence of the values, which are comparable to the accelerated method of Nesterov. With this respect, as a main advantage of our approach, we can handle nonsmooth functions Φ.

1.1. Introducing the dynamics. The (RIMS) γ,λ system is a natural development of some recent studies concerning rapid inertial dynamics for convex optimization and monotone equilibrium problems. We will rely heavily on the techniques developed in [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF] concerning the general damping coefficient γ(t), and in [START_REF] Attouch | Convergence of inertial dynamics and proximal algorithms governed by maximal monotone operators[END_REF] concerning the general Yosida regularization parameter λ(t).

1.1.1. General damping coefficient γ(t). Some simple observations lead to the introduction of quantities that play a central role in our analysis. Taking A = 0, then A λ = 0, and (RIMS) γ,λ boils down to the linear differential equation ẍ(t) + γ(t) ẋ(t) = 0.

Let us multiply this equality by the integrating factor Throughout the paper, we always assume that condition (H 0 ) is satisfied. For s ≥ t 0 , we then define the quantity Γ(s) by The function s → Γ(s) plays a key role in the asymptotic behavior of the trajectories of (RIMS) γ,λ . This was brought to light by the authors in the potential case, see [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF] (no regularization process was used in this work). The theorem below gathers the main results obtained in [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF] for a gradient operator A = ∇Φ.

p(t
It enlights the basic assumptions on the function γ(t) which give rates of convergence of the values.

Theorem (Attouch and Cabot [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF]). Let Φ : H → R be a convex function of class C 1 such that argmin Φ = ∅. Let us assume that γ : [t 0 , +∞[→ R + is a continuous function satisfying:

(i) +∞ t0 ds p(s) < +∞; (ii) There exist t 1 ≥ t 0 and m < 3 2 such that γ(t)Γ(t) ≤ m for every t ≥ t 1 ; (iii) +∞ t0 Γ(s) ds = +∞.

Then every solution trajectory x : [t 0 , +∞[→ H of (IGS) γ ẍ(t) + γ(t) ẋ(t) + ∇Φ(x(t)) = 0, converges weakly toward some x * ∈ argmin Φ, and satisfies the following rates of convergence:

Φ(x(t)) -min as t → +∞.

The (IGS) γ system was previously studied by Cabot, Engler and Gadat [START_REF] Cabot | On the long time behavior of second order differential equations with asymptotically small dissipation[END_REF][START_REF] Cabot | Second order differential equations with asymptotically small dissipation and piecewise flat potentials[END_REF] in the case of a vanishing damping coefficient γ(t) and for a possibly nonconvex potential Φ. The importance of the dynamics (IGS) γ in the case γ(t) = α/t (α > 1) was highlighted by Su, Boyd and Candés in [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF]. They showed that taking α = 3 gives a continuous version of the accelerated gradient method of Nesterov. The corresponding rate of convergence for the values is at most of order O(1/t 2 ) as t → +∞. Let us show how this result can be obtained as a consequence of the above general theorem. Indeed, taking γ(t) = α/t gives after some elementary computation

Γ(t) = t t 0 α +∞ t t 0 τ α dτ = t α τ -α+1 -α + 1 +∞ t = t α -1 .
Then, the condition γ(t)Γ(t) ≤ m with m < 3 2 is equivalent to α > 3. As a consequence, for γ(t) = α/t and α > 3, we obtain the convergence of the trajectories of (IGS) γ and the rates of convergence Φ(x(t)) -min

H Φ = o 1 t 2 and ẋ(t) = o 1 t as t → +∞.
This result was first established in [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing damping[END_REF] and [START_REF] May | Asymptotic for a second order evolution equation with convex potential and vanishing damping term[END_REF]. Because of its importance, a rich literature has been devoted to the algorithmic versions of these results, see [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing damping[END_REF][START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1 k 2[END_REF][START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF][START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage/Thresholding Algorithm[END_REF][START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF] and the references therein.

The above theorem relies on energetical arguments that are not available in the general framework of monotone operators. It ensues that the expected results in this context are weaker than in the potential case, and require different techniques. That's where the Yosida regularization comes into play.

1.1.2. General regularization parameter λ(t). Our approach is in line with Attouch and Peypouquet [START_REF] Attouch | Convergence of inertial dynamics and proximal algorithms governed by maximal monotone operators[END_REF] who studied the system (RIMS) γ,λ with a general maximally monotone operator, and in the particular case γ(t) = α/t (the importance of this system has been stressed just above). This approach can be traced back to Álvarez-Attouch [START_REF] Álvarez | The heavy ball with friction dynamical system for convex constrained minimization problems, Optimization[END_REF] and Attouch-Maingé [START_REF] Attouch | Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects[END_REF] who studied the equation

ẍ(t) + γ ẋ(t) + A(x(t)) = 0,
where A is a cocoercive operator. Several variants of the above equation were considered by Bot and Csetnek (see [START_REF] Bot | Second order forward-backward dynamical systems for monotone inclusion problems[END_REF] for the case of a time-dependent coefficient γ(t), and [START_REF] Bot | Approaching monotone inclusion problems via second order dynamical systems with linear and anisotropic damping[END_REF] for a linear anisotropic damping). Cocoercivity plays an important role, not only to ensure the existence of solutions, but also in analyzing their long-term behavior. Attouch-Maingé [START_REF] Attouch | Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects[END_REF] proved the weak convergence of the trajectories to zeros of A if the cocoercivity parameter λ and the damping coefficient γ satisfy the condition λγ 2 > 1.

Taking into account that for λ > 0, the operator A λ is λ-cocoercive and that A -1 λ (0) = A -1 (0) (see Appendix A.1), we immediately deduce that, under the condition λγ 2 > 1, each trajectory of ẍ(t) + γ ẋ(t) + A λ (x(t)) = 0 converges weakly to a zero of A. In the quest for a faster convergence, in the case γ(t) = α/t, Attouch-Peypouquet introduced a time-dependent regularizing parameter λ(•) satisfying

λ(t) × α 2 t 2 > 1
for t ≥ t 0 . So doing, in the case of a general maximal monotone operator, they were able to prove the asymptotic convergence of the trajectories to zeros of A. Our approach will consist in extending these results to the case of a general damping coefficient γ(t), taking advantage of the techniques developed in the above mentioned papers [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF] and [START_REF] Attouch | Convergence of inertial dynamics and proximal algorithms governed by maximal monotone operators[END_REF].

1.2. Organization of the paper. The paper is divided into three parts. Part A concerns a general maximally monotone operator A. We show that a suitable tuning of the damping parameter and of the Yosida regularization parameter, gives the weak convergence of the trajectories. Then, we specialize our results to some important cases, including the case of the continuous version of the Nesterov method, that is, γ(t) = α t . In part B, we examine the ergodic convergence properties of the trajectories. In part C, we consider the case where A is the subdifferential of a closed convex proper function Φ : H → R ∪ {+∞}. In this case, we will obtain rates of convergence of the values. In the Appendix we have collected several lemmas related to Yosida's approximation, to Moreau's envelopes and to the study of scalar differential inequalities that play a central role in the Lyapunov analysis of our system.

PART A: DYNAMICS FOR A GENERAL MAXIMALLY MONOTONE OPERATOR

In this part, A : H → 2 H is a general maximally monotone operator such that zerA = ∅, and t 0 is a fixed real number.

Convergence results

Let us first establish the existence and uniqueness of a global solution to the Cauchy problem associated with equation (RIMS) γ,λ .

x ∈ C 2 ([t 0 , +∞[, H) to equation (RIMS) γ,λ , satisfying the initial conditions x(t 0 ) = x 0 and ẋ(t 0 ) = v 0 .
Proof. The argument is standard and consists in writing (RIMS) γ,λ as a first-order system in H × H. By setting

X(t) = x(t) ẋ(t) and F (t, u, v) = v -γ(t)v -A λ(t) (u)
, equation (RIMS) γ,λ amounts to the first-order differential system Ẋ(t) = F (t, X(t)). Owing to the To establish the weak convergence of the trajectories of (RIMS) γ,λ , we will apply Opial lemma [START_REF]Weak convergence of the sequence of successive approximations for nonexpansive mappings[END_REF], that we recall in its continuous form. Lemma 2.2 (Opial). Let S be a nonempty subset of H, and let x : [t 0 , +∞[→ H. Assume that (i) for every z ∈ S, lim t→+∞ x(t) -z exists;

(ii) every weak sequential limit point of x(t), as t → +∞, belongs to S. Then x(t) converges weakly as t → +∞ to a point in S.

We associate to the continuous function

γ : [t 0 , +∞[→ R + the function p : [t 0 , +∞[→ R * + given by p(t) = e t t 0 γ(τ ) dτ for every t ≥ t 0 . Under assumption (H 0 ), the function Γ : [t 0 , +∞[→ R *
+ is then defined by Γ(s) = +∞ s du p(u) p(s) for every s ≥ t 0 . Besides the function Γ, to analyze the asymptotic behavior of the trajectory of the system (RIMS) γ,λ we will also use the quantity Γ(s, t), which is defined by, for any s, t p(s). Suppose that there exists ε ∈]0, 1[ such that for t large enough,

(H 1 ) (1 -ε)λ(t)γ(t) ≥ 1 + d dt (λ(t)γ(t)) Γ(t).
Then for any global solution x(.) of (RIMS) γ,λ , we have

(i) +∞ t0 λ(s)γ(s) ẋ(s)
It ensues that ḧ(t) + γ(t) ḣ(t) = ẋ(t) 2 + ẍ(t) + γ(t) ẋ(t), x(t) -z = ẋ(t) 2 -A λ(t) (x(t)), x(t) -z . (4)
Since z ∈ zerA = zerA λ(t) , we have A λ(t) (z) = 0. We then deduce from the λ(t)-cocoercivity of

A λ(t) that A λ(t) (x(t)), x(t) -z ≥ λ(t) A λ(t) (x(t)) 2 , whence (5) ḧ(t) + γ(t) ḣ(t) ≤ ẋ(t) 2 -λ(t) A λ(t) (x(t)) 2 .
Writing that A λ(t) (x(t)) = -ẍ(t) -γ(t) ẋ(t), we have

λ(t) A λ(t) (x(t)) 2 = λ(t) ẍ(t) + γ(t) ẋ(t) 2 = λ(t) ẍ(t) 2 + λ(t)γ(t) 2 ẋ(t) 2 + 2 λ(t)γ(t) ẍ(t), ẋ(t) ≥ λ(t)γ(t) 2 ẋ(t) 2 + λ(t)γ(t) d dt ẋ(t) 2 = λ(t)γ(t) 2 - d dt (λ(t)γ(t)) ẋ(t) 2 + d dt (λ(t)γ(t) ẋ(t) 2 ).
In view of (5), we infer that

