Hedy Attouch
email: hedy.attouch@umontpellier.fr

Jalal Fadili
email: jalal.fadili@greyc.ensicaen.fr

Vyacheslav Kungurtsev
email: vyacheslav.kungurtsev@fel.cvut.cz

ON THE EFFECT OF PERTURBATIONS IN FIRST-ORDER OPTIMIZATION METHODS WITH INERTIA AND HESSIAN DRIVEN DAMPING

Keywords: 2020 Mathematics Subject Classification. 37N40, 46N10, 49M30, 65B99, 65K05, 65K10, 90B50, 90C25 Hessian driven damping, damped inertial dynamics, accelerated convex optimization, convergence rates, Lyapunov analysis, perturbation, errors

Second-order continuous-time dissipative dynamical systems with viscous and Hessian driven damping have inspired effective first-order algorithms for solving convex optimization problems. While preserving the fast convergence properties of the Nesterov-type acceleration, the Hessian driven damping makes it possible to significantly attenuate the oscillations. To study the stability of these algorithms with respect to perturbations, we analyze the behaviour of the corresponding continuous systems when the gradient computation is subject to exogenous additive errors. We provide a quantitative analysis of the asymptotic behaviour of two types of systems, those with implicit and explicit Hessian driven damping. We consider convex, strongly convex, and non-smooth objective functions defined on a real Hilbert space and show that, depending on the formulation, different integrability conditions on the perturbations are sufficient to maintain the convergence rates of the systems. We highlight the differences between the implicit and explicit Hessian damping, and in particular point out that the assumptions on the objective and perturbations needed in the implicit case are more stringent than in the explicit case.

1. Introduction. The continuous-time dynamic perspective of optimization algorithms, which can be viewed as temporal discretization schemes thereof, offers an insightful and powerful framework for the study of the behaviour of these algorithms. In this paper, we study inertial systems involving both viscous and Hessian-driven damping, where the first-order gradient information is only accessible up to some exogenous additive error .

1.1. Problem statement. Throughout the paper, we make the following standing assumptions:

f is a convex function on a real Hilbert space H, and S := argmin H f = ∅.

We will study perturbed versions of two second-order ordinary differential equations (ODE). They differ from each other in that the Hessian driven damping appears explicitly in one and implicitly in the other.

1.1.1. Explicit Hessian. The first system we look at, which was proposed in [START_REF] Attouch | First order optimization algorithms via inertial systems with Hessian driven damping[END_REF] (see also [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF]), takes the form ẍ(t) + γ(t) ẋ(t) + β(t) d dt (∇f (x(t))) + b(t)∇f (x(t)) = 0, (ISEHD)

where f ∈ C 1 (H), γ, β, b : [t 0 , +∞[→ R + are continuous functions, and t 0 > 0 is the initial time. The coefficients (γ, β, b) have a physical interpretation corresponding to natural phenomena:

• γ(t) is the viscous damping coefficient,

• β(t) is the Hessian-driven damping coefficient (which will be made clear),

• b(t) is the time scaling coefficient (see [START_REF] Attouch | Fast proximal methods via time scaling of damped inertial dynamics[END_REF]).

We term the above ODE an Inertial System with Explicit Hessian Damping (ISEHD for short), since d dt (∇f (x(t))) = ∇ 2 f (x(t)) ẋ(t), when f is of class C 2 (H). Throughout the paper, we consider (ISEHD) with the particular choice of parameters γ(t) = α t , α ≥ 0, β(t) ≡ β > 0 and b(t) ≡ 1.

This choice of the viscous damping parameter γ(t) = α t is justified by its direct link with the accelerated gradient method of Nesterov [START_REF] Nesterov | A method of solving a convex programming problem with convergence rate O(1/k 2 )[END_REF][START_REF] Nesterov | Introductory Lectures on Convex Optimization: A Basic Course[END_REF], as shown in [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with timedependent viscosity[END_REF], [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF], [START_REF] Aujol | Optimal convergence rates for Nesterov acceleration[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]. Related systems have been considered in [START_REF] Lin | A control-theoretic perspective on optimal high-order optimization[END_REF] from the closed loop control perspective and in [START_REF] Shi | Understanding the acceleration phenomenon via high-resolution differential equations[END_REF] by means of high-resolution of differential equations.

1.1.2. Implicit Hessian. The second system we consider, inspired by [START_REF] Alecsa | An extension of the second order dynamical system that models Nesterov's convex gradient method[END_REF] (see also [START_REF] Muehlebach | A dynamical systems perspective on nesterov acceleration[END_REF] for a related autonomous system in the case of a strongly convex function f ), is ẍ(t) + α t ẋ(t) + ∇f x(t) + β(t) ẋ(t) = 0, (ISIHD)

where α ≥ 3 and β(t) = γ + β t , γ, β ≥ 0. We coin this ODE an Inertial System with Implicit Hessian Damping (ISIHD for short). The rationale justifying our use of the term "implicit" comes from the observation that by a Taylor expansion (as t → +∞ we have ẋ(t) → 0 which justifies using Taylor expansion), one has ∇f (x(t) + β(t) ẋ(t)) ≈ ∇f (x(t)) + β(t)∇ 2 f (x(t)) ẋ(t), hence making the Hessian damping appear indirectly in (ISIHD). This ODE was found to have a smoothing effect on the energy error and oscillations.

1.1.3. Exogenous additive error. We are interested in the situation where ∇f (x(t)) is always evaluated with an exogenous additive error e(t). With the choice of parameters made above, the perturbed dynamics of (ISEHD) and (ISIHD) are written ẍ(t) + α t ẋ(t) + β d dt ∇f (x(t)) + e(t) + ∇f (x(t)) + e(t) = 0, (ISEHD-Pert) ẍ(t) + α t ẋ(t) + ∇f x(t) + β(t) ẋ(t) + e(t) = 0.

(ISIHD-Pert)

For system (ISEHD-Pert), the overall perturbation error affecting the system is β ė(t) + e(t). Because the Hessian appears explicitly, both the error on the gradient and its derivative appear. It can then be anticipated that assumptions regarding both e(t) and ė(t), in particular their integrability, will be instrumental in deriving any convergence guarantees. On the other hand, in the system (ISIHD-Pert) with implicit Hessian damping, the error perturbation e(t) appears without its time derivative. Naturally, we will see in this case that convergence results will be derived without any assumptions on the time derivative of the error. While this may be seen as an advantage at first glance, this comes at a price. Indeed, as we will also see, to maintain fast convergence guarantees, the integrability requirements on the error e(t) will be more stringent for (ISIHD-Pert) than for (ISEHD-Pert), i.e., higher-order moments of e(t) will be required to be finite. We anticipate that when it comes to discrete algorithms, the assumptions on the objective and perturbations needed in the implicit case are more stringent than in the explicit case. We plan to study these questions in a future work. Note that similar questions arise when the perturbation is attached to a Tikhonov regularization term with an asymptotically vanishing coefficient [START_REF] Bot | Tikhonov regularization of a second order dynamical system with Hessian damping[END_REF]. One of our motivations for the above additive perturbation model originates from optimization, where the gradient may be accessed only inaccurately, either because of physical or computational reasons. The prototype example we think of is

f (x) = E ξ [F (x, ξ)],
where E ξ [•] is expectation with respect to the random variable ξ, and F (•, ξ) ∈ C 1 (H) for any ξ. This is a popular setting in numerous applications (imaging, statistical learning, etc.), where computing ∇f (x) (or even f (x)) is either impossible or computationally very expensive. Rather, one draws m independent samples of ξ, say (ξ i ) 1≤i≤m , and compute the average estimate

∇f (x) = 1 m m i=1 ∇F (x, ξ i ).
In our notation, the error is then e(t) = ∇f (x(t)) -∇f (x(t)). Under independence and mild assumptions, one has by the law of iterated logarithm that e(t) = O log(log(m)) m almost surely. Thus, to make this error vanish or even integrable, one has to take m(.) an increasing function of t at least at the rate O(t 2(1+δ) ) for δ > 0. At this stage, some readers may have expected a smallness condition on the perturbation rather than integrability conditions, i.e., of e(t) = O(r(t)) for some r(t) such that lim t→∞ r(t) = 0. Of course, integrability is stronger as it obviously implies that the error function, whenever it converges, will vanish asymptotically. However, one has to keep in mind that our goal is not only to establish convergence (and rate of convergence) of the objective values to a "perturbation dominated region" around the optimal value, but to show additional convergence guarantees, in particular the important matter of weak convergence of the trajectory. Deriving convergence of trajectories is typically much more challenging than the objective and cannot be proved under a mere smallness condition on the error. 1.2. Contributions. In [START_REF] Attouch | First order optimization algorithms via inertial systems with Hessian driven damping[END_REF] (resp. [START_REF] Alecsa | An extension of the second order dynamical system that models Nesterov's convex gradient method[END_REF]), which studied the unperturbed system (ISEHD) (resp. (ISIHD)), fast convergence rates were obtained for the objective, velocities and gradients. Our main contribution in this paper is to analyze the robustness and stability of these systems, by quantifying their convergence properties in the presence of errors. We do this both in the general convex case and in the strongly convex case. We also study the case where the function f is non-smooth convex by proposing a first order formulation in time and space, with existence results and Lyapunov analysis. The main motivation for our work is to pave the way for the design and study of provably accelerated optimization algorithms that appropriately discretize the above dynamics while handling inexact evaluations of the gradient with deterministic and/or stochastic errors. The extension to the discrete setting of the results here will be the focus of a forthcoming paper.