ḧ(t) + γ(t) ḣ(t) ≤ -λ(t)γ(t) 2 - d dt (λ(t)γ(t)) -1 ẋ(t) 2 - d dt (λ(t)γ(t) ẋ(t) 2 ). Let's use Lemma B.1 (i) with g(t) = -λ(t)γ(t) 2 -d dt (λ(t)γ(t)) -1 ẋ(t) 2 -d dt (λ(t)γ(t) ẋ(t) 2 ). Set- ting k(t) := h(t 0 ) + ḣ(t 0 ) t t0 du p(u)
, we obtain for every t ≥ t 0 ,

h(t) ≤ k(t) - t t0 Γ(s, t) λ(s)γ(s) 2 - d ds (λ(s)γ(s)) -1 ẋ(s) 2 + d ds (λ(s)γ(s) ẋ(s) 2 ) ds = k(t) - t t0 Γ(s, t) λ(s)γ(s) 2 - d ds (λ(s)γ(s)) -1 ẋ(s) 2 ds -Γ(s, t)λ(s)γ(s) ẋ(s) 2 t t0 + t t0 d ds Γ(s, t) λ(s)γ(s) ẋ(s) 2 ds.
Let us observe that Γ(t, t) = 0 and that

d ds Γ(s, t) = d ds t s du p(u) p(s) = -1 + γ(s)Γ(s, t).
Then it follows from the above inequality that

h(t) ≤ k(t) - t t0 λ(s)γ(s) -Γ(s, t) 1 + d ds (λ(s)γ(s)) ẋ(s) 2 ds +Γ(t 0 , t)λ(t 0 )γ(t 0 ) ẋ(t 0 ) 2 .
Since Γ(t 0 , t) ≤ Γ(t 0 ) and h(t) ≥ 0, we deduce that (6)

t t0 λ(s)γ(s) -Γ(s, t) 1 + d ds (λ(s)γ(s)) ẋ(s) 2 ds ≤ C 1 ,
with

C 1 := h(t 0 ) + | ḣ(t 0 )| +∞ t0 du p(u) + Γ(t 0 )λ(t 0 )γ(t 0 ) ẋ(t 0 ) 2 . Now observe that Γ(s, t) 1 + d ds (λ(s)γ(s)) ≤ Γ(s, t) 1 + d ds (λ(s)γ(s)) ≤ Γ(s) 1 + d ds (λ(s)γ(s)) .
We then infer from (6) that

t t0 λ(s)γ(s) -Γ(s) 1 + d ds (λ(s)γ(s)) ẋ(s) 2 ds ≤ C 1 .
By assumption, inequality (H 1 ) holds true for t large enough, say t ≥ t 1 . It ensues that for t ≥ t 1 ,

t t1 ελ(s)γ(s) ẋ(s) 2 ds ≤ C 1 -C 2 ,
with

C 2 = t1 t0 λ(s)γ(s) -Γ(s) 1 + d ds (λ(s)γ(s)) ẋ(s) 2 ds.
Taking the limit as t → +∞, we find

+∞ t1 λ(s)γ(s) ẋ(s) 2 ds ≤ 1 ε (C 1 -C 2 ) < +∞.
By using again (H 1 ), we deduce that +∞ t1 Γ(s) ẋ(s) 2 ds < +∞.

(ii) Let us come back to inequality [START_REF] Attouch | Asymptotic behavior of coupled dynamical systems with multiscale aspects[END_REF]. Using Lemma B.1 (i) with g(t) = ẋ(t) 2 -λ(t) A λ(t) (x(t)) 2 , we obtain for every t ≥ t 0 ,

h(t) ≤ h(t 0 ) + ḣ(t 0 ) t t0 du p(u) + t t0 Γ(s, t) ẋ(s) 2 -λ(s) A λ(s) (x(s)) 2 ds.
Since h(t) ≥ 0 and Γ(s, t) ≤ Γ(s), we deduce that

t t0 Γ(s, t)λ(s) A λ(s) (x(s)) 2 ds ≤ h(t 0 ) + ḣ(t 0 ) t t0 du p(u) + t t0 Γ(s) ẋ(s) 2 ds.
Recalling from (i) that +∞ t0

Γ(s) ẋ(s) 2 ds < +∞, we infer that for every t ≥ t 0 ,

t t0 Γ(s, t)λ(s) A λ(s) (x(s)) 2 ds ≤ C 3 ,
where we have set

C 3 := h(t 0 ) + | ḣ(t 0 )| +∞ t0 du p(u) + +∞ t0 Γ(s) ẋ(s) 2 ds.
Since Γ(s, t) = 0 for s ≥ t, this yields in turn

+∞ t0 Γ(s, t)λ(s) A λ(s) (x(s)) 2 ds ≤ C 3 .
Letting t tend to +∞, the monotone convergence theorem then implies that

+∞ t0 Γ(s)λ(s) A λ(s) (x(s)) 2 ds ≤ C 3 < +∞.
(iii) From inequality (5), we derive that

ḧ(t) + γ(t) ḣ(t) ≤ ẋ(t) 2 on [t 0 , +∞[. Recall from (i) that +∞ t0
Γ(s) ẋ(s) 2 ds < +∞. Applying Lemma B.1 (ii) with g(t) = ẋ(t) 2 , we infer that lim t→+∞ h(t) exists. Thus, we have obtained that lim t→+∞ x(t) -z exists for every z ∈ zerA, whence in particular the boundedness of the trajectory x(•).

(iv) Using that the operator A λ(t) is 1 λ(t) -Lipschitz continuous and that A λ(t) (z) = 0, we obtain that This proves the first inequality of (iv). For the second one, take the norm of each member of the equality ẍ(t) = -γ(t) ẋ(t) -A λ(t) (x(t)). The triangle inequality yields

(7) A λ(t) (x(t)) ≤ 1 λ(t) x(t) -z ≤ C 4 λ(t
ẍ(t) ≤ γ(t) ẋ(t) + A λ(t) (x(t)) .
The announced majorization of ẍ(t) then follows from ( 7) and ( 8).

(v) Recall the estimate of (ii) that we write as ( 9)

+∞ t0 Γ(s) λ(s) u(s) 2 ds < +∞, with the function u : [t 0 , +∞[→ H defined by u(t) = λ(t)A λ(t) (x(t))
. By applying [START_REF] Attouch | Convergence of inertial dynamics and proximal algorithms governed by maximal monotone operators[END_REF]Lemma A.4] with γ = λ(t), δ = λ(s), x = x(t) and y = x(s) with s, t ≥ t 0 , we find

λ(t)A λ(t) (x(t)) -λ(s)A λ(s) (x(s)) ≤ 2 x(t) -x(s) + 2 x(t) -z |λ(t) -λ(s)| λ(t) .
This shows that the map t → λ(t)A λ(t) (x(t)) is locally Lipschitz continuous, hence almost everywhere differentiable on [t 0 , +∞[. Dividing by t -s with t = s, and letting s tend to t, we infer that

u(t) = d dt (λ(t)A λ(t) (x(t))) ≤ 2 ẋ(t) + 2 x(t) -z | λ(t)| λ(t) ,
for almost every t ≥ t 0 . In view of (8), we deduce that for almost every t large enough,

u(t) ≤ 2 C 5 p(t) t t0 p(s) λ(s) ds + 2 C 4 | λ(t)| λ(t) , with C 4 = sup t≥t0 x(t) -z < +∞.
Recalling the assumption (H 2 ), we obtain the existence of C 6 ≥ 0 such that for almost every t large enough

u(t) ≤ C 6 Γ(t) λ(t) .
Then we have

d dt u(t) 3 ≤ 3 u(t) u(t) 2 ≤ 3 C 6 Γ(t) λ(t) u(t) 2 .
Taking account of estimate [START_REF] Baillon | Une remarque sur le comportement asymptotique des semigroupes non linéaires[END_REF], this shows that

d dt u(t) 3 + ∈ L 1 (t 0 , +∞).
From a classical result, this implies that lim t→+∞ u(t) 3 exists, which entails in turn that lim t→+∞ u(t) exists. Using again the estimate ( 9), together with the assumption (H 3 ), we immediately conclude that lim t→+∞ u(t) = 0. (vi) To prove the weak convergence of x(t) as t → +∞, we use the Opial lemma with S = zerA. Item (iii) shows the first condition of the Opial lemma. For the second one, let t n → +∞ be such that x(t n ) x weakly as n → +∞. By (v), we have lim n→+∞ λ(t n )A λ(tn) (x(t n )) = 0 strongly in H. Since the function λ is minorized by some positive constant on [t 0 , +∞[, we also have lim n→+∞ A λ(tn) (x(t n )) = 0 strongly in H. Passing to the limit in

A λ(tn) (x(t n )) ∈ A x(t n ) -λ(t n )A λ(tn) (x(t n )) ,
and invoking the graph-closedness of the maximally monotone operator A for the weak-strong topology in H × H, we find 0 ∈ A(x). This shows that x ∈ zerA, which completes the proof.

(vii) Let us now assume that +∞ t0 Γ(s) λ(s) ds < +∞. Recalling inequality [START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1 k 2[END_REF], we deduce that

+∞ t0 Γ(s) A λ(s) (x(s)) ds < +∞.
By applying Lemma B.2 with F (t) = -A λ(t) (x(t)), we obtain that +∞ t0 ẋ(s) ds < +∞, and hence x(t) converges strongly as t → +∞ toward some x ∞ ∈ H.

Remark 2.4. When +∞ t0 Γ(s)
λ(s) ds < +∞, the trajectories of (RIMS) γ,λ have a finite length, and hence are strongly convergent. However, the limit point is not a zero of the operator A in general.

Let us now particularize Theorem 2.3 to the case of a constant parameter λ > 0. In this case, the operator arising in equation (RIMS) γ,λ is constant and equal to the λ-cocoercive operator A λ . On the other hand, it is well-known that every λ-cocoercive operator B : H → H can be viewed as the Yosida regularization A λ of some maximally monotone operator A : H → 2 H , see [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF]Proposition 23.20]. This leads to the following statement.

(1 -ε)λγ(s) ≥ (1 + λ| γ(s)|)Γ(s). (10)
Then for any global solution x(.) of

ẍ(t) + γ(t) ẋ(t) + B(x(t)) = 0, t ≥ t 0 ,
we have

(i) +∞ t0
γ(s) ẋ(s) 2 ds < +∞, and as a consequence

+∞ t0 Γ(s) ẋ(s) 2 ds < +∞. (ii) +∞ t0 Γ(s) B(x(s)) 2 ds < +∞.
(iii) For any z ∈ zerB, lim t→+∞ x(t) -z exists, and hence x(•) is bounded.

(iv) There exists C ≥ 0 such that for t large enough,

ẋ(t) ≤ C ∆(t) and ẍ(t) ≤ C γ(t)∆(t) + C.
Assuming that +∞ t0 Γ(s) ds = +∞, and that ∆(t) = O(Γ(t)) as t → +∞, the following holds

(v) lim t→+∞ B(x(t)) = 0. (vi) There exists x ∞ ∈ zerB such that x(t) x ∞ weakly in H as t → +∞.
Finally assume that +∞ t0 Γ(s) ds < +∞. Then we obtain

(vii) +∞ t0
ẋ(s) ds < +∞, and hence x(•) converges strongly toward some x ∞ ∈ H.

Assume now that the function γ is constant, say γ(t) ≡ γ > 0. In this case, it is easy to check that ( 11)

Γ(t) ∼ 1 γ and ∆(t) ∼ 1 γ as t → +∞, see Proposition 3.1.
As a consequence of Corollary 2.5, we then obtain the following result that was originally discovered by Attouch-Maingé [START_REF] Attouch | Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects[END_REF].

Corollary 2.6 (Attouch-Maingé [START_REF] Attouch | Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects[END_REF]). Let λ > 0 and let B : H → H be a λ-cocoercive operator such that zerB = ∅. Let γ > 0 be such that λγ 2 > 1. Then for any global solution x(.) of

(12) ẍ(t) + γ ẋ(t) + B(x(t)) = 0, t ≥ t 0 ,
we have

(i) +∞ t0 ẋ(s) 2 ds < +∞. (ii) +∞ t0 B(x(s)) 2 ds < +∞.
(iii) For any z ∈ zerB, lim t→+∞ x(t) -z exists, and hence x(•) is bounded.

(iv) lim t→+∞ ẋ(t) = 0 and lim t→+∞ ẍ(t) = 0. (v) lim t→+∞ B(x(t)) = 0. (vi) There exists x ∞ ∈ zerB such that x(t)
x ∞ weakly in H as t → +∞.