1.3. Related works. Due to the importance of the subject in optimization and control, several articles have been devoted to the study of perturbations in dissipative inertial systems and in the corresponding accelerated first order algorithms. The subject was first considered in the case of a fixed viscous damping, [START_REF] Attouch | Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria[END_REF][START_REF] Haraux | On a second order dissipative ode in Hilbert space with an integrable source term[END_REF]. Then it was studied within the framework of the accelerated gradient method of Nesterov, and of the corresponding inertial dynamics with vanishing viscous damping, see [START_REF] Shi | Understanding the acceleration phenomenon via high-resolution differential equations[END_REF][START_REF] Attouch | Accelerated forward-backward algorithms with perturbations: Application to Tikhonov regularization[END_REF][START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF][START_REF] Aujol | Stability of over-relaxations for the forward-backward algorithm, application to FISTA[END_REF][START_REF] Schmidt | Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization[END_REF][START_REF] Villa | Accelerated and inexact forward-backward[END_REF]. In the presence of the additional Hessian driven damping, first results have been obtained in [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF][START_REF] Attouch | First order optimization algorithms via inertial systems with Hessian driven damping[END_REF][START_REF] Attouch | Convergence of iterates for first-order optimization algorithms with inertia and hessian driven damping[END_REF] in the case of a smooth function. To the best of our knowledge, our work is the first to consider these questions in full generality and in presence of perturbations.

1.4. Contents. In Section 2, we prove that the two systems are well-posed both in the smooth and non-smooth cases. In Section 3, we study the convex case, and establish convergence rates for both systems under appropriate integrability assumptions on the error. In Section 4, we consider the strongly convex case. Section 5 is devoted to studying non-smooth f . In Section 6, we present some numerical illustrations of the results. In Section 7, we draw key conclusions and present some perspectives. 1.5. Main notations. H is a real Hilbert space, •, • is the scalar product on H and • is the corresponding norm. Γ 0 (H) is the class of proper, lower semicontinuous (lsc) and convex functions from

H to R ∪ {+∞}. A function g : H → R ∪ {+∞} is µ-strongly convex (µ > 0) if g -µ 2 • 2 is convex. For g ∈ Γ 0 (H)
, its domain is dom(g) := {x ∈ H : g(x) < +∞}. ∂g denotes the (convex) subdifferential operator of g. When g is differentiable at x ∈ H, then ∂g(x) = {∇g(x)}. We also denote dom(∂g) := {x ∈ H : ∂g(x) = ∅}.

C s (D) is the class of s-continuously differentiable functions on D, D will be specified in the context. For T > t 0 and p ≥ 1, L p (t 0 , T ; H) is the Lebesgue space of measurable functions

x : t ∈ [t 0 , T ] → x(t) ∈ H such that T t0 x(t)
p dt < +∞. W 1,1 (t 0 , T ; H) is the Sobolev space of functions x(.) ∈ L 1 (t 0 , T ; H) with distributional derivative ẋ(•) ∈ L 1 (t 0 , T ; H). We will also invoke the notion of a strong solution to a differential inclusion, see [24, Definition 3.1], that will be given precisely in Section 2.1.2. The reader interested mostly in quantitative convergence estimates can skip the corresponding section. We take t 0 > 0 as the origin of time. This is justified by the singularity of the viscous damping coefficient α t at the origin. This is not restrictive since we are interested in asymptotic analysis. We denote S to be the set of minimizers of f , i.e., S := argmin x∈H f (x) which is assumed nonempty, and x to be an arbitrary element of S (unique in the case of strongly convex f ). We denote f := inf x∈H f (x).

2. Well-posedness. When β > 0, the presence of Hessian driven damping in the inertial dynamics makes it possible to reformulate the equations as first-order systems both in time and in space, without explicit evaluation of the Hessian. This will allow us to extend the existence of trajectories and the convergence results to the case f ∈ Γ 0 (H), by simply replacing the gradient of f with the subdifferential ∂f . This approach was initiated in [START_REF] Alvarez | A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics[END_REF] and used in [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF] for the unperturbed case.

Explicit Hessian damping.

2.1.1. Formulation as a first-order system. Let us start by establishing this equivalence in the case of a smooth function f .

Theorem 2.1. Let f : H → R be a C 2 (H) function and e : [t 0 , +∞[→ H be C 1 (H). Suppose that α ≥ 0, β > 0. Let (x 0 , ẋ0 ) ∈ H × H.
The following statements are equivalent:

1.

x : [t 0 , +∞[→ H is a solution trajectory of (ISEHD-Pert) with the initial conditions x(t 0 ) = x 0 , ẋ(t 0 ) = ẋ0 . 2. (x, y) : [t 0 , +∞[→ H × H is a solution trajectory of the first-order system

   ẋ(t) + β(∇f (x(t)) + e(t)) -1 β -α t x(t) + 1 β y(t) = 0 ẏ(t) -1 β -α t + αβ t 2 x(t) + 1 β y(t) = 0, (1)
with initial conditions x(t 0 ) = x 0 , y(t 0 ) = -β( ẋ0 + β∇f (x 0 )) + (1 -βα/t 0 )x 0 - β 2 e(t 0 ). Proof. 2. ⇒ 1.
Differentiating the first equation of (1) gives

ẍ(t) + β ∇ 2 f (x(t)) ẋ(t) + ė(t) - α t 2 x(t) - 1 β - α t ẋ(t) + 1 β ẏ(t) = 0. (2)
Replacing ẏ(t) by its expression as given by the second equation of (1) gives

ẍ(t) + β ∇ 2 f (x(t)) ẋ(t) + ė(t) - α t 2 x(t) - 1 β - α t ẋ(t) + 1 β 1 β - α t + αβ t 2 x(t) - 1 β y(t) = 0. ( 3
)
Then replace y(t) by its expression as given by the first equation of ( 1)

ẍ(t) + β ∇ 2 f (x(t)) ẋ(t) + ė(t) - α t 2 x(t) - 1 β - α t ẋ(t) + 1 β 1 β - α t + αβ t 2 x(t) + ẋ(t) + β(∇f (x(t)) + e(t)) - 1 β - α t x(t) = 0.
After simplification of the above expression, we obtain (ISEHD-Pert).

1. ⇒ 2. Define y(t) by the first equation of [START_REF] Adly | Finite convergence of proximal-gradient inertial algorithms combining dry friction with Hessian-driven damping[END_REF]. Differentiating y(t) and using equation (ISEHD-Pert) allows one to eliminate ẍ(t), which finally gives the second equation of (1).