Proof. Since γ(t) ≡ γ > 0, we have the equivalences [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF] as t → +∞. It ensues that condition [START_REF] Balti | Asymptotic for the perturbed heavy ball system with vanishing damping term[END_REF] is guaranteed by λγ 2 > 1. All points are then obvious consequences of Corollary 2.5, except for (iv). Corollary 2.5 (iv) shows that the acceleration ẍ is bounded on [t 0 , +∞[. Taking account of (i), we deduce classically that lim t→+∞ ẋ(t) = 0. In view of equation ( 12) and the fact that lim t→+∞ B(x(t)) = 0 by (v), we conclude that lim t→+∞ ẍ(t) = 0.

Application to particular classes of functions γ and λ

We now look at special classes of functions γ and λ, for which we are able to estimate precisely the quantities +∞ t ds p(s) and t t0 p(s) λ(s) ds as t → +∞. This consists of the differentiable functions γ, λ

: [t 0 , +∞[→ R * + satisfying (13) lim t→+∞ γ(t) γ(t) 2 = -c and lim t→+∞ d dt (λ(t)γ(t)) λ(t)γ(t) 2 = -c ,
for some c ∈ [0, 1[ and c > -1. Some properties of the functions γ satisfying the first condition above were studied by Attouch-Cabot [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF], in connection with the asymptotic behavior of the inertial gradient system (IGS) γ . The next proposition extends some of these properties. We now show that the key condition (H 1 ) of Theorem 2.3 takes a simple form for functions γ and λ satisfying conditions [START_REF] Bot | Second order forward-backward dynamical systems for monotone inclusion problems[END_REF].

Proposition 3.2. Let γ, λ : [t 0 , +∞[→ R * + be two differentiable functions satisfying conditions (13) for some c ∈ [0, 1[ and c ∈] -1, 1[ such that |c | < 1 -c. Then condition (H 1 ) is equivalent to (15) lim inf t→+∞ λ(t)γ(t) 2 > 1 1 -c -|c | .
Proof. The inequality arising in condition (H 1 ) can be rewritten as

(16) (1 -ε)λ(t) γ(t) Γ(t) - d dt (λ(t)γ(t)) ≥ 1.
The assumption lim t→+∞ γ(t)

γ(t) 2 = -c implies that Γ(t) ∼ 1 (1-c) γ(t) as t → +∞, see Proposition 3.1 (i). It ensues that (17) λ(t) γ(t) Γ(t) = (1 -c)λ(t)γ(t) 2 + o(λ(t)γ(t) 2 ) as t → +∞.
On the other hand, we deduce from the second condition of ( 13) that

(18) d dt (λ(t)γ(t)) = |c |λ(t)γ(t) 2 + o(λ(t)γ(t) 2 ) as t → +∞.
In view of ( 17) and ( 18), inequality [START_REF] Brézis | Nonlinear ergodic theorems[END_REF] amounts to

λ(t)γ(t) 2 [(1 -ε)(1 -c) -|c | + o(1)] ≥ 1 as t → +∞. Therefore condition (H 1 ) is equivalent to the existence of ε ∈]0, 1 -c -|c |[ such that λ(t)γ(t) 2 [1 -c -|c | -ε ] ≥ 1,
for t large enough. This last condition is equivalent to [START_REF] Brézis | Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution[END_REF], which ends the proof.

∈] -1, 1[ such that |c | < 1 -c. Assume moreover that lim inf t→+∞ λ(t)γ(t) 2 > 1 1 -c -|c | .
Then for any global solution x(.) of (RIMS) γ,λ , we have

(i) +∞ t0 λ(s)γ(s) ẋ(s) 2 ds < +∞. (ii) +∞ t0 λ(s) γ(s) A λ(s) (x(s)) 2 ds < +∞.
(iii) For any z ∈ zerA, lim t→+∞ x(t) -z exists, and hence x(•) is bounded.

(iv) ẋ(t) = O 1 λ(t)γ(t)
and

ẍ(t) = O 1 λ(t)
as t → +∞.

Assuming that

+∞ t0 1 λ(s)γ(s) ds = +∞ and that | λ(t)| = O 1 γ(t)
as t → +∞, the following holds

(v) lim t→+∞ λ(t)A λ(t) (x(t)) = 0. (vi)
Γ(t) ∼ 1 (1 -c) γ(t) and 1 p(t) t t0 p(s) λ(s) ds ∼ 1 (1 + c )λ(t)γ(t) as t → +∞.
It ensues that the first condition of (H 2 ) is automatically satisfied, while the second one is given by

| λ(t)| = O 1 γ(t)
as t → +∞. Condition (H 3 ) is implied by the assumption +∞ t0 1 λ(s)γ(s) ds = +∞. Items (i)-(vii) follow immediately from the corresponding points in Theorem 2.3.

Let us now particularize to the case γ(t) = α t q and λ(t) = β t r , for some α, β > 0, q ≥ -1 and r ∈ R.

Corollary 3.4. Let A : H → 2 H be a maximally monotone operator such that zerA = ∅. Assume that γ(t) = α t q and λ(t) = β t r for every t ≥ t 0 > 0. Suppose that (q, r) ∈ ] -1, +∞[×R is such that 2q + r ≥ 0, and that (α,

β) ∈ R * + × R * + satisfies α 2 β > 1 if 2q + r = 0 (no condition if 2q + r > 0)
. Then for any global solution x(.) of (RIMS) γ,λ , we have

(i) +∞ t0 s q+r ẋ(s) 2 ds < +∞. (ii) +∞ t0 s r-q A λ(s) (x(s)) 2 ds < +∞.
(iii) For any z ∈ zerA, lim t→+∞ x(t) -z exists, and hence x(•) is bounded.

(iv) ẋ(t) = O 1 t q+r and ẍ(t) = O 1 t r as t → +∞. Assuming that q + r ≤ 1, the following holds (v) lim t→+∞ t r A λ(t) (x(t)) = 0. (vi) If r ≥ 0, there exists x ∞ ∈ zerA such that x(t)
x ∞ weakly in H as t → +∞. Finally assume that q + r > 1. Then we obtain (vii) +∞ t0 ẋ(s) ds < +∞, and hence x(•) converges strongly toward some x ∞ ∈ H.

Proof. Since q > -1, the first (resp. second) condition of ( 13) is satisfied with c = 0 (resp. c = 0). On the other hand, we have λ(t)γ(t

) 2 = α 2 β t 2q+r , hence lim t→+∞ λ(t)γ(t) 2 = +∞ if 2q + r > 0 α 2 β if 2q + r = 0.
It ensues that the condition lim inf t→+∞ λ(t)γ(t) 2 > 1 is guaranteed by the hypotheses of Corollary 3.4. Conditions When q = r = 0, the functions γ and λ are constant: γ(t) ≡ α > 0 and λ(t) ≡ β > 0. We then recover the result of [6, Theorem 2.1] with the key condition α 2 β > 1. To finish, let us consider the case q = -1, thus leading to a damping parameter of the form γ(t) = α t . This case was recently studied by Attouch and Peypouquet [START_REF] Attouch | Convergence of inertial dynamics and proximal algorithms governed by maximal monotone operators[END_REF] in the framework of Nesterov's accelerated methods.

Corollary 3.5. Let A : H → 2 H be a maximally monotone operator such that zerA = ∅. Let r ≥ 2, α > r and β ∈ R * + be such that β >

1 α(α-r) if r = 2 (no condition on β if r > 2). Assume that γ(t) = α t
and λ(t) = β t r for every t ≥ t 0 > 0. Then for any global solution x(.) of (RIMS) γ,λ , we have

(i) +∞ t0 s r-1 ẋ(s) 2 ds < +∞. (ii) +∞ t0 s r+1 A λ(s) (x(s)) 2 ds < +∞.
(iii) For any z ∈ zerA, lim t→+∞ x(t) -z exists, and hence x(•) is bounded.

(iv) ẋ(t) = O 1 t r-1 and ẍ(t) = O 1 t r as t → +∞.
Assuming that r = 2, the following holds (v) lim t→+∞ t 2 A λ(t) (x(t)) = 0. (vi) There exists x ∞ ∈ zerA such that x(t)

x ∞ weakly in H as t → +∞. Finally assume that r > 2. Then we obtain (vii) +∞ t0 ẋ(s) ds < +∞, and hence x(•) converges strongly toward some x ∞ ∈ H.

Proof. The first (resp. second) condition of ( 13) is satisfied with c = 1 α (resp. c = 1-r α ). Since r ≥ 2 and α > r, we have

c ∈]0, 1/2[ and |c | = r -1 α < α -1 α = 1 -c.
On the other hand, observe that λ(t)γ(t

) 2 = α 2 β t r-2 , hence lim t→+∞ λ(t)γ(t) 2 = +∞ if r > 2 α 2 β if r = 2. Condition lim inf t→+∞ λ(t)γ(t) 2 > 1 1-c-|c | is automatically satisfied if r > 2, while it amounts to α 2 β > 1 1 -1 α -r-1 α = α α -r ⇐⇒ β > 1 α(α -r) if r = 2.
Items (i)-(vii) follow immediately from the corresponding points in Corollary 3.3.

Taking r = 2 in the previous corollary, we recover the result of [8, Theorem 2.1] as a particular case.

PART B: ERGODIC CONVERGENCE RESULTS

Let A : H → 2 H be a maximally monotone operator. The trajectories associated to the semigroup of contractions generated by A are known to converge weakly in average toward some zero of A, cf. the seminal paper by Brezis and Baillon [START_REF] Baillon | Une remarque sur le comportement asymptotique des semigroupes non linéaires[END_REF]. Our purpose in this part of the paper is to study the ergodic convergence of the solutions of the system (RIMS) γ,λ . When the regularizing parameter λ(•) is minorized by some positive constant, it is established in part A that the trajectories of (RIMS) γ,λ do converge weakly toward a zero of A, see Theorem 2.3 (vi). Our objective is to show that weak ergodic convergence can be expected when the regularization parameter λ(t) tends toward 0 as t → +∞. The key ingredient is the use of some suitable ergodic variant of the Opial lemma.

4. Weak ergodic convergence of the trajectories 4.1. Ergodic variants of Opial's lemma. Ergodic versions of the Opial lemma were derived by Brézis-Browder [START_REF] Brézis | Nonlinear ergodic theorems[END_REF] and Passty [START_REF] Passty | Ergodic convergence to a zero of the sum of monotone operators in Hilbert space[END_REF] Λ(s, t) x(s) ds.

Lemma B.4 in the appendix shows that the map x is well-defined, bounded and that convergence of x(t) as t → +∞ implies convergence of x(t) toward the same limit (Cesaro property). The extension of Opial lemma to a general averaging process satisfying [START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage/Thresholding Algorithm[END_REF] and ( 20) is given hereafter. This result was established in [START_REF] Attouch | Asymptotic behavior of coupled dynamical systems with multiscale aspects[END_REF] for the particular case corresponding to Λ(s, t) = 1 t if s ≤ t and Λ(s, t) = 0 if s > t. Proposition 4.1. Let S be a nonempty subset of H and let x : [t 0 , +∞[→ H be a continuous map, supposed to be bounded on [t 0 , +∞[. Let Λ : [t 0 , +∞[×[t 0 , +∞[→ R + be a measurable function satisfying [START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage/Thresholding Algorithm[END_REF] and [START_REF] Haraux | Systèmes dynamiques dissipatifs et applications[END_REF], and let x : [t 0 , +∞[→ H be the averaged trajectory defined by [START_REF] Imbert | Convex Analysis techniques for Hopf-Lax formulae in Hamilton-Jacobi equations[END_REF]. Assume that (i) for every z ∈ S, lim t→+∞ x(t) -z exists;

(ii) every weak sequential limit point of x(t), as t → +∞, belongs to S. Then x(t) converges weakly as t → +∞ to a point in S.

Proof. From Lemma B.4 (i), the map x is bounded, therefore it is enough to establish the uniqueness of weak limit points. Let ( x(t n )) and ( x(t m )) be two weakly converging subsequences satisfying respectively x(t n )

x 1 as n → +∞ and x(t m )

x 2 as m → +∞. From (ii), the weak limit points x 1 and x 2 belong to S. In view of (i), we deduce that lim t→+∞ x(t) -x 1 2 and lim t→+∞ x(t) -x 2 2 exist. Writing that

x(t) -x 1 2 -x(t) -x 2 2 = 2 x(t) - x 1 + x 2 2 , x 2 -x 1 ,
we infer that lim t→+∞ x(t), x 2 -x 1 exists. Observe that

x(t), x 2 -x 1 = +∞ t0 Λ(s, t) x(s) ds, x 2 -x 1 = +∞ t0 Λ(s, t) x(s), x 2 -x 1 ds.
By applying Lemma B.4 (ii) to the real-valued map t → x(t), x 2 -x 1 , we deduce that lim t→+∞ x(t), x 2x 1 exists. This implies that

lim n→+∞ x(t n ), x 2 -x 1 = lim m→+∞ x(t m ), x 2 -x 1 , which entails that x 1 , x 2 -x 1 = x 2 , x 2 -x 1 .
We conclude that x 2 -x 1 2 = 0, which ends the proof. Assume that (i) for every z ∈ S, lim t→+∞ x(t) -z exists;

(ii) every weak sequential limit point of x(t), as t → +∞, belongs to S. Then x(t) converges weakly as t → +∞ to a point in S.

Proof. Just check that conditions [START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage/Thresholding Algorithm[END_REF] and [START_REF] Haraux | Systèmes dynamiques dissipatifs et applications[END_REF] Γ(s) ds = +∞ we deduce that lim t→+∞ t t0 Γ(u, t) du = +∞, see [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF]. We deduce from the above inequality that lim t→+∞ T t0 Λ(s, t) ds = 0, hence property (20) is satisfied. It ensues that Proposition 4.1 can be applied, which ends the proof. 4.2. Ergodic convergence of the trajectories. To each solution x(.) of (RIMS) γ,λ , we associate the averaged trajectory x(.) defined by

x(t) = 1 t t0 Γ(s, t) ds t t0
Γ(s, t) x(s) ds.

We show that under suitable conditions, every averaged trajectory x(.) converges weakly as t → +∞ toward some zero of the operator A. x ∞ weakly in H as n → +∞. Let us fix (z, q) ∈ gphA and define the function h : [t 0 , +∞[→ R + by h(t) = 1 2 x(t) -z 2 . Since q ∈ A(z) and A λ(t) (x(t)) ∈ A x(t) -λ(t)A λ(t) (x(t)) , the monotonicity of A implies that x(t) -λ(t)A λ(t) (x(t)) -z, A λ(t) (x(t)) -q ≥ 0, hence

x(t) -z, A λ(t) (x(t)) ≥ λ(t) A λ(t) (x(t)) 2 + x(t) -λ(t)A λ(t) (x(t)) -z, q ≥ x(t) -λ(t)A λ(t) (x(t)) -z, q .
Recalling equality (4), we obtain for every t ≥ t 0 ,

ḧ(t) + γ(t) ḣ(t) ≤ ẋ(t) 2 -x(t) -λ(t)A λ(t) (x(t)) -z, q .
Using Lemma B.1 (i) with g(t) = ẋ(t) 2 -x(t) -λ(t)A λ(t) (x(t)) -z, q , we obtain for every t ≥ t 0 ,

h(t) ≤ h(t 0 ) + ḣ(t 0 ) t t0 du p(u) + t t0 Γ(s, t) ẋ(s) 2 -x(s) -λ(s)A λ(s) (x(s)) -z, q ds.
Since h(t) ≥ 0 and Γ(s, t) ≤ Γ(s), we deduce that

t t0 Γ(s, t) x(s) -λ(s)A λ(s) (x(s)) -z, q ds ≤ h(t 0 ) + ḣ(t 0 ) t t0 du p(u) + t t0 Γ(s) ẋ(s) 2 ds.
Recalling the assumption +∞ t0 du p(u) < +∞ and the estimate +∞ t0

Γ(s) ẋ(s) 2 ds < +∞ (see Theorem 2.3 (i)), we infer that for every t ≥ t 0 , [START_REF] Nesterov | A method of solving a convex programming problem with convergence rate O(1/k 2 )[END_REF] t t0 Γ(s, t) x(s) -λ(s)A λ(s) (x(s)) -z, q ds ≤ C, where we have set

C := h(t 0 ) + | ḣ(t 0 )| +∞ t0 du p(u) + +∞ t0 Γ(s) ẋ(s) 2 ds.
It ensues that

t t0 Γ(s, t) x(s) -z, q ds ≤ C + t t0 Γ(s, t) λ(s)A λ(s) (x(s)), q ds ≤ C + q t t0 Γ(s, t)λ(s) A λ(s) (x(s)) ds.
This can be rewritten as

t t0 Γ(s, t)(x(s) -z) ds, q ≤ C + q t t0 Γ(s, t)λ(s) A λ(s) (x(s)) ds.
Dividing by t t0 Γ(s, t) ds, we find [START_REF]Weak convergence of the sequence of successive approximations for nonexpansive mappings[END_REF] x(t) -z, q ≤ C t t0 Γ(s, t) ds

+ q t t0 Γ(s, t) ds t t0 Γ(s, t)λ(s) A λ(s) (x(s)) ds.
The assumption +∞ t0

Γ(s) ds = +∞ implies that lim t→+∞ t t0 Γ(s, t) ds = +∞, see [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF]. On the other hand, we have lim t→+∞ λ(t)A λ(t) (x(t)) = 0 by Theorem 2.3 (v). From the Cesaro property, we infer that 1 t t0 Γ(s, t) ds t t0 Γ(s, t)λ(s) A λ(s) (x(s)) ds → 0 as t -→ +∞, see Lemma B.4 (ii). Taking the limit as t → +∞ in inequality [START_REF]Weak convergence of the sequence of successive approximations for nonexpansive mappings[END_REF], we then obtain lim sup t→+∞ x(t) -z, q ≤ 0.

Recall that the sequence (t n ) is such that x(t n )

x ∞ weakly in H as n → +∞, hence x(t n ) -z, q → x ∞ -z, q as n → +∞. From what precedes, we deduce that x ∞ -z, q ≤ 0 for every (z, q) ∈ gphA. Since the operator A is maximally monotone, we infer that 0 ∈ A(x ∞ ). We have proved that x ∞ ∈ zerA, which shows that condition (ii) of Corollary 4.2 is satisfied.

Let us now consider the alternate averaged trajectory x defined by In view of assumption ( 25), we then obtain [START_REF] Passty | Ergodic convergence to a zero of the sum of monotone operators in Hilbert space[END_REF]. By applying Lemma B.5, we infer that lim t→+∞ x(t)x(t) = 0. On the other hand, Theorem 4.3 shows that there exists x ∞ ∈ zerA such that x(t) x ∞ weakly in H as t → +∞. We then conclude that x(t)

x(t) = 1 t t0 Γ(
x ∞ weakly in H as t → +∞. Now assume that the function Γ :

[t 0 , +∞[→ R + is such that Γ(s) ∼ Γ(s)
λ(t)γ(t) 2 > 1; (b) | λ(t)| = O (1/γ(t)) as t → +∞; (c) +∞ t0 ds λ(s)γ(s) = +∞; (d) +∞ t0 ds γ(s) = +∞.
Then for any global solution x(.) of (RIMS) γ,λ , there exists x ∞ ∈ zerA such that Let us now particularize to the case γ(t) = α t q and λ(t) = β t r , for some α, β > 0, q ∈] -1, 1] and r ∈ R.

Corollary 4.6. Let A : H → 2 H be a maximally monotone operator such that zerA = ∅. Assume that γ(t) = α t q and λ(t) = β t r for every t ≥ t 0 > 0. Let (q, r) ∈ ] -1, 1] × R be such that q + r ≤ 1 and 2q + r ≥ 0, and let (α,

β) ∈ R * + × R * + be such that α 2 β > 1 if 2q + r = 0 (no condition if 2q + r > 0)
. Then for any global solution x(.) of (RIMS) γ,λ , there exists x ∞ ∈ zerA such that

1 t t0 ds s q t t0 x(s) s q ds
x ∞ weakly in H as t → +∞.

Proof. The conditions of ( 27) are guaranteed by q > -1. On the other hand, we have λ(t)γ(t) = +∞ amount respectively to q + r ≤ 1, which holds true by assumption. The condition +∞ t0 ds γ(s) = +∞ is implied by q ≤ 1. Then just apply Corollary 4.5.

PART C: THE SUBDIFFERENTIAL CASE

Let us particularize our study to the case A = ∂Φ, where Φ : H → R ∪ {+∞} is a convex lower semicontinuous proper function. Then A λ = ∇Φ λ is equal to the gradient of Φ λ : H → R, which is the Moreau envelope of Φ of index λ > 0. Let us recall that, for all x ∈ H

(30) Φ λ (x) = inf ξ∈H Φ(ξ) + 1 2λ x -ξ 2 .
In this case, we will study the rate of convergence of the values, when the time t goes to +∞, of the trajectories of the second-order differential equation

(RIGS) γ,λ ẍ(t) + γ(t) ẋ(t) + ∇Φ λ(t) (x(t)) = 0,
called the Regularized Inertial Gradient System with parameters γ, λ. As a main feature, the above system involves two time-dependent positive parameters: the Moreau regularization parameter λ(t), and the damping parameter γ(t). System (RIGS) γ,λ comes as a natural development of several recent studies concerning fast inertial dynamics and algorithms for convex optimization. Indeed, when Φ is a smooth convex function, it was highlighted that the fact of taking a vanishing damping coefficient γ(t) in system (IGS) γ ẍ(t) + γ(t) ẋ(t) + ∇Φ(x(t)) = 0, is a key property for obtaining fast optimization methods. Precisely Su, Boyd and Candès [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF] showed that, in the particular case γ(t) = 3 t , (IGS) γ is a continuous version of the fast gradient method initiated by Nesterov [START_REF] Nesterov | A method of solving a convex programming problem with convergence rate O(1/k 2 )[END_REF], with Φ(x(t)) -min H Φ = O( 1 t 2 ) in the worst case. Attouch and Peypouquet [START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1 k 2[END_REF] and May [START_REF] May | Asymptotic for a second order evolution equation with convex potential and vanishing damping term[END_REF] have improved this result by showing that Φ(x(t)) -min

H Φ = o( 1 t 2 ) for γ(t) = α t with α > 3.
Recently, in the case of a general damping function γ(•), the study of the speed of convergence of trajectories of (IGS) γ was developed by Attouch-Cabot in [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF]. Note that a main advantage of (RIGS) γ,λ over (IGS) γ is that Φ is just assumed to be lower semicontinuous (not necessarily smooth). In line with these results, by jointly adjusting the tuning of the two parameters in (RIGS) γ,λ , we will obtain fast convergence results for the values.