2.1.2. Existence and uniqueness of a solution. Capitalizing on the result of Theorem 2.1, the following first order formulation assists in providing a meaning to our system when f ∈ Γ 0 (H). It is obtained by substituting the subdifferential ∂f for the gradient ∇f in the first-order formulation (1). Definition 2.1. Let α ≥ 0, β > 0 and f ∈ Γ 0 (H). Given (x 0 , y 0 ) ∈ dom(f )×H, the Cauchy problem for the perturbed inertial system with explicit generalized Hessian driven damping is defined by

         ẋ(t) + β(∂f (x(t)) + e(t)) -1 β -α t x(t) + 1 β y(t) 0 ẏ(t) -1 β -α t + αβ t 2 x(t) + 1 β y(t) = 0 x(t 0 ) = x 0 , y(t 0 ) = y 0 . (4)
Let us formulate (4) in a condensed form as an evolution equation in the product space H × H. Setting Z(t) = (x(t), y(t)) ∈ H × H, (4) can be equivalently written

Ż(t) + ∂G(Z(t)) + D(t, Z(t)) 0, Z(t 0 ) = (x 0 , y 0 ), (5)
where G ∈ Γ 0 (H × H) is the function defined by G(Z) = βf (x), and the time-

dependent operator D : [t 0 , +∞[×H × H → H × H is given by D(t, Z) = βe(t) - 1 β - α t x + 1 β y, - 1 β - α t + αβ t 2 x + 1 β y . (6)
The differential inclusion ( 5) is governed by the sum of the maximal monotone operator ∂G (a convex subdifferential) and the time-dependent affine continuous operator D(t, •). The existence and uniqueness of a global solution for the corresponding Cauchy problem is a consequence of the general theory of evolution equations governed by maximally monotone operators. In this setting, we need to invoke the notion of strong solution that we make precise now. Definition 2.2. Given g ∈ Γ 0 (H), and an operator D : [t 0 , +∞[×H → H, we say that z : [t 0 , T ] → H is a strong solution trajectory on [t 0 , T ] of the differential inclusion ż(t) + ∂g(z(t)) + D(t, z(t)) 0, (7) if the following properties are satisfied:

1. z is continuous on [t 0 , T ] and absolutely continuous on any compact subset of ]t 0 , T ]; 2. z(t) ∈ dom(∂g) for almost every t ∈]t 0 , T ], and ( 7) is verified for almost every t ∈]t 0 , T ]. z : [t 0 , +∞[→ H is a global strong solution of [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with timedependent viscosity[END_REF], if it is a strong solution on [t 0 , T ] for all T > t 0 .

The existence and uniqueness of a global strong solution of the Cauchy problem (4) is established in the following theorem. Theorem 2.2. Let f ∈ Γ 0 (H), α ≥ 0 and β > 0. Suppose that e ∈ L 2 (t 0 , T ; H) for every T > t 0 . Then, for any Cauchy data (x 0 , y 0 ) ∈ dom(f ) × H, there exists a unique global strong solution (x, y) : [t 0 , +∞[→ H × H of (4) satisfying the initial condition x(t 0 ) = x 0 , y(t 0 ) = y 0 . Moreover, this solution exhibits the following properties:

(i) y ∈ C 1 ([t 0 , +∞[), and ẏ(t) -1 β -α t + αβ t 2
x(t) + 1 β y(t) = 0, for t ≥ t 0 ; (ii) x is absolutely continuous on [t 0 , T ] and ẋ ∈ L 2 (t 0 , T ; H) for all T > t 0 ;

(iii) x(t) ∈ dom(∂f ) for all t > t 0 ;

(iv) x is Lipschitz continuous on any compact subinterval of ]t 0 , +∞[; (v) the function t → f (x(t)) is absolutely continuous on [t 0 , T ] for all T > t 0 ;

(vi) there exists a function ξ

: [t 0 , +∞[→ H such that (a) ξ(t) ∈ ∂f (x(t)) for all t > t 0 ; (b) ẋ(t) + βξ(t) + βe(t) -1 β -α t x(t) + 1 β y(t) = 0 for almost every t > t 0 ; (c) ξ ∈ L 2 (t 0 , T ; H) for all T > t 0 ; (d) d dt f (x(t)) = ξ(t), ẋ(t) for almost every t > t 0 .
Proof. It is sufficient to prove that (x, y) is a strong solution of (4) on [t 0 , T ] and that the properties hold on [t 0 , T ] for all T > t 0 . So let us fix T > t 0 . As we have already noticed, (4) can be written in the form (5) which is a Lipschitz perturbation of the differential inclusion governed by the subdifferential of a proper lsc convex function.

A direct application of [START_REF] Brézis | Opérateurs Maximaux Monotones Dans Les Espaces de Hilbert et Équations D'évolution[END_REF]Proposition 3.12] gives the existence and uniqueness of a strong global solution Z = (x, y) : [t 0 , T ] → H × H to (5), or equivalently to (4), with initial condition Z(t 0 ) = (x(t 0 ), y(t 0 )) = (x 0 , y 0 ). Verification of items (iii) to (vi) follows the same lines as the proof of [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF]Theorem 4.4]. Of particular importance is the generalized derivation chain rule given in (vi)(d), which follows from [START_REF] Brézis | Opérateurs Maximaux Monotones Dans Les Espaces de Hilbert et Équations D'évolution[END_REF]Lemma 3.3] after checking that the corresponding assumptions are met thanks to (ii), (vi)(a) and (vi)(c).

Under sufficient differentiability properties of the data, we recover a classical solution, i.e., x(•) is a C 2 ([t 0 , +∞[) function, all the derivatives involved in the equation (ISEHD-Pert) are taken in the sense of classical differential calculus, and the equation (ISEHD-Pert) is satisfied for all t ≥ t 0 . This is made precise in the following statement.

Corollary 2.1. Assume that f is a convex C 2 (H) function and e belongs to C 1 ([t 0 , +∞[). Then, for any t 0 > 0, and any Cauchy data (x 0 , ẋ0 ), the system (ISEHD-Pert) with α, β ≥ 0 admits a unique classical global solution x : [t 0 , +∞[→ H satisfying (x(t 0 ), ẋ(t 0 )) = (x 0 , ẋ0 ).

Proof. Under the above regularity assumptions, the first equation of the first order system (4)

ẋ(t) + β(∇f (x(t)) + e(t)) - 1 β - α t x(t) + 1 β y(t) = 0 implies that ẋ is a C 1 ([t 0 , +∞[)
β(t) > 0.
Let us introduce the new function

y(t) := x(t) + β(t) ẋ(t), (8)
whose time derivation gives

ẏ(t) = ẋ(t) + β(t)ẍ(t) + β(t) ẋ(t). (9)
From (ISIHD-Pert) we know that

ẍ(t) = -α(t) ẋ(t) -∇f (y(t)) -e(t). (10)
By combining ( 9) and ( 10) we obtain

ẏ(t) = ẋ(t) + β(t) (-α(t) ẋ(t) -∇f (y(t)) -e(t)) + β(t) ẋ(t) = 1 -α(t)β(t) + β(t) ẋ(t) -β(t) (∇f (y(t) + e(t)) . (11)
From ( 8) and the fact that inf [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF]

t∈[t0,+∞[ β(t) > 0 we get ẋ(t) = 1 β(t) (y(t) -x(t)). Re- placing ẋ(t) in
) = x 0 , ẋ(t 0 ) = ẋ0 . 2. (x, y) : [t 0 , +∞[→ H × H is a solution trajectory of the first-order system    ẋ(t) + 1 β(t) x(t) -1 β(t) y(t) = 0. ẏ(t) + β(t) (∇f (y(t)) + e(t)) + 1 β(t) 1 -α(t)β(t) + β(t) (x(t) -y(t)) = 0 ( 12
)
with initial conditions x(t 0 ) = x 0 , y(t 0 ) = x 0 + β(t 0 ) ẋ0 .

2.2.2.

Existence and uniqueness of a solution. Existence and uniqueness of a global strong solution for the Cauchy problem associated with the unperturbed problem (ISIHD) was shown in [START_REF] Alecsa | An extension of the second order dynamical system that models Nesterov's convex gradient method[END_REF] when ∇f is Lipschitz continuous using the Cauchy-Lipschitz theorem. This result can be easily extended to (ISIHD-Pert). Rather, we take a different path here and proceed as in Section 2.1.2, so that we can extend the above formulation to the case where f ∈ Γ 0 (H), by replacing the gradient ∇f with the subdifferential ∂f .

Definition 2.3. Let α(t) ≥ 0, β(t) > 0, f ∈ Γ 0 (H). Given (x 0 , y 0 ) ∈ H × dom(f ), the Cauchy problem associated with the perturbed inertial system with implicit generalized Hessian driven damping is defined by

       ẋ(t) + 1 β(t) x(t) -1 β(t) y(t) = 0 ẏ(t) + β(t) (∂f (y(t)) + e(t)) + 1 β(t) 1 -α(t)β(t) + β(t) (x(t) -y(t)) 0 x(t 0 ) = x 0 , y(t 0 ) = y 0 . ( 13
)
We reformulate [START_REF] Attouch | Fast proximal methods via time scaling of damped inertial dynamics[END_REF] [START_REF] Brézis | Opérateurs Maximaux Monotones Dans Les Espaces de Hilbert et Équations D'évolution[END_REF]. Instead, one can appeal to the theory of evolution equations governed by general time-dependent subdifferentials as proposed in [START_REF] Attouch | Strong solutions for parabolic variational inequalities[END_REF] for example. In fact, for a system in the simpler form [START_REF] Attouch | Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria[END_REF], one can argue more easily, by making the change of time variable t = τ (s) with β(τ (s)) τ (s) = 1. Lemma A.4 then shows that ( 14) is equivalent to

Ẇ (s) + ∂G(W (s)) + F(s, W (s)) 0, (16)
where W (s) = Z(τ (s)), and

F(s, W (s)) = 1 β(τ (s)) D(τ (s), W (s)
) is affine continuous in its second argument. Provided that β ∈ L 1 (t 0 , +∞; R), this defines a proper change of variable in time. With the formulation [START_REF] Attouch | Newton-like inertial dynamics and proximal algorithms governed by maximally monotone operators[END_REF], we are brought back to the appropriate form to argue as before and invoke the results of [START_REF] Brézis | Opérateurs Maximaux Monotones Dans Les Espaces de Hilbert et Équations D'évolution[END_REF]. We leave the details to the reader for the sake of brevity.