Convergence rates and weak convergence of the trajectories

The following assumptions and notations will be needed throughout this section:

           Φ : H → R ∪ {+∞} convex, lower semicontinuous, proper, bounded from below, argmin Φ = ∅; γ : [t 0 , +∞[→ R + continuous, with t 0 ∈ R; λ : [t 0 , +∞[→ R * + continuously differentiable, nondecreasing; x : [t 0 , +∞[→ H the solution to (RIGS) γ,λ , with initial conditions x(t 0 ) = x 0 , ẋ(t 0 ) = v 0 ; ξ(t) = prox λ(t)Φ (x(t)) for t ≥ t 0 .
5.1. Preliminaries on Moreau envelopes. For classical facts about the Moreau envelopes we refer the reader to [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF][START_REF] Brézis | Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution[END_REF][START_REF] Parikh | Proximal algorithms[END_REF][START_REF] Peypouquet | Convex optimization in normed spaces: theory, methods and examples[END_REF]. We point out the following properties that will be useful in the sequel:

(i) λ ∈]0, +∞[ → Φ λ (x) is nonincreasing for all x ∈ H; (ii) inf H Φ = inf H Φ λ for all λ > 0;
(iii) argmin Φ = argmin Φ λ for all λ > 0.

It turns out that it is convenient to consider the Moreau envelope as a function of the two variables x ∈ H and λ ∈]0, +∞[. Its differentiability properties with respect to (x, λ) play a crucial role in our analysis. a. Let us first recall some classical facts concerning the differentiability properties with respect to x of the Moreau envelope x → Φ λ (x). The infimum in ( 30) is achieved at a unique point

(31) prox λΦ (x) = argmin ξ∈H Φ(ξ) + 1 2λ x -ξ 2 , which gives Φ λ (x) = Φ(prox λΦ (x)) + 1 2λ x -prox λΦ (x) 2 .
Writing the optimality condition for (31), we get

prox λΦ (x) + λ∂Φ (prox λΦ (x)) x, that is prox λΦ (x) = (I + λ∂Φ) -1 (x).
Thus, prox λΦ is the resolvent of index λ > 0 of the maximally monotone operator ∂Φ. As a consequence, the mapping prox λΨ : H → H is firmly expansive. For any λ > 0, the function x → Φ λ (x) is continuously differentiable, with

∇Φ λ (x) = 1 λ (x -prox λΦ (x)) .
Equivalently

∇Φ λ = 1 λ I -(I + λ∂Φ) -1 = (∂Φ) λ
which is the Yosida approximation of the maximally monotone operator ∂Φ. As such, ∇Φ λ is Lipschitz continuous, with Lipschitz constant 1 λ , and Φ λ ∈ C 1,1 (H). b. A less known result is the C 1 -regularity of the function λ → Φ λ (x), for any x ∈ H. Its derivative is given by

(32) d dλ Φ λ (x) = - 1 2 ∇Φ λ (x) 2 .
This result is known as the Lax-Hopf formula for the above first-order Hamilton-Jacobi equation, see [2, Remark 3.32; Lemma 3.27], and [START_REF] Imbert | Convex Analysis techniques for Hopf-Lax formulae in Hamilton-Jacobi equations[END_REF]. A proof is given in Lemma A.1 for the convenience of the reader. As a consequence of the semi-group property satisfied by the orbits of the autonomous evolution equation (32), for any x ∈ H, λ > 0 and µ > 0,

(Φ λ ) µ (x) = Φ (λ+µ) (x). (33)
5.2. Preliminary estimates. Let us introduce functions W , h z , of constant use in this section.

Global energy.

The global energy of the system W : [t 0 , +∞[→ R + is given by

W (t) = 1 2 ẋ(t) 2 + Φ λ(t) (x(t)) -min H Φ.
Since inf H Φ = inf H Φ λ , we have W ≥ 0. From (RIGS) γ,λ and property (32), we immediately obtain the following equality

Ẇ (t) = -γ(t) ẋ(t) 2 - λ(t) 2 ∇Φ λ(t) (x(t)) 2 . ( 34
)
As a direct consequence of (34), we obtain the following results.

Proposition 5.1. The function W is nonincreasing, and hence W ∞ := lim t→+∞ W (t) exists. In addition,

sup t≥t0 ẋ(t) < +∞, ∞ t0 γ(t) ẋ(t) 2 dt < +∞ and ∞ t0 λ(t) ∇Φ λ(t) (x(t)) 2 dt < +∞.
Proof. From (34), and λ nondecreasing, we deduce that Ẇ (t) ≤ 0. Hence, W is nonincreasing. Since W is nonnegative, W ∞ := lim t→+∞ W (t) exists. After integrating (34) from t 0 to t, we get

W (t) -W (t 0 ) + t t0 γ(s) ẋ(s) 2 ds + 1 2 t t0 λ(s) ∇Φ λ(s) (x(s)) 2 ds ≤ 0.
By definition of W , and using again that inf

H Φ = inf H Φ λ , it follows that 1 2 ẋ(t) 2 + t t0 γ(s) ẋ(s) 2 ds + 1 2 t t0 λ(s) ∇Φ λ(s) (x(s)) 2 ds ≤ W (t 0 ).
This being true for any t ≥ t 0 , we get the conclusion.

Φ) -Γ(t) ∇Φ λ(t) (x(t)), x(t) -x = 2Γ(t) Γ(t)(Φ λ(t) (x(t)) -min H Φ) -Γ(t) ∇Φ λ(t) (x(t)), x(t) -x .
In the above calculation, we have neglected the term -Γ(t) 2 λ(t) 2 ∇Φ λ(t) (x(t)) 2 which is less or equal than zero, because λ(•) is a nondecreasing function. To obtain the last equality, we have used again the equality -Γ(t)γ(t) + Γ(t) + 1 = 0. Let us now use the convexity of Φ λ(t) and equality (37) to obtain

Ė(t) ≤ -(Γ(t) -2Γ(t) Γ(t)) (Φ λ(t) (x(t)) -min H Φ) = -Γ(t)(3 -2γ(t)Γ(t)) (Φ λ(t) (x(t)) -min H Φ).
When (K 1 ) is satisfied, we have 3 -2γ(t)Γ(t) ≥ 0. Since Γ(t) and Φ λ(t) (x(t)) -min H Φ are nonnegative, we deduce that Ė(t) ≤ 0. (i) For every t ≥ t 1 , we have 2 , we obtain that lim t→+∞ h(t) exists. This shows the first point of the Opial lemma. Let us now verify the second point. Let x(t k ) converge weakly to x ∞ as k → +∞. Point (i) implies that ξ(t k ) also converges weakly to x ∞ as k → +∞. Since the function Φ is convex and lower semicontinuous, it is semicontinuous for the weak topology, hence satisfies

Φ λ(t) (x(t)) -min H Φ ≤ E(t 1 ) Γ(t)
[t 0 , +∞[→ R + defined by g(t) = ẋ(t)
Φ(x ∞ ) ≤ lim inf t→+∞ Φ(ξ(t k )) = lim t→+∞ Φ(ξ(t)) = min H Φ,
cf. the last point of Theorem 5.6. It ensues that x ∞ ∈ argmin Φ, which establishes the second point of the Opial lemma, and ends the proof. This will immediately give our result, since, by the derivation chain rule,

d dλ Φ λ (x) = d dλ 1 λ × λΦ λ (x) = 1 λ Φ(J λ (x)) - 1 λ 2 λΦ λ (x) = - 1 λ Φ λ (x) -Φ(J λ (x)) = - 1 2λ 2 x -J λ (x) 2 = - 1 2 ∇Φ λ (x) 2 .
To obtain (47), take two values λ 1 and λ 2 of the parameter λ, and compare the corresponding values of the function λ → λΦ λ (x). By the formulation (46) of λΦ λ (x) as an infimal value, we have

λ 1 Φ λ1 (x) -λ 2 Φ λ2 (x) ≤ λ 1 Φ(J λ2 (x)) + 1 2 x -J λ2 (x) 2 -λ 2 Φ(J λ2 (x)) - 1 2 x -J λ2 (x) 2 = (λ 1 -λ 2 )Φ(J λ2 (x)).
Exchanging the roles of λ 1 and λ 2 , we obtain

(λ 1 -λ 2 )Φ(J λ1 (x)) ≤ λ 1 Φ λ1 (x) -λ 2 Φ λ2 (x) ≤ (λ 1 -λ 2 )Φ(J λ2 (x)).
Then note that the mapping λ → Φ(J λ (x)) is continuous. This follows from (46) and the continuity of the mappings λ → Φ λ (x) and λ → J λ (x). Indeed, these mappings are locally Lipschitz continuous. This is a direct consequence of the resolvent equations (33), see [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF]Proposition 23.28] for further details. Then divide the above formula by λ 1 -λ 2 (successively examining the two cases λ 1 < λ 2 , then λ 2 < λ 1 ). Letting λ 1 → λ 2 , and using the continuity of λ → Φ(J λ (x)) gives the differentiability of the mapping λ → λΦ λ (x), and formula (47). Then, writing Φ λ (x) = 1 λ (λΦ λ (x)), and applying the derivation chain rule gives (45). The continuity of λ → ∇Φ λ (x) gives the continuous differentiability of λ → Φ λ (x).

Appendix B. Some auxiliary results

In this section, we present some auxiliary lemmas that are used throughout the paper. The following result allows us to establish some majorization and also the convergence as t → +∞ of a real-valued function satisfying some differential inequality. Γ(s) w(s) ds = 0.

Lemma

The conclusion follows from the two above relations.

Given a Banach space (X , . ) and a bounded map x : [t 0 , +∞[→ X , the next lemma gives basic properties of the averaged trajectory x defined by [START_REF] Imbert | Convex Analysis techniques for Hopf-Lax formulae in Hamilton-Jacobi equations[END_REF].

Lemma B.4. Let us give (X , . ) a Banach space, Λ : [t 0 , +∞[×[t 0 , +∞[→ R + a measurable function satisfying [START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage/Thresholding Algorithm[END_REF], and x : [t 0 , +∞[→ X a bounded map. Then we have (i) For every t ≥ t 0 , the vector x(t) = +∞ t0 Λ(s, t) x(s) ds is well-defined. The map x is bounded and sup t≥t0 x(t) ≤ sup t≥t0 x(t) . (ii) Assume moreover that the function Λ satisfies [START_REF] Haraux | Systèmes dynamiques dissipatifs et applications[END_REF]. If lim t→+∞ x(t) = x ∞ for some x ∞ ∈ X , then lim t→+∞ x(t) = x ∞ .

Proof. (i) Let us set M = sup t≥t0 x(t) < +∞. In view of [START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage/Thresholding Algorithm[END_REF], observe that for every t ≥ t 0 , Λ(s, t) x(s) ds, and hence x(t) ≤ M in view of (54). (ii) Assume that lim t→+∞ x(t) = x ∞ for some x ∞ ∈ X . Observe that for every t ≥ t 0 ,

x(t) -x ∞ = +∞ t0
Λ(s, t) (x(s) -x ∞ ) ds by using ( 19)

≤ +∞ t0
Λ(s, t) x(s) -x ∞ ds. (55) Fix ε > 0 and let T ≥ t 0 be such that x(t) -x ∞ ≤ ε for every t ≥ T . From (55), we obtain

x(t) -x ∞ ≤ sup t∈[t0,T ] x(t) -x ∞ T t0

Corollary 2 . 5 .

 25 Let λ > 0 and let B : H → H be a λ-cocoercive operator such that zerB = ∅. Given a differentiable function γ : [t 0 , +∞[→ R + satisfying (H 0 ), let Γ, ∆ : [t 0 , +∞[→ R + be the functions respectively defined by Γ(s) = p(s) u) du . Assume that there exists ε ∈]0, 1[ such that for s large enough,

  )γ(s) = +∞ and | λ(t)| = O (1/γ(t)) as t → +∞ amount respectively to q + r ≤ 1. Items (i)-(vii) are immediate consequences of the corresponding points in Corollary 3.3.

Corollary 5 . 4 .

 54 Let γ : [t 0 , +∞[→ R + be a continuous function satisfying (H 0 ) and (K 1 ).

+∞ t0 Λ

 t0 (s, t) x(s) ds ≤ M +∞ t0 Λ(s, t) ds = M.Since X is complete, we classically deduce that the integral +∞ t0 Λ(s, t) x(s) ds is convergent. From the definition of x(t), we then have x(t) ≤ +∞ t0

  ) = e We obtain p(t) ẋ(t) = ẋ(t 0 ) for every t ≥ t 0 . By integrating again, we findx(t) = x(t 0 ) +

		t t 0	γ(τ ) dτ
	and integrate on [t 0 , t]. t t0	ds p(s)	ẋ(t 0 ).
	+∞ t0	ds p(s)	< +∞.

It ensues immediately that the trajectory x(.) converges if and only if ẋ(t 0 ) = 0 or (H 0 )

  Proposition 2.1. Let A : H → 2 H be a maximally monotone operator, and let γ : [t 0 , +∞[→ R + and λ : [t 0 , +∞[→ R

* + be continuous functions. Then, for any x 0 ∈ H, v 0 ∈ H, there exists a unique global solution

  + be the function defined by Γ(s) =

							monotone
	convergence theorem then implies that
			t				+∞
	(3)	lim t→+∞	t0	Γ(s, t) ds = lim t→+∞	t0	Γ(s, t) ds =
						+∞	du
						s	p(u)
	∈ [t 0 , +∞[,
	(2)	Γ(s, t) =	s	t	du p(u)	p(s) if s ≤ t, and Γ(s, t) = 0 if s > t.

For each s ∈ [t 0 , +∞[, the quantity Γ(s, t) tends increasingly toward Γ(s) as t → +∞. The +∞ t0 Γ(s) ds, since Γ(s, t) = 0 for s ≥ t. Let us state the main result of this section. Theorem 2.3. Let A : H → 2 H be a maximally monotone operator such that zerA = ∅. Let γ : [t 0 , +∞[→ R + and λ : [t 0 , +∞[→ R * + be differentiable functions. Assuming (H 0 ), let Γ : [t 0 , +∞[→ R

  2 ds < +∞, and as a consequence

													+∞
													Γ(s) ẋ(s) 2 ds < +∞.
													t0
	(ii)
	(iv) There exists a positive constant C such that for t large enough,
		ẋ(t) ≤	C p(t)	t t0	p(s) λ(s)	ds and		ẍ(t) ≤ C	γ(t) p(t)	t t0	p(s) λ(s)	ds +	C λ(t)	.
	Assuming that
	(H 2 )	λ(t) p(t)	t t0	p(s) λ(s)	ds = O(Γ(t)) and | λ(t)| = O(Γ(t)) as t → +∞,
	(H 3 )						+∞ t0	Γ(s) λ(s)	ds = +∞,
	the following holds
	(v) lim t→+∞ λ(t)A λ(t) (x(t)) = 0.
	(vi) If λ(•) is minorized by some positive constant on [t 0 , +∞[, then there exists x ∞ ∈ zerA such that
	x(t)	x ∞ weakly in H as t → +∞.
	Finally assume that (H 3 ) is not satisfied, i.e.	+∞ t0	Γ(s) λ(s)	ds < +∞. Then we obtain
	+∞
	(vii)	ẋ(s) ds < +∞, and hence x(•) converges strongly toward some x ∞ ∈ H.
	t0
	Proof. (i) Let z ∈ zerA, and let us set h(t) = 1 2 x(t) -z 2 for every t ≥ t 0 . By differentiating, we find
	for every t ≥ t 0 ,
		ḣ(t) = ẋ(t), x(t) -z and ḧ(t) = ẋ(t) 2 + ẍ(t), x(t) -z .

+∞ t0 λ(s)Γ(s) A λ(s) (x(s)) 2 ds < +∞.

(iii) For any z ∈ zerA, lim t→+∞ x(t) -z exists, and hence x(•) is bounded.

  Combining Theorem 2.3 and Propositions 3.1 and 3.2, we obtain the following result.

Corollary 3.3. Let A : H → 2 H be a maximally monotone operator such that zerA = ∅. Let γ, λ : [t 0 , +∞[→ R * + be two differentiable functions satisfying conditions (13) for some c ∈ [0, 1[ and c

  in a discrete setting. In order to give a continuous ergodic version, let us consider a measurable function Λ : [t 0 , +∞[×[t 0 , +∞[→ R + satisfying the following assumptions To each bounded map x : [t 0 , +∞[→ H, we associate the averaged map x : [t 0 , +∞[→ H by

		+∞
	(19)		Λ(s, t) ds = 1 for every t ≥ t 0 ,
		t0
			T
	(20)	lim t→+∞	t0	Λ(s, t) ds = 0 for every T ≥ t 0 .
				+∞
	(21)		x(t) =
				t0

  of Proposition 4.1 are satisfied for the function Λ : [t 0 , +∞[×[t 0 , +∞[→ R + given by[START_REF] May | Asymptotic for a second order evolution equation with convex potential and vanishing damping term[END_REF]. Property[START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage/Thresholding Algorithm[END_REF] clearly holds true. Observe that for every T ≥ t 0 , ds is finite and independent of t. On the other hand, from the assumption

	T t0	Λ(s, t) ds =	T t0 Γ(s, t) ds t t0 Γ(u, t) du	≤	T t0 Γ(s) ds t0 Γ(u, t) du t	.
	The quantity t0 Γ(s) +∞ T
	t0

  + satisfies (H 0 ). For s, t ≥ t 0 , let Γ(s) and Γ(s, t) be the quantities respectively defined by (1) and (2). Assume that conditions (H 1 )-(H 2 )-(H 3 ) hold, together with

	+∞
	t0

Theorem 4.3. Let A : H → 2 H be a maximally monotone operator such that zerA = ∅ and let λ : [t 0 , +∞[→ R * + be a differentiable function. Suppose that the differentiable function γ : [t 0 , +∞[→ R Γ(s) ds = +∞. Then for any global solution x(.) of (RIMS) γ,λ , there exists x ∞ ∈ zerA such that x(t) = 1 t t0 Γ(s, t) ds t t0 Γ(s, t)x(s) ds x ∞ weakly in H as t → +∞. Proof. We apply Corollary 4.2 with S = zerA. Condition (i) of Corollary 4.2 is realized in view of Theorem 2.3 (iii). Let us now assume that there exist x ∞ ∈ H and a sequence (t n ) such that t n → +∞ and x(t n )

  Then for any global solution x(.) of (RIMS) γ,λ , there exists x ∞ ∈ zerA such that The latter result still holds true if the function Γ in the above quotient is replaced with a function Γ : [t 0 , +∞[→ R + such that Γ(s) ∼ Γ(s) as s → +∞.Proof. We are going to show that lim t→+∞ x(t) -x(t) = 0, where x is the averaged trajectory of Theorem 4.3. For that purpose, we use Lemma B.5 with the functions Λ 1 , Λ 2 : [t 0 , +∞[×[t 0 , +∞[→ R +

	x(t) = x respectively defined by t 1 t t0 Γ(s) ds t0 Γ(s)x(s) ds
		Λ 1 (s, t) =	Γ(s, t) t0 Γ(u, t) du t	, Λ 2 (s, t) =	Γ(s) t0 Γ(u) du t	, if s ≤ t,
	and Λ 1 (s, t) = Λ 2 (s, t) = 0 if s > t. The functions Λ 1 and Λ 2 clearly satisfy property (19). Let us now
	check that
							+∞
	(26)			lim t→+∞	t0	|Λ 1 (s, t) -Λ 2 (s, t)| ds = 0.
										t t0 Γ(u) du	+	Γ(s, t) -Γ(s) t t0 Γ(u) du
	and hence
		|Λ +∞ t	ds p(s) t0 Γ(s) ds t t0 p(s) ds t	.
	Theorem 4.4. Under the hypotheses of Theorem 4.3, assume moreover that
	(25)	t	+∞	ds p(s)		t t0	p(s) ds = o	t t0	Γ(s) ds	as t → +∞.

s) ds t t0 Γ(s)x(s) ds, for every t ≥ t 0 . The next result gives sufficient conditions that ensure the weak convergence of x(t) as t → +∞ toward a zero of A. ∞ weakly in H as t → +∞.

For s ≤ t, we have

Λ 1 (s, t) -Λ 2 (s, t) = Γ(s, t) t t0 Γ(u, t) du t t0 (Γ(u) -Γ(u, t)) du 1 (s, t) -Λ 2 (s, t)| ≤ Γ(s, t) t t0 Γ(u, t) du t t0 |Γ(u) -Γ(u, t)| du t t0 Γ(u) du + |Γ(s, t) -Γ(s)| t t0 Γ(u) du .

By integrating on [t 0 , t], we find t t0 |Λ 1 (s, t) -Λ 2 (s, t)| ds ≤ 2 t t0 |Γ(s, t) -Γ(s)| ds t t0 Γ(s) ds . Recalling that Λ 1 (s, t) = Λ 2 (s, t) = 0 for s > t, this implies that +∞ t0 |Λ 1 (s, t) -Λ 2 (s, t)| ds ≤ 2 t t0 |Γ(s, t) -Γ(s)| ds t t0 Γ(s) ds . From the expression of Γ(s) and Γ(s, t), see (1) and (2), we immediately deduce that +∞ t0 |Λ 1 (s, t) -Λ 2 (s, t)| ds ≤ 2

  as s → +∞. Let us denote by Λ 2 the function defined by (s, t) = 0 if s > t. The corresponding averaged trajectory is denoted by x. By arguing as above, we obtain that

	and Λ 2 +∞ t0	| Λ 2 (s, t) -Λ 2 (s, t)| ds ≤ 2	t t0 | Γ(s) -Γ(s)| ds t0 Γ(s) ds t	.
	Then, using the estimate
		t				t
		| Γ(s) -Γ(s)| ds = o	Γ(s) ds	as t → +∞,
		t0				t0
	we deduce that		+∞
				| Λ 2 (s, t) -Λ 2 (s, t)| ds -→ 0 as t → +∞.
		t0
	In view of Lemma B.5, this implies that lim t→+∞ x(t) -x(t) = 0, which ends the proof.
	Let us now apply Theorem 4.4 to the class of differentiable functions γ, λ : [t 0 , +∞[→ R * + satisfying
	(27)	lim t→+∞	γ(t) γ(t) 2 = 0 and	lim t→+∞	d dt (λ(t)γ(t)) λ(t)γ(t) 2 = 0.
				Λ 2 (s, t) =	Γ(s) t0 Γ(u) du t	if s ≤ t,

Corollary 4.5. Let A : H → 2 H be a maximally monotone operator such that zerA = ∅. Let γ, λ : [t 0 , +∞[→ R * + be two differentiable functions satisfying conditions

[START_REF] Peypouquet | Convex optimization in normed spaces: theory, methods and examples[END_REF]

. Assume that (a) lim inf t→+∞

  as t → +∞. It ensues that the first condition of (H 2 ) is automatically satisfied, while the second one is given by (b). Condition (H 3 ) is implied by the assumption (c). In the same way, condition ds = +∞ is guaranteed by the assumption (d). It remains to establish condition (25) of Theorem 4.4. By applying Proposition 3.1 (ii) with λ(t) ≡ 1 and c = 0, we obtain +∞, the above result holds true with the function 1/γ in place of Γ, see the last assertion of Theorem 4.4.