Smooth convex case.

3.1. Explicit Hessian damping. Consider first the explicit Hessian system (ISEHD-Pert), where we assume that f ∈ C 2 (H), and recall the specific choices of γ(t) = α t , α > 0, β(t) ≡ β and b(t) ≡ 1. We will develop a Lyapunov analysis to study the dynamics of (ISEHD-Pert). Some of our arguments are inspired by the works of [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF] and [START_REF] Attouch | First order optimization algorithms via inertial systems with Hessian driven damping[END_REF]. Throughout this section we use the shorthand notation g(t) := e(t) + β ė(t) [START_REF] Attouch | Continuous Newton-like inertial dynamics for monotone inclusions[END_REF] for the overall contribution of the errors terms. We will first establish the minimization property which is valid by simply assuming the integrability of the error term and its derivative. Then, by reinforcing these hypotheses, we will obtain rapid convergence results, and the convergence of trajectories.

3.1.1. Minimizing properties. Define u : [t 0 , +∞[→ H by u(t) := x(t) + β t t0
∇f (x(s))ds, which will be instrumental in the proof of the following theorem. Note that, in the following statement, it is simply assumed that f is bounded from below, the set S := argmin H f may be empty. Then, for any solution trajectory x : [t 0 , +∞[→ H of (ISEHD-Pert), we have

Theorem 3.1. Let f : H → R be a C 2 (H)
(i) sup t≥t0 u(t) < +∞; (ii) +∞ t 0 1 t ẋ(t) 2 dt < +∞, +∞ t 0 ∇f (x(t)) 2 dt < +∞, +∞ t 0 1 t u(t) 2 dt < +∞; (iii) lim t→+∞ u(t) = 0; lim t→+∞ ẋ(t) = 0; lim t→+∞ ∇f (x(t)) = 0; (iv) lim t→+∞ f (x(t)) = inf H f .
Proof. Recall f := inf H f . Since our analysis is asymptotic, there is no restriction in assuming that t ≥ t 1 := max(t 0 , 2αβ). We will then prove the statements in terms of t 1 and passing to t 0 is immediate thanks to the properties of the solution x(t) in Theorem 2.2. Claim (i). For T ≥ t ≥ t 1 , define the function

W T (t) := 1 2 u(t) 2 + f (x(t)) -f - T t u(τ ), g(τ ) dτ.
Observe that W T is well-defined under our assumptions. Thus, taking the derivative in time and using (ISEHD-Pert), we get,

ẆT (t) = u(t), ü(t) + g(t) + ẋ(t), ∇f (x(t)) = ẋ(t) + β∇f (x(t)), ẍ(t) + β∇ 2 f (x(t)) ẋ(t) + g(t) + ẋ(t), ∇f (x(t)) = ẋ(t) + β∇f (x(t)), - α t ẋ(t) -∇f (x(t)) + ẋ(t), ∇f (x(t)) = - α t ẋ(t) 2 -β ∇f (x(t)) 2 - αβ t ẋ(t), ∇f (x(t)) ≤ - α 2t ẋ(t) 2 -β 1 - αβ 2t ∇f (x(t)) 2 ≤ - α 2t ẋ(t) 2 - β 2 ∇f (x(t)) 2 , (18)
where we used Young inequality and the fact that t ≥ t 1 > αβ. This implies that W T is non-increasing and in turn that

W T (t) ≤ W T (t 1 ) for t ∈ [t 1 , T ], i.e. 1 2 u(t) 2 + f (x(t)) -f - T t u(τ ), g(τ ) dτ ≤ 1 2 u(t 1 ) 2 + f (x(t 1 )) -f - T t1 u(τ ), g(τ ) dτ.
Therefore,

1 2 u(t) 2 ≤ 1 2 u(t 1 ) 2 + f (x(t 1 )) -f + t t1 u(τ ) g(τ ) dτ.
Applying the Gronwall Lemma A.3, we get

sup t≥t 1 u(t) ≤ u(t1) 2 + 2(f (x(t1)) -f ) 1/2 + +∞ t 1 e(τ ) dτ + β +∞ t 1 ė(τ ) dτ < +∞,
hence proving the first claim. Claim (ii). Now define for all t ≥ t 1

W (t) := 1 2 u(t) 2 + (f (x(t)) -f ) - +∞ t u(τ ), g(τ ) dτ.
This is again a well-defined function thanks to the first claim, and the integrability of g.

Moreover, W (•) is bounded from below, inf t≥t1 W (t) ≥ -sup t≥t1 u(t) +∞ t1 e(τ ) dτ + β +∞ t1 ė(τ ) dτ > -∞. (19)
Observe that Ẇ (t) = ẆT (t). This together with (18) yields

Ẇ (t) + α 2t ẋ(t) 2 + β 2 ∇f (x(t)) 2 ≤ 0. (20)
Integrating and using that W is bounded from below, we obtain the first two claims. From u(t) = ẋ(t) + β∇f (x(t)), we deduce that u(t) 2 ≤ 2( ẋ(t) 2 + β 2 ∇f (x(t)) 2 ). After integration, we get the last claim

+∞ t1 1 t u(t) 2 dt ≤ 2 +∞ t1 1 t ẋ(t) 2 dt + +∞ t1 β 2 t ∇f (x(t)) 2 dt ≤ 2 +∞ t1 1 t ẋ(t) 2 dt + β 2 t 1 +∞ t1 ∇f (x(t)) 2 dt < +∞.

Claim (iii) and (iv). Define

h : t ∈ [t 1 , +∞[ → 1 2 u(t) -z 2 for arbitrary z ∈ H. We then have ḧ(t) + α t ḣ(t) = u(t) 2 + u(t) -z, ü(t) + α t u(t) = u(t) 2 + u(t) -z, ẍ(t) + β∇ 2 f (x(t)) ẋ(t) + α t ẋ(t) + αβ t ∇f (x(t)) = u(t) 2 -u(t) -z, g(t) + 1 - αβ t ∇f (x(t)) = u(t) 2 -1 - αβ t x(t) -z, ∇f (x(t)) -u(t) -z, g(t) -β 1 - αβ t t t1 ∇f (x(s))ds, ∇f (x(t)) = u(t) 2 -1 - αβ t x(t) -z, ∇f (x(t)) -u(t) -z, g(t) -β 1 - αβ t İ(t),
where

I(t) := 1 2 t t1 ∇f (x(s))ds 2 .
From the convexity of f and Cauchy-Schwarz inequality we get

ḧ(t)+ α t ḣ(t)+ 1 - αβ t (f (x(t)) -f (z))+β 1 - αβ t İ(t) ≤ u(t) 2 + u(t) -z g(t) .
Inserting W (t) into this expression, we get,

ḧ(t) + α t ḣ(t) + 1 - αβ t W (t) + f -f (z) + β 1 - αβ t İ(t) ≤ 3 2 - αβ 2t u(t) 2 + u(t) -z g(t) -1 - αβ t +∞ t u(τ ), g(τ ) dτ. ( 21
)
According to ( 20) and ( 19) W (•) is nonincreasing and bounded from below. Therefore, it converges to some W ∞ ∈ R as t → +∞. Since u is bounded, and g is integrable, we have that

τ → u(τ ), g(τ ) is integrable on [t 0 , +∞[. Therefore lim t→+∞ +∞ t u(τ ), g(τ ) dτ = 0.
By definition of W (t), this implies that, as t → +∞

1 2 u(t) 2 + (f (x(t)) -f ) → W ∞ .
If W ∞ = 0, since the two terms that enter the above expression (potential energy and kinetic energy) are nonnegative, we obtain that each of them tends to zero as t → +∞. This gives the claims (iii) and (iv). To prove that W ∞ = 0, we argue by contradiction, and show that assuming W ∞ > 0 leads to a contradiction. Since W (•) is nonincreasing, we then have

W (t) ≥ W ∞ > 0. Take z ∈ H such that f (z) < f + 1 2 W ∞ . Then W (t) + f -f (z) > W ∞ - 1 2 W ∞ = 1 2 W ∞ .
Returning to [START_REF] Aujol | Optimal convergence rates for Nesterov acceleration[END_REF] we deduce that, for t ≥ t 1

ḧ(t) + α t ḣ(t) + 1 2 1 - αβ t W ∞ + β 1 - αβ t İ(t) ≤ 3 2 - αβ 2t u(t) 2 + u(t) -z g(t) -1 - αβ t +∞ t u(τ ), g(τ ) dτ. ( 22
)
Since t > 2αβ, we have 1 -αβ t > 1 2 . Therefore, after rearranging the terms in ( 22), we obtain

1 4 W∞ ≤ 3 2 u(t) 2 + u(t) -z g(t) + sup t≥t 1 u(t) +∞ t g(s) ds - 1 t α d dt (t α ḣ(t)) -β 1 - αβ t İ(t).
Multiplying both sides by 1 t , and integrating between t 1 and τ

> t 1 , 1 4 W ∞ log τ t 1 ≤ 3 2 τ t1 1 t u(t) 2 dt + τ t1 g(t) u(t) -z t dt + sup t≥t1 u(t) τ t1 1 t +∞ t g(s) ds dt - τ t1 1 t α+1 d dt (t α ḣ(t))dt -β τ t1 1 t - αβ t 2 İ(t)dt. ( 23
)
Throughout the rest of the proof, we will use the inequality

u(t) -z ≤ u(t 1 ) -z + t t1 u(s) ds ≤ u(t 1 ) -z + t sup s≥t1 u(s) .
Let us examine successively the different terms which enter the second member of [START_REF] Bot | Tikhonov regularization of a second order dynamical system with Hessian damping[END_REF]. The first term is bounded according to claim (ii). The second term is also bounded since

τ t 1 g(t) u(t) -z t dt ≤ u(t1) -z t1 + sup t≥t 1 u(t) +∞ t 1 ( e(t) + β ė(t) ) dt < +∞.
The third term can be handled by integration by parts,

τ t 1 1 t +∞ t g(s) ds dt = log τ ∞ τ g(s) ds -log t1 +∞ t 1 g(s) ds+ τ t 1 g(t) log t dt.
For the fourth term, set

K(τ ) = - τ t1 1 t α+1 d dt (t α ḣ(t)
)dt and integrate by parts twice to get,

K(τ ) = - 1 t ḣ(t) τ t 1 -(α + 1) τ t 1 1 t 2 ḣ(t)dt = - 1 t ḣ(t) τ t 1 - (1 + α) τ 2 h(τ ) + (1 + α) t 2 1 h(t1) -2(1 + α) τ t 1 h(t) t 3 dt ≤ - 1 t ḣ(t) τ t 1 + (1 + α) t 2 1 h(t1) ≤ 1 t1 | u(t1), u(t1) -z | + 1 τ | u(τ ), u(τ ) -z | + (1 + α) t 2 1 h(t1) ≤ C + sup t≥t 1 u(t) u(t1) -z t1 + sup t≥t 1 u(t) < +∞.
For the last term, we infer from Lemma A.2 that

sup τ >t1 -β τ t1 1 t - αβ t 2 İ(t)dt < +∞.
Overall, we have shown that there exists a constant C > 0 such that (23) reads

1 4 W ∞ log τ t 1 ≤ C + sup t≥t1 u(t) log τ ∞ τ g(s) ds + τ t1 g(t) log tdt .
Observe that lim τ →+∞ ∞ τ g(s) ds = 0 since g is integrable. Then, divide the last inequality by log τ t1 and let τ → ∞. According to Lemma A.5, we get that W ∞ ≤ 0. Thus W ∞ = 0, hence the contradiction. The last statement, lim t→+∞ ẋ(t) = 0, is obtained by following an argument similar to the one above, which now uses the perturbed version of the classical energy function, namely

W 0 (t) := 1 2 ẋ(t) 2 + (f (x(t)) -f ) - +∞ t ẋ(τ ), g(τ ) dτ.
We do not detail this proof for the sake of brevity. Then, according to ∇f (x(t)) = 3.1.2. Fast convergence rates. We now move on to showing fast convergence of the objective. For this, we will need to strengthen the integrability assumption on the errors. We will denote in this section the two functions

w(t) := 1 - β t and δ(t) := t 2 w(t). ( 24
) Theorem 3.2. Let f ∈ C 2 (H), e ∈ C
(i) f (x(t)) -inf H f = o 1 t 2 as t → +∞. (ii) +∞ t0 t 2 ∇f (x(t)) 2 dt < +∞. (iii) +∞ t0 t f (x(t)) -min H f dt < +∞.
(iv) for any x ∈ S,

+∞ t0 t ∇f (x(t)), x(t) -x dt < +∞. (v) +∞ t0 t ẋ(t) 2 dt < +∞. (vi) ẋ(t) = o t -1 as t → +∞.
Proof. Following an argument similar to that of Theorem 3.1, since our analysis is asymptotic, there is no restriction in assuming that t 0 > β(α-2) α-3 . This gives

w(t) ≥ 1 α-2 > 0 for all t ≥ t 0 . Define, v(t) := (α -1)(x(t) -x ) + t( ẋ(t) + β∇f (x(t))).
Take T > t 0 , and define for all

t 0 ≤ t ≤ T E(t) := δ(t)(f (x(t)) -f (x )) + 1 2 v(t) 2 - T t τ v(τ ), g(τ ) dτ.
This is a well-defined differentiable function. Taking its derivative in time yields

Ė(t) = δ(t)(f (x(t)) -f (x )) + δ(t) ∇f (x(t)), ẋ(t) + v(t), v(t) + tg(t) .
From (ISEHD-Pert), we have,

v(t) = α ẋ(t) + β∇f (x(t)) + t ẍ(t) + β∇ 2 f (x(t)) ẋ(t) = α ẋ(t) + β∇f (x(t)) + t - α t ẋ(t) -∇f (x(t)) -g(t) = -t 1 - β t ∇f (x(t)) -tg(t).
Let us inject this expression into the scalar product v(t), v(t) + tg(t) . After developing and rearranging, and taking into account the definition of w and δ (see ( 24)), we obtain

Ė(t) = δ(t)(f (x(t))-f (x ))-(α-1)tw(t) ∇f (x(t)), x(t) -x -βδ(t) ∇f (x(t)) 2 . ( 25
) Convexity of f then yields Ė(t) + βδ(t) ∇f (x(t)) 2 + (α -1)tw(t) -δ(t) (f (x(t)) -f (x )) ≤ 0. ( 26
)
By assumption on the parameters, we have for any

t ≥ t 0 > β(α-2) α-3 , (α -1)tw(t) -δ(t) = t (α -3) - β t (α -2) ≥ ct (27)
with c = (α -3) -β t0 (α -2) > 0. Thefore, [START_REF] Chambolle | On the convergence of the iterates of the "fast iterative shrinkage thresholding algorithm[END_REF] implies that E is non-increasing. In turn, by definition of E

δ(t) (f (x(t)) -f (x )) + 1 2 v(t) 2 ≤ C - t t0 τ v(τ ), g(τ ) dτ (28)
with

C = δ(t 0 )(f (x 0 ) -f (x )) + 1 2 v(t 0 ) 2 .
The above argument is valid for all t 0 ≤ t ≤ T and arbitrary T , therefore for all t ≥ t 0 . Neglecting the nonnegative term δ(t) (f (x(t)) -f (x )), we infer from (28)

1 2 v(t) 2 ≤ C + t t0 v(τ ) (τ g(τ ) ) dτ.
Lemma A.3 (Gronwall lemma) then gives

v(t) ≤ (2C) 1/2 + t t0 τ ( e(τ ) + β ė(τ ) ) dτ, (29)
and thus sup

t≥t0 v(t) < +∞. ( 30
)
By reinjecting this inequality into (28) we obtain

t 2 α -2 (f (x(t)) -f (x )) ≤ δ(t) (f (x(t)) -f (x )) ≤ C + sup t≥t0 v(t) t t0 (τ e(τ ) + βτ ė(τ ) ) dτ < +∞ (31) hence proving f (x(t)) -inf H f = O 1 t 2
as t → +∞. We will see a little later how to refine this estimate and go from a capital O to a small o to prove the statement (i). Then, integrating [START_REF] Chambolle | On the convergence of the iterates of the "fast iterative shrinkage thresholding algorithm[END_REF], and using the fact that E(t) is bounded from below by [START_REF] Nesterov | A method of solving a convex programming problem with convergence rate O(1/k 2 )[END_REF] and the assumptions on the errors, we get

β +∞ t0 t 2 w(t) ∇f (x(t)) 2 dt ≤ C, and c +∞ t0 t(f (x(t)) -f (x ))dt ≤ +∞ t0 (α -1)tw(t) + δ(t) (f (x(t)) -f (x ))dt ≤ C,
for some constant C > 0. This shows the integral estimates (ii) and (iii).