	It ensues that condition (25) amounts to 1 γ(t) 2 = o	t t0 Γ(s) ds as t → +∞, which is in turn equivalent
	to
	(29)				1 γ(t) 2 = o	t t0	ds γ(s)	as t → +∞.
	Since lim t→+∞ γ(t)/γ(t) 2 = 0, we have -γ(t)/γ(t) 3 = o(1/γ(t)) as t → +∞. By integrating on [t 0 , t], we
	obtain
				1 2γ(t) 2	t t0	=	t t0	-	γ(s) γ(s) 3 ds = o	t t0	ds γ(s)	as t → +∞,
	because	+∞ t0	ds γ(s) = +∞ by assumption. It ensues that condition (29) is fulfilled, hence all the hypotheses
	of Theorem 4.4 are satisfied. We deduce that there exists x ∞ ∈ zerA such that
				1 t0 Γ(s) ds t	t t0	Γ(s)x(s) ds	x ∞ weakly in H as t → +∞.
	Since Γ(t) ∼ 1/γ(t) as t →
			t	1 γ(s) ds x +∞ t t0 t t0 x(s) γ(s) ds ds p(s) ∼ 1 p(t)γ(t) and	t t0	p(s) λ(s)	ds ∼	p(t) λ(t)γ(t)	as t → +∞,
	thus implying that Γ(t) ∼ 1 γ(t) +∞ t0 Γ(s) t t0	p(s) ds ∼	p(t) γ(t)	as t → +∞.
	In view of the first equivalence of (28), we infer that
				t	+∞	ds p(s)		t t0	p(s) ds ∼	1 γ(t) 2 as t → +∞.

∞ weakly in H as t → +∞.

Proof. Let us check that the assumptions of Theorem 4.4 are satisfied. Assumption (H 0 ) is verified in view of Proposition 3.1 (i) applied with c = 0. Since lim inf t→+∞ λ(t)γ(t) 2 > 1, condition (H 1 ) holds true by Proposition 3.2 used with c = c = 0. On the other hand, Proposition 3.1 shows that

[START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method: theory and insights[END_REF]

  5.2.2. Anchor. Given z ∈ H, we define h z : [t 0 , +∞[→ R by Lemma 5.2. For each z ∈ H and all t ≥ t 0 , we have ḧz (t) + γ(t) ḣz (t) + x(t) -z, ∇Φ λ(t) (x(t)) = ẋ(t) 2 Rate of convergence of the values. Let x : [t 0 , +∞[→ H be a solution of (RIGS) γ,λ . Let us fix x ∈ argmin Φ, and set h= h x , that is, h : [t 0 , +∞[→ R + satisfies h(t) = 1 2 x(t) -x 2 . We define the function p : [t 0 , +∞[→ R + by p(t) = eThe following rate of convergence analysis is based on the decreasing properties of the function E, that will serve us as a Lyapunov function. Proposition 5.3 (Decay of E). Let γ : [t 0 , +∞[→ R + be a continuous function satisfying (H 0 ). The energy function E : [t 0 , +∞[→ R + satisfies for every t ≥ t 0 ,

	Proof. By differentiating the function E, as expressed in (38), we obtain
	h z (t) = Ė(t) = Γ(t) 2 Ẇ (t) + 2Γ(t) Γ(t)W (t) + (1 + Γ(t)) ḣ(t) + Γ(t) ḧ(t). 1 x(t) -z 2 . 2 We have the following: Taking into account the expression of W and Ẇ , along with equalities (35) and (37), we obtain
	Ė(t) = Γ(t) 2 Ẇ (t) + 2Γ(t) Γ(t)W (t) + Γ(t)( ḧ(t) + γ(t) ḣ(t))
	(35) (36) In particular, if z ∈ argmin Φ, then = -Γ(t) 2 γ(t) ẋ(t) 2 + λ(t) 2 ḧz (t) + γ(t) ḣz (t) + Φ λ(t) (x(t)) -Φ λ(t) (z) ≤ ẋ(t) 2 . ∇Φ λ(t) (x(t)) 2 + 2Γ(t) Γ(t) 1 2 ẋ(t) 2 + Φ λ(t) (x(t)) -min H +Γ(t) ẋ(t) 2 -∇Φ λ(t) (x(t)), x(t) -x	Φ
		≤	ḧz (t) + γ(t) ḣz (t) ≤ ẋ(t) 2 . ẋ(t) 2 Γ(t)(-Γ(t)γ(t) + Γ(t) + 1) + 2Γ(t) Γ(t)(Φ λ(t) (x(t)) -min H
	Proof. First observe that
			ḣz (t) = x(t) -z, ẋ(t)		and	ḧz (t) = x(t) -z, ẍ(t) + ẋ(t) 2 .
	By (RIGS) γ,λ and the convexity of Φ λ(t) , it ensues that
		ḧz (t) + γ(t) ḣz (t) = ẋ(t) 2 + x(t) -z, -∇Φ λ(t) (x(t)) ≤ ẋ(t) 2 + Φ λ(t) (z) -Φ λ(t) (x(t)),
	which is precisely (35)-(36). The last statement follows from the fact that argmin Φ λ = argmin Φ for all
	λ > 0.
	5.3. t t 0	γ(τ ) dτ . Under the assumption
				(H 0 )	+∞ t0	ds p(s)	< +∞,
	the function Γ : [t 0 , +∞[→ R + is defined by Γ(t) = p(t)	+∞ t	ds p(s) . Clearly, the function Γ is of class C 1
	and satisfies
	(37)			Γ(t) = γ(t)Γ(t) -1, t ≥ t 0 .
	Let us define the function E : [t 0 , +∞[→ R by
	(38)	E(t) = Γ(t) 2 W (t) + h(t) + Γ(t) ḣ(t)
			= Γ(t) 2 1 2	ẋ(t) 2 + Φ λ(t) (x(t)) -min H	Φ +	1 2	x(t) -x 2 + Γ(t) ẋ(t), x(t) -x
	(39)		= Γ(t) 2 Φ λ(t) (x(t)) -min H	Φ +	1 2	x(t) -x + Γ(t) ẋ(t) 2 .
	(40)		Ė(t) + Γ(t) (3 -2γ(t)Γ(t)) Φ λ(t) (x(t)) -min H	Φ ≤ 0.
	Under the assumption
	(K 1 )		There exists t 1 ≥ t 0 such that γ(t)Γ(t) ≤ 3/2 for every t ≥ t 1 ,
	then we have Ė(t) ≤ 0 for every t ≥ t 1 .

  There exist t 1 ≥ t 0 and m < 3/2 such that γ(t)Γ(t) ≤ m for every t ≥ t 1 .Proof. (i) From Proposition 5.3, the function E is nonincreasing on [t 1 , +∞[. It ensues that E(t) ≤ E(t 1 ) for every t ≥ t 1 . Taking into account the expression (39), we deduce that for every t ≥ t 1 , Proposition 5.5. Let γ : [t 0 , +∞[→ R + be a continuous function satisfying (H 0 ) and (K + 1 ). Then, we have Let θ : [t 0 , +∞[→ R + be a differentiable test function, and let t 1 ≥ t 0 be given by the assumption (K + 1 ). Let us multiply the inequality (42) by θ(t) and integrate on [t 1 , t] Theorem 5.6. Let γ : [t 0 , +∞[→ R + be a continuous function satisfying (H 0 ), (K + 1 ), along with(K 2 )In particular, we obtain lim t→+∞ Φ(ξ(t)) = min H Φ, and lim t→+∞ ẋ(t) = 0.

	Since E(t) ≥ 0 and γ(t)Γ(t) ≤ m for every t ≥ t 1 , this implies that
						t	+∞
				(3 -2m)	t1	Γ(s) (Φ λ(s) (x(s)) -min t0 H Γ(s) ds = +∞.	Φ) ds ≤ E(t 1 ).
	The inequality (41) is obtained by letting t tend toward infinity. Let x(.) be a solution of (RIGS) γ,λ . Then we have
						1/2
	(i) As a consequence, setting ξ(t) = prox λ(t)Φ (x(t)), we have +∞ t0 +∞ Φ λ(t) (x(t)) -min H Φ = o 1 t t0 Γ(s) ds and ẋ(t) = o Γ(t) ẋ(t) 2 dt < +∞, and hence t0 Γ(t) W (t) dt < +∞; t0 Γ(s) ds 1 t	as t → +∞.
	(ii) Proof. By (34) and λ nondecreasing we have +∞ t0 γ(t) t t0 Γ(s) ds ẋ(t) 2 dt < +∞. (44) Φ(ξ(t)) -min H Φ = o 1 t t0 Γ(s) ds and	x(t) -ξ(t) = o	λ(t) t0 Γ(s) ds t	1/2	as t → +∞.
	Ẇ (t) ≤ -γ(t) ẋ(t) 2 . Proof. From Proposition 5.5 (i), we have (42) +∞ t0 Γ(t) W (t) dt < +∞. On the other hand, the energy
	function W is nonincreasing by Proposition 5.1. By applying Lemma B.3 in the Appendix, we obtain
	t t1 The announced estimates follow immediately. θ(s) Ẇ (s) ds + that W (t) = o t t0 Γ(s) ds t t1 θ(s)γ(s) ẋ(s) 2 ds ≤ 0. 1 as t → +∞. Integrating by parts yields
					t	t
	(43)	θ(t)W (t) +	θ(s)γ(s) ẋ(s) 2 ds ≤ θ(t 1 )W (t 1 ) +	θ(s)W (s) ds.
					t1	t1
	Using the expression of W and rearranging the terms, we find
		t				t
	θ(t)W (t) +	t1	2 . θ(s)γ(s) -θ(s)/2 ẋ(s) 2 ds ≤ θ(t 1 )W (t 1 ) +	t1	θ(s)(Φ λ(s) (x(s)) -min
	As a consequence, setting ξ(t) = prox λ(t)Φ (x(t)), we have
			Φ(ξ(t)) -min Φ ≤	E(t 1 ) Γ(t) 2 and	x(t) -ξ(t) 2 ≤	2λ(t) Γ(t) 2 E(t 1 ).
	(ii) Assume moreover that λ(t) t Proof. (i) Since sup t≥t0 t 0
	(K + 1 )
	Then we have (41) satisfies the following differential inequality +∞ Γ(t) (Φ λ(t) (x(t)) -min H ḧ(t) + γ(t) ḣ(t) ≤ ẋ(t) 2 . Φ) dt ≤ E(t 1 ) 3 -2m t1 From Proposition 5.5 (i), we have +∞ t0 Γ(s) ẋ(s)	< +∞.
	Under (K + 1 ), we have the estimate	+∞ t1	Γ(s)(Φ λ(s) (x(s)) -min H Φ) ds < +∞, see Corollary 5.4 (ii). The
	Γ(t) 2 (Φ λ(t) (x(t)) -min H announced estimates follow immediately. Φ) ≤ E(t 1 ) and (ii) Take now θ(t) = t t0 Γ(s) ds. Recalling that W (t) ≥ 0, inequality (43) then implies that for every 1 x(t) -x + Γ(t) ẋ(t) 2 ≤ E(t 1 ). 2 t ≥ t 1 , The first assertion follows immediately. t s t1 t (ii) Now assume (K + 1 ). By integrating (40) on [t 1 , t], we find γ(s) Γ(u) du ẋ(s) 2 ds ≤ Γ(s) ds W (t 1 ) + Γ(s)W (s) ds.
	t1		t0	t		t0	t1
	E(t) + It suffices then to recall that t1	Φ λ(s) (x(s)) -min H +∞ t1 Γ(s)W (s) ds < +∞ under hypothesis (K + Φ Γ(s)(3 -2γ(s)Γ(s)) ds ≤ E(t 1 ). 1 ), see point (i).