Let us turn to statement (iv). We embark from [START_REF] Castera | An inertial newton algorithm for deep learning[END_REF] to write, for some ρ ∈]0, 1[ to be chosen shortly,

Ė(t) = δ(t)(f (x(t)) -f (x )) -(1 -ρ)(α -1)tw(t) ∇f (x(t)), x(t) -x -ρ(α -1)tw(t) ∇f (x(t)), x(t) -x -βδ(t) ∇f (x(t)) 2 ≤ -(1 -ρ)(α -1)tw(t) -δ(t) (f (x(t)) -f (x )) -ρ(α -1)tw(t) ∇f (x(t)), x(t) -x -βδ(t) ∇f (x(t)) 2 .
To conclude, it remains to check that (1

-ρ)(α -1)tw(t) -δ(t) is non-negative. Since t 0 > β(α-2)
α-3 , we deduce by a continuity argument the existence of some ε > 0 such that t 0 > β(α-2-ε) α-3-ε . Then, take ρ = ε/(α-1) ∈]0, 1[. In view of the assumption on the parameters, we have

(1 -ρ)(α -1)tw(t) -δ(t) = (α -1 -ε)tw(t) -δ(t) = t((α -3 -ε)w(t) -t ẇ(t)) = t (α -3 -ε) - (α -2 -ε)β t ≥ t 0 (α -3 -ε) - (α -2 -ε)β t 0 ≥ 0.
For claim (v), we multiply (ISEHD-Pert) by t 2 ẋ(t) to get

t 2 ẍ(t), ẋ(t) + αt ẋ(t) 2 + t 2 β ẋ(t), ∇ 2 f (x(t)) ẋ(t) + t 2 ∇f (x(t)), ẋ(t) + t 2 g(t), ẋ(t) = 0.
With the chain rule, Cauchy-Schwarz inequality and convexity of f , we obtain

1 2 t 2 d dt ẋ(t) 2 + αt ẋ(t) 2 + t 2 d dt (f (x(t)) -f ) ≤ tg(t) t ẋ(t) . (32)
Integrating by parts on [t 0 , t] we get,

t 2 2 ẋ(t) 2 + (α -1) t t 0 s ẋ(s) 2 ds ≤ C0 + 2 t t 0 s(f (x(s)) -f )ds + t t 0 sg(s) s ẋ(s) ds (33)
for some non-negative constant C 0 , where we have used claim (i) of Theorem

) 34
Using this in [START_REF] Polyak | Introduction to Optimization[END_REF], we also get that

+∞ t0 t ẋ(t) 2 dt < +∞. ( 35
)
We finally turn to statement (vi). We embark from [START_REF]Weak convergence of the sequence of successive approximations for nonexpansive mappings[END_REF], use [START_REF] Schmidt | Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization[END_REF], and integrate on [s, t] to see that

t 2 1 2 ẋ(t) 2 + (f (x(t)) -f ) -s 2 1 2 ẋ(s) 2 + (f (x(s)) -f ) + (α -1) t s τ ẋ(τ ) 2 dτ -2 t s τ (f (x(τ )) -f )dτ -C t s τ g(τ ) dτ ≤ 0,
where C = sup t≥t0 t ẋ(t) . This means that the function

G(t) = t 2 1 2 ẋ(t) 2 + (f (x(t)) -f ) + (α -1) t t0 τ ẋ(τ ) 2 dτ -2 t t0 τ (f (x(τ )) -f )dτ -C t t0 τ g(τ ) dτ
is non-increasing on [t 0 , +∞[. Since it is bounded from below by assumption on the errors and claim (iii), lim t→+∞ G(t) exists. This together with assertions (iii) and (v) shows that the limit

0 ≤ L := lim t→+∞ t 2 1 2 ẋ(t) 2 + (f (x(t)) -f )
exists. Suppose that L > 0. Then, there exists s ≥ t 0 such that

+∞ t0 t 2 ẋ(t) 2 + t(f (x(t)) -f ) dt ≥ +∞ s t 2 1 2 ẋ(t) 2 + (f (x(t)) -f ) t -1 dt ≥ +∞ s L 2t dt = +∞,
leading to a contradiction with claims (iii) and (v). This proves (vi) and completes the proof of (i) with small o instead of capital O. Remark 3.3. The choice of the viscous damping parameter α is important for optimality of the convergence rates obtained. For the subcritical case α ≤ 3, β = 0 and e ≡ 0, it has been shown by [START_REF] Apidopoulos | Convergence rate of inertial Forward-Backward algorithm beyond Nesterov's rule[END_REF] and [START_REF] Attouch | Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3[END_REF] that the convergence rate of the objective values is O t -2α 3 , and these rates are optimal, that is, they can be attained, or approached arbitrarily closely. For α ≥ 3, the optimal rate O t -2 is achieved for the function f (x) = x r with r → +∞ [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF], and for α < 3, the optimal rate O t - Proof. Keeping in mind that the goal is to apply Opial's Lemma (see Lemma A.1), we will now show that lim t→+∞ x(t)-x exists. Following an argument similar to that of Theorem 3.1 and 3.2, since our analysis is asymptotic, there is no restriction in assuming that t 0 > β(α-2) α-3 . Hence the existence of ε > 0 such that t 0 ≥ β(α-2-ε) α-3-ε for some ε ∈]0, α-3[. Recall the Lyapunov function E from the proof of Theorem 3.2, and define its generalized version

E ε (t) := (δ(t) + εβt) (f (x(t)) -f (x )) + 1 2 v ε (t) 2 + ε(α -1 -ε) 2 x(t) -x 2 - T t τ v ε (τ ), g(τ ) dτ, (36)
where

v ε (t) = (α -1 -ε)(x(t) -x ) + t( ẋ(t) + β∇f (x(t))).
One can check, arguing as for E, that for t 0 ≤ t ≤ T

Ėε (t) = δ(t) + εβ (f (x(t)) -f (x )) -(α -1 -ε)tw(t) ∇f (x(t)), x(t) -x -βδ(t) ∇f (x(t)) 2 -εt ẋ(t) 2 .
Convexity of f then entails

Ėε(t) + (α -1 -ε)tw(t) -δ(t) (f (x(t)) -f (x )) + βδ(t) ∇f (x(t)) 2 + εt ẋ(t) 2 ≤ 0.
The assumption on the parameters gives

(α -1 -ε)tw(t) -δ(t) = t((α -3 -ε)w(t) -t ẇ(t)) ≥ t0 (α -3 -ε) - (α -2 -ε)β t0 ≥ 0.
Thus, ignoring the non-negative terms in this inequality entails that E ε (•) is a decreasing function on [t 0 , T [. According to the boundedness of E ε (•), an argument similar to that developed in Theorem 5 gives that

sup t≥t0 v ε (t) < +∞, (37)
with a bound which is independent of ε and T . Comparing with [START_REF] Nesterov | A method of solving a convex programming problem with convergence rate O(1/k 2 )[END_REF] gives

sup t≥t0 x(t) < +∞, ( 38
)
where we use that the previous argument is valid for arbitrary t ≤ T , hence for all t ≥ t 0 . As a consequence, the energy functions E(•) and E ε (•) with T = +∞ are well-defined on [t 0 , +∞[, and are then Lyapunov functions for the dynamical system (ISEHD-Pert). Both E(t) and E ε (t) thus have limits as t → +∞, and so does their difference

E ε (t) -E(t) = εβt(f (x(t)) -f (x )) - ε(α -1) 2 x(t) -x 2 -εt ẋ(t), x(t) -x -εt ∇f (x(t)), x(t) -x + ε +∞ t τ x(τ ) -x , g(τ ) dτ.
By Theorem 3.2(i), the first term converges to 0 as t → +∞. By the integrability assumptions on the errors and boundedness of x(t) (see (38)), the last term also converges to 0 as t → +∞. We have then shown that the limit as t goes to infinity of

p(t) := α -1 2 x(t) -x 2 + t ẋ(t), x(t) -x + t ∇f (x(t)), x(t) -x exists. Set q(t) := α -1 2 x(t) -x 2 + (α -1) t t0
∇f (x(s)), x(s) -x ds.

We obviously have

p(t) = q(t) + t α -1 q(t) -(α -1) t t0
∇f (x(s)), x(s) -x ds.

By Theorem 3.2(iv), and since ∇f (x(s)), x(s) -x is non-negative, we have that

lim t→+∞ t t0 ∇f (x(s)), x(s) -x ds (39)
exists. Overall, we have shown that

lim t→+∞ q(t) + t α -1 q(t)
exists. Since α > 1, it follows from [19, Lemma 7.2] that lim t→+∞ q(t) exists.

Using again (39), we deduce that lim t→+∞ x(t) -x exists for any x ∈ S. From claim (i) of Theorem 3.2 (see also Lemma 3.1(iv)), it follows that for any sequence (x(t n )) n∈N which converges weakly to, say, x, we have

f (x) ≤ lim inf n→+∞ f (x(t n )) = lim t→+∞ f (x(t)) = f , i.e., x ∈ S.
Consequently, all the conditions of Lemma A.1 are satisfied, hence the weak convergence of the trajectories.

Remark 3.4. In [10, Theorem 2.2], the authors proved weak convergence of the trajectory for the perturbation-free system (ISEHD). Theorem 3.3 shows that weak convergence is preserved under perturbations provided that they verify reasonable integrability results. Again, the proof necessitates new estimates and bounds to cope with the presence of errors.

Remark 3.5. The condition α > 3 is known to play an important role to show that each trajectory converges weakly to a minimizer. The case α = 3, which corresponds to Nesterov's historical algorithm when β = 0 and e ≡ 0, is critical. In fact, even for those inertial systems with α = 3, convergence of the trajectories remains an open problem (except in one dimension where it holds as shown in [START_REF] Attouch | Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3[END_REF]).

3.2. Implicit Hessian damping. We now turn to the second-order ODE (ISIHD-Pert) where f ∈ C 1 (H), e ∈ C([t 0 , +∞[) and β(t) = γ + β t , γ, β ≥ 0. Let us denote for brevity f := inf H f . Given x ∈ S, we consider the function

E(t) = a(t) f (x(t) + β(t) ẋ(t)) -f + 1 2 b(t)(x(t) -x ) + c(t) ẋ(t) 2 + d(t) 2 x(t) -x 2 - +∞ t c(τ ) b(τ )(x(τ ) -x ) + c(τ ) ẋ(τ ), e(τ ) dτ - +∞ t a(τ )β(τ ) ∇f (x(τ ) + β(τ ) ẋ(τ )) , e(τ ) dτ (40)
parametrized by some functions a(t), b(t), c(t) and d(t) to be specified later.

3.2.1. Lyapunov function. We first show that for proper choices of (a(t), b(t), c(t), d(t)) as a function of the problem parameters (α, γ, β), E can serve as a Lyapunov function for (ISIHD-Pert). We will denote for short α(t) = α t .

Lemma 3.1. Assume that f ∈ C 1 (H), e ∈ C([t 0 , +∞[), and

                   ȧ(t) -b(t)c(t) ≤ 0, -a(t)β(t) ≤ 0, -a(t)α(t)β(t) + a(t) β(t) + a(t) -c(t) 2 + b(t)c(t)β(t) = 0, ḃ(t)b(t) + ḋ(t) 2 ≤ 0, ḃ(t)c(t) + b(t)(b(t) + ċ(t) -c(t)α(t)) + d(t) = 0, c(t)(b(t) + ċ(t) -c(t)α(t)) ≤ 0. ( 41
)
Then

Ė(t) ≤ ( ȧ(t) -b(t)c(t))(f (x(t) + β(t) ẋ(t)) -f ) -a(t)β(t) ∇f (x(t) + β(t) ẋ(t)) 2 + ḃ(t)b(t) + ḋ(t) 2 x(t) -x 2 + c(t)(b(t) + ċ(t) -c(t)α(t)) ẋ(t) 2 ≤ 0. ( 42
)
Proof. Recall that since f ∈ C 1 (H) and e ∈ C([t 0 , +∞[), (ISIHD-Pert) has a unique classical global solution x; see paragraph after [START_REF] Attouch | Strong solutions for parabolic variational inequalities[END_REF]. We now proceed as in the proof of Theorems 3.1 and 3.2, and first consider the function E T where the integrals involving the error terms are calculated on [t, T ], T < +∞. This shows that E is well-posed under our assumptions. We can then compute the time derivative of E and use the chain rule to get

Ė(t) = ȧ(t) f (x(t) + β(t) ẋ(t)) -f + a(t) ∇f (x(t) + β(t) ẋ(t)) , ẋ(t) + β(t) ẋ(t) + β(t)ẍ(t) + (b(t) + ċ(t)) ẋ(t) + c(t)ẍ(t) + ḃ(t)(x(t) -x ), b(t)(x(t) -x ) + c(t) ẋ(t) + ḋ(t) 2 x(t) -x 2 + d(t) ẋ(t), x(t) -x + c(t) b(t)(x(t) -x ) + c(t) ẋ(t), e(t) + a(t)β(t) ∇f (x(t) + β(t) ẋ(t)) , e(t) (43)
Using (ISIHD-Pert) in the second term of (43), we get

a(t) ∇f (x(t) + β(t) ẋ(t)) , ẋ(t) + β(t) ẋ(t) + β(t)ẍ(t) = a(t) ∇f (x(t) + β(t) ẋ(t)) , 1 + β(t) -α(t)β(t) ẋ(t) -β(t)∇f (x(t) + β(t) ẋ(t)) -β(t)e(t) = -a(t)β(t) ∇f (x(t) + β(t) ẋ(t)) 2 + 1 + β(t) -α(t)β(t) a(t) ∇f (x(t) + β(t) ẋ(t)) , ẋ(t) -β(t)a(t) ∇f (x(t) + β(t) ẋ(t)) , e(t) . ( 44
)
We expand the third term in (43) as

(b(t) + ċ(t)) ẋ(t) + c(t)ẍ(t) + ḃ(t)(x(t) -x ), b(t)(x(t) -x ) + c(t) ẋ(t) = (b(t) + ċ(t) -c(t)α(t)) ẋ(t) -c(t)∇f (x(t) + β(t) ẋ(t)) , b(t)(x(t) -x ) + c(t) ẋ(t) + -c(t)e(t) + ḃ(t)(x(t) -x ), b(t)(x(t) -x ) + c(t) ẋ(t) = -c(t) ∇f (x(t) + β(t) ẋ(t)), b(t)(x(t) -x ) + c(t) ẋ(t) + ḃ(t)(x(t) -x ), b(t)(x(t) -x ) + c(t) ẋ(t) +c(t)(b(t) + ċ(t) -c(t)α(t)) ẋ(t) 2 -c(t) e(t), b(t)(x(t) -x ) + c(t) ẋ(t) . ( 45
)
Plugging ( 44) and ( 45) into (43), we get,

Ė(t) = ȧ(t) f (x(t) + β(t) ẋ(t)) -f -a(t)β(t) ∇f (x(t) + β(t) ẋ(t)) 2 + c(t)(b(t) + ċ(t) -c(t)α(t)) ẋ(t) 2 + ḃ(t)b(t) + ḋ(t) 2 x(t) -x 2 + b(t) 2 + b(t) ċ(t) + ḃ(t)c(t) -b(t)c(t)α(t) + d(t) ẋ(t), x(t) -x + -a(t)α(t)β(t) + a(t) β(t) + a(t) -c(t) 2 ∇f (x(t) + β(t) ẋ(t)) , ẋ(t) -b(t)c(t) ∇f (x(t) + β(t) ẋ(t)) , x(t) -x . (46)
Since,

∇f (x(t) + β(t) ẋ(t)), x(t) -x = ∇f (x(t) + β(t) ẋ(t)), x(t) + β(t) ẋ(t) -x -∇f (x(t) + β(t) ẋ(t)), β(t) ẋ(t)
and using the convex (sub)differential inequality on f , we can write

-b(t)c(t) ∇f (x(t) + β(t) ẋ(t), x(t) -x ≤ -b(t)c(t)(f (x(t) + β(t) ẋ(t)) -f ) + b(t)c(t)β(t) ∇f (x(t) + β(t) ẋ(t)), ẋ (t)
and we arrive at

Ė(t) ≤ ( ȧ(t) -b(t)c(t))(f (x(t) + β(t) ẋ(t)) -f ) -a(t
+ (b(t) 2 + b(t) ċ(t) + ḃ(t)c(t) -b(t)c(t)α(t) + d(t)) ẋ(t), x(t) -x + c(t)(b(t) + ċ(t) -c(t)α(t)) ẋ(t) 2 .
Thus, conditions (41) guarantee that Ė(t) ≤ 0, in particular they imply (42).

Following the discussion of [START_REF] Alecsa | An extension of the second order dynamical system that models Nesterov's convex gradient method[END_REF]Remark 11], in the rest of the section, we take

β(t) = γ + β t , γ, β ≥ 0, b(t) ≡ b ∈]0, α -1], α > 1, c(t) = t and d(t) ≡ b(α -1 -b). (47)
Such a choice is reminescent of that in [START_REF] Su | A differential equation for modeling nesterov's accelerated gradient method: Theory and insights[END_REF]. The choices of d(t) and b(t) comply with the fourth, fifth and sixth conditions of (41). To satisfy the third condition, one has to take

a(t) = t 2 1 + (α -b)γt -β(α + 1 -b) t 2 -αγt -β(α + 1) . (48)
Clearly, for t large enough, one has a(t) ≥ t 2 and β(t) ≥ γ/2. Thus, the second condition is in force. One can also verify that the first inequality is satisfied for t large enough provided that b > 2 (and thus α > 3) when γ > 0, and b = 2 (with α = 3) when γ = 0.

3.2.2. Fast convergence rates. We start with the following boundedness properties.

Lemma 3.2. Let E(t) = a(t) f (x(t) + β(t) ẋ(t)) -f + 1 2 b(x(t) -x ) + t ẋ(t) 2 + b(α -1 -b) 2 x(t) -x 2 .
Choose the parameters according to (47)-( 48

E(t) ≤ E(t 0 ) + t t 0 τ b(x(τ ) -x ) + τ ẋ(τ ) + a(τ )β(τ )∇f (x(τ ) + β(τ ) ẋ(τ )) , e(τ ) dτ = E(t 0 ) + t t 0 τ b(x(τ ) -x ) + τ ẋ(τ ) + a(τ )β(τ ) ∇f (x(τ ) + β(τ ) ẋ(τ )) -∇f (x ) , e(τ ) dτ ≤ E(t 0 ) + t t 0 τ b(x(τ ) -x ) + τ ẋ(τ ) + a(τ )β(τ ) ∇f (x(τ ) + β(τ ) ẋ(τ )) -∇f (x ) e(τ ) dτ ≤ E(t 0 ) + t t 0 τ b(x(τ ) -x ) + τ ẋ(τ ) + a(τ )β(τ )L x(τ ) -x + β(τ ) ẋ(τ ) e(τ ) dτ ≤ E(t 0 ) + t t 0 τ b(x(τ ) -x ) + τ ẋ(τ ) + La(τ )β(τ ) x(τ ) -x + La(τ )β(τ ) 2 ẋ(τ ) e(τ ) dτ. (50)
Denote d = b(α -1 -b). We have d > 0. Moreover, a(t) > 0 for t ≥ t 0 . One can then drop the first term in E(t), and (50) becomes, for any t ≥ t 0 ,

1 2 b(x(t) -x ) + t ẋ(t) 2 + d 2 x(t) -x 2 ≤ E(t 0 ) + t t 0 τ b(x(τ ) -x ) + τ ẋ(τ ) + √ d La(τ )β(τ ) √ d x(τ ) -x + τ La(τ )β(τ ) 2 t 0 ẋ(τ ) e(τ ) dτ ≤ E(t 0 ) + t t 0 b(x(τ ) -x ) + τ ẋ(τ ) + √ d x(τ ) -x + τ ẋ(τ ) max 1, d -1/2 , t -1 0 m(τ ) e(τ ) dτ ≤ E(t 0 ) + t t 0 2 b(x(τ ) -x ) + τ ẋ(τ ) + (b + √ d) x(τ ) -x max 1, d -1/2 , t -1 0 m(τ ) e(τ ) dτ ≤ E(t 0 ) + t t 0 b(x(τ ) -x ) + τ ẋ(τ ) + √ d x(τ ) -x Cm(τ ) e(τ ) dτ, for some constant C ≥ max 1, d -1/2 , t -1 0 max 2, 1 + b α-1-b . Now, Jensen's inequality yields 1 4 b(x(t) -x ) + t ẋ(t) + √ d x(t) -x 2 ≤ 1 2 b(x(t) -x ) + t ẋ(t) 2 + d 2 x(t) -x 2 ≤ E(t 0 ) + t t0 b(x(τ ) -x ) + τ ẋ(τ ) + √ d x(τ ) -x C|m(τ )| e(τ ) dτ.
Using the Gronwall Lemma A. Clearly the condition m(•)e(•) ∈ L 1 (t 0 , +∞; H) is equivalent to t 2 e(t) ∈ L 1 (t 0 , +∞; H).

From Lemma 3.2, we obtain the following convergence rates and integral estimates.

Theorem 3.4. Under the assumptions of Lemma 3.2, the following holds:

(i) f (x(t) + β(t) ẋ(t)) -min H f = O 1 t 2 as t → +∞; (ii) ẋ(t) = O 1 t as t → +∞; (iii) +∞ t0 t f (x(t)) -min H f dt < +∞; (iv) +∞ t0 t 2 ∇f (x(t) + β(t) ẋ(t)) 2 dt < +∞; (v) +∞ t0 t ẋ(t) 2 dt < +∞.
Proof. Claim (ii) follows from Lemma 3.2. Discarding the non-negative terms in E(t), Lemma 3.2 together with the fact that a(t) ≥ t 2 for t large enough, also gives

f (x(t) + β(t) ẋ(t)) -min H f = O 1 t 2 .
To show the remaining integral estimates, consider the function E(•) in (40) with the choices (47)-(48) of c(t), d(t), b(t) and a(t), where b ∈]2, α -1[. We first argue similarly to [START_REF] Alecsa | An extension of the second order dynamical system that models Nesterov's convex gradient method[END_REF] to show that for t large enough, we have ȧ(t) -bt ≤ -(α-3)t 2 , since α > 3 and b > 2. In addition, for (a possibly different) t large enough, it is straightforward to see that a(t) (γ + β/t) ≥ t 2 γ/2. With these bounds, (42) reads, for t large enough,

Ė(t) ≤ - (α -3)t 2 (f (x(t)+β(t) ẋ(t))-f )-t 2 γ/2 ∇f (x(t) + β(t) ẋ(t)) 2 -t(α-1-b) ẋ(t) 2 . ( 52
)
Integrating (52), and using that E is bounded thanks to Lemma 3.2, we get statements (iv)-(v) and

+∞ t0 t f (x(t) + β(t) ẋ(t)) -min H f dt < +∞. ( 53
)
Let β 0 = γ + |β| /t 0 . By the gradient descent lemma,

f (x(t)) -f (x(t) + β(t) ẋ(t)) ≤ -β(t) ∇f (x(t) + β(t) ẋ(t)), ẋ(t) + L 2 β(t) 2 ẋ(t) 2 (54) ≤ β 0 ∇f (x(t) + β(t) ẋ(t)) ẋ(t) + L 2 β 2 0 ẋ(t) 2 .
By Cauchy-Schwarz inequality, we have

+∞ t0 t f (x(t)) -min H f dt ≤ +∞ t0 t f (x(t) + β(t) ẋ(t)) -min H f dt + β 0 +∞ t0 t ∇f (x(t) + β(t) ẋ(t)) 2 dt 1/2 +∞ t0 t ẋ(t) 2 dt 1/2 + Lβ 2 0 2 +∞ t0 t ẋ(t) 2 dt.
In view of (53) and claims (iv)-(v), statement (iii) follows.

3.2.3. Convergence of the trajectories. We now turn to showing weak convergence of the trajectories to a minimizer.

Theorem 3.5. Suppose that the assumptions of Lemma 3.2 hold. Then x(t) converges weakly to a minimizer of f .

Proof. As in the explicit case, we invoke Opial's Lemma A.1. Recall that the trajectory x(•) is bounded by Lemma 3.2. Therefore, for any sequence (x(t n )) n∈N which converges weakly to, say, x, as t n → +∞, Theorem 3.4(i)-(ii) entails that

f (x) ≤ lim inf n→+∞ f (x(t n ) + β(t n ) ẋ(t n )) = lim t→+∞ f (x(t) + β(t) ẋ(t)) = f ,
i.e., each weak cluster point of x(t n ) belongs to S. To get weak convergence of the trajectory, it remains to show that lim t→+∞ x(t) -x exists. Let h : t ∈ [t 0 , +∞[ → 1 2 x(t) -x 2 . Under the assumptions on f and e, x is the unique classical global solution to (ISIHD-Pert), i.e., x ∈ C 2 ([t 0 , +∞[). Thus so is h and ḣ(t) = ẋ(t), x(t) -x and ḧ(t) = ẍ