H Φ) ds.

(i) Choosing θ(t) = Γ(t)

2

, the above equality gives for every t ≥ t 1 ,

Γ(t) 2 W (t) + t t1 Γ(s)[Γ(s)γ(s) -Γ(s)] ẋ(s) 2 ds ≤ Γ(t 1 ) 2 W (t 1 ) + 2 t t1 Γ(s) Γ(s)(Φ λ(s) (x(s)) -min H Φ) ds.

Recalling that Γ = γΓ -1, we deduce that

Γ(t) 2 W (t) + t t1 Γ(s) ẋ(s) 2 ds ≤ Γ(t 1 ) 2 W (t 1 ) + 2 t t1 Γ(s)(γ(s)Γ(s) -1)(Φ λ(s) (x(s)) -min H Φ) ds.

By assumption (K + 1 ), we have γ(t)Γ(t) ≤ 3/2 for every t ≥ t 1 . Since W (t) ≥ 0, it ensues that

t t1 Γ(s) ẋ(s) 2 ds ≤ Γ(t 1 ) 2 W (t 1 ) + t t1 Γ(s)(Φ λ(s) (x(s)) -min H Φ)

ds. Theorem 5.7. Let γ : [t 0 , +∞[→ R + be a continuous function satisfying (H 0 ), (K + 1 ), and (K 2 ). Suppose that λ : [t 0 , +∞[→ R * + is nondecreasing and satisfies sup t≥t0 λ(t) t t0 Γ(s) ds < +∞.

Then, for every solution x(.) of (RIGS) γ,λ the following properties hold:

(i) lim t→+∞ ξ(t) -x(t) = 0, where ξ(t) = prox λ(t)Φ (x(t));

(ii) x(t) converges weakly as t → +∞ toward some x * ∈ argmin Φ.

Γ(s) ds < +∞, the second estimate of (44) implies that lim t→+∞ ξ(t) -x(t) = 0.

(ii) We apply the Opial lemma, see Lemma 2.2. Let us fix x ∈ argmin Φ, and show that lim t→+∞ x(t)-x exists. For that purpose, let us set h(t) = 1 2 x(t) -x 2 . Recall from Lemma 5.2 that the function h 2 ds < +∞. By applying Lemma B.1 with g :

  B.1. Let γ : [t 0 , +∞[→ R + be a continuous function satisfying ) dτ . Let g : [t 0 , +∞[→ R be a continuous function. Assume that h : [t 0 , +∞[→ R + is a function of class C 2 satisfying . Then the nonnegative part ḣ+ of ḣ belongs to L 1 (t 0 , +∞), and hence lim t→+∞ h(t) exists. Proof. (i) Let us multiply each member of inequality (48) by p(t) = e ) dτ and integrate on [t 0 , t]. By integrating again on [t 0 , t], we find h(t) ≤ h(t 0 ) + ḣ(t 0 ) |g(s)| ds < +∞. We easily deduce from (50) that for every t ≥ t 0 , < +∞ by assumption, we deduce from (51) and (52) that ḣ+ ∈ L 1 (t 0 , +∞). Hence lim t→+∞ h(t) exists. Let us now state a vector-valued version of Lemma B.1. Lemma B.2. Let γ : [t 0 , +∞[→ R + be a continuous function satisfying < +∞, where the function p is defined by p(t) = e ) dτ . Let F : [t 0 , +∞[→ H be a measurable map such that +∞ t0 Γ(t) F (t) dt < +∞. Assume that x : [t 0 , +∞[→ H is a map of class C 2 satisfying ) dτ and integrate on [t 0 , t]. We obtain for every t ≥ t 0 , By integrating and applying Fubini theorem as in the proof of Lemma B.1, we find F (s) ds < +∞. The strong convergence of x(t) as t → +∞ follows immediately. Owing to the next lemma, we can estimate the rate of convergence of a function w : [t 0 , +∞[→ R + supposed to be nonincreasing and summable with respect to a weight function Γ. Lemma B.3. Let Γ : [t 0 , +∞[→ R + be a measurable function such that +∞ t0 Γ(t) dt = +∞. Assume that w : [t 0 , +∞[→ R + is nonincreasing and satisfies Proof. Let F : [t 0 , +∞[→ R + be the function defined by F (t) = t t0 Γ(s) ds. It follows from the hypothesis +∞ t0 Γ(s) ds = +∞ that the function F is an increasing bijection from [t 0 , +∞[ onto [0, +∞[. For every t ≥ t 0 , let us set α(t) = F -1 ( 1 2 F (t)). By definition, we have

	+∞ t0 p(s)g(s) ds. ds p(s) < +∞, where the +∞ t ds p(s) t t 0 γ(τ We obtain function p is defined by p(t) = e t t 0 γ(τ (48) ḧ(t) + γ(t) ḣ(t) ≤ g(t) on [t 0 , +∞[. (i) For every t ≥ t 0 , we have (49) h(t) ≤ h(t 0 ) + ḣ(t 0 ) t t0 du p(u) + t t0 t s du p(u) (ii) Assume that +∞ t0 t t0 du p(u) + t t0 1 p(u) u t0 p(s) g(s) ds du. From Fubini theorem, we have t t0 1 p(u) u t0 p(s) g(s) ds du = t t0 t s du p(u) p(s)g(s) ds, and the inequality (49) follows immediately. (ii) Let us now assume that +∞ t0 Γ(s) (51) ḣ+ (t) ≤ | ḣ(t 0 )| 1 p(t) + 1 p(t) t t0 p(s) |g(s)| ds. By applying Fubini theorem, we find +∞ t0 1 p(t) t t0 p(s) |g(s)| ds dt = +∞ t0 +∞ s dt p(t) p(s) |g(s)| ds = +∞ t0 Γ(s) |g(s)| ds < +∞. (52) Since +∞ t0 dt p(t) +∞ t0 ds p(s) t t 0 γ(τ (53) t t 0 γ(τ ẋ(t) = ẋ(t 0 ) 1 p(t) + 1 p(t) t t0 p(s) F (s) ds. Taking the norm of each member, we deduce that ẋ(t) ≤ ẋ(t 0 ) 1 p(t) + 1 p(t) t t0 p(s) F (s) ds. +∞ t0 ẋ(t) dt ≤ ẋ(t 0 ) +∞ t0 dt t0 p(t) + +∞ t0 Γ(t)w(t) dt < +∞. Then we have w(t) = o 1 t t0 Γ(s) ds as t → +∞. α(t) t0 Γ(s) ds = 1 2 t t0 Γ(s) ds, hence t α(t) Γ(s) ds = 1 2 t t0 Γ(s) ds. Recalling that the function w is nonincreasing, we obtain t α(t) Γ(s) w(s) ds ≥ w(t) t α(t) Γ(s) ds = 1 2 w(t) t t0 Γ(s) ds. By assumption, we have +∞ t0 Γ(s)w(s) ds < +∞. Since lim t→+∞ α(t) = +∞, we deduce that t Γ(s) +∞ lim t→+∞ α(t)
	(50)	ḣ(t) ≤ ḣ(t 0 )	1 p(t)	+	1 p(t)	t t0	p(s) g(s) ds.

Γ(s) |g(s)| ds < +∞, where Γ : [t 0 , +∞[→ R + is given by Γ(t) = p(t) ẍ(t) + γ(t) ẋ(t) = F (t) on [t 0 , +∞[.

Then ẋ ∈ L 1 (t 0 , +∞), and hence x(t) converges strongly as t → +∞.

Proof. Let us multiply (53) by p(t) = e

  with M = sup t≥t0 x(t) -x ∞ < +∞. Taking the upper limit as t → +∞, we deduce from property (20) that lim sup Since this is true for every ε > 0, we conclude that lim t→+∞ x(t) -x ∞ = 0.Lemma B.5. Let (X , . ) be a Banach space, and let x : [t 0 , +∞[→ X be a continuous map, supposed to be bounded on [t 0 , +∞[. Let Λ 1 , Λ 2 : [t 0 , +∞[×[t 0 , +∞[→ R + be measurable functions satisfying[START_REF] Chambolle | On the convergence of the iterates of the Fast Iterative Shrinkage/Thresholding Algorithm[END_REF]. (s, t) -Λ 2 (s, t)| ds = 0.Let us consider the averaged trajectories x 1 , x 2 : [t 0 , +∞[→ X defined byx 1 (t) =Then we have lim t→+∞ x 1 (t) -x 2 (t) = 0.Proof. Let M ≥ 0 be such that x(t) ≤ M for every t ≥ t 0 . Observe thatx 1 (t) -x 2 (t) =

	Assume that
			+∞
	(56)	lim t→+∞ |Λ 1 +∞ t0	+∞
		Λ 1 (s, t) x(s) ds and x 2 (t) =	Λ 2 (s, t) x(s) ds.
		t0		t0
				+∞
				Λ(s, t) ds
				T
		T
		≤ M	Λ(s, t) ds + ε,
		t0

Λ(s, t) ds

+ ε t→+∞ x(t) -x ∞ ≤ ε. +∞ t0 (Λ 1 (s, t) -Λ 2 (s, t))x(s) ds ≤ +∞ t0 |Λ 1 (s, t) -Λ 2 (s, t) | x(s) ds ≤ M +∞ t0 |Λ 1 (s, t) -Λ 2 (s, t))| ds -→ 0 as t → +∞,

in view of (56).

Appendix A. Yosida regularization and Moreau envelopes A.1. Yosida regularization of an operator A. A set-valued mapping A from H to H assigns to each x ∈ H a set A(x) ⊂ H, hence it is a mapping from H to 2 H . Every set-valued mappping A : H → 2 H can be identified with its graph defined by

The set {x ∈ H : 0 ∈ A(x)} of the zeros of A is denoted by zerA. An operator A : H → 2 H is said to be monotone if for any (x, u), (y, v) ∈ gphA, one has y -x, v -u ≥ 0. It is maximally monotone if there exists no monotone operator whose graph strictly contains gphA. If a single-valued operator A : H → H is continuous and monotone, then it is maximally monotone, cf. [START_REF] Brézis | Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution[END_REF]Proposition 2.4].

Given a maximally monotone operator A and λ > 0, the resolvent of A with index λ and the Yosida regularization of A with parameter λ are defined by

respectively. The operator J λA : H → H is nonexpansive and eveywhere defined (indeed it is firmly non-expansive). Moreover, A λ is λ-cocoercive: for all x, y ∈ H we have

This property immediately implies that A λ : H → H is 1 λ -Lipschitz continuous. Another property that proves useful is the resolvent equation (see, for example, [15, Proposition 2.6] or [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF]Proposition 23.6])

which is valid for any λ, µ > 0. This property allows to compute simply the resolvent of A λ by

for any λ, µ > 0. Also note that for any x ∈ H, and any λ > 0

Finally, for any λ > 0, A and A λ have the same solution set S := A -1 λ (0) = A -1 (0). For a detailed presentation of the properties of the maximally monotone operators and the Yosida approximation, the reader can consult [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF] or [START_REF] Brézis | Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution[END_REF]. A.2. Differentiability properties of the Moreau envelopes. Lemma A.1. For each x ∈ H, the real-valued function λ → Φ λ (x) is continuously differentiable on ]0, +∞[, with

Proof. By definition of Φ λ , we have

where the infimum in the above expression is achieved at J λ (x) := (I + λ∂Φ) -1 (x). Let us prove that (47) d dλ λΦ λ (x) = Φ(J λ (x)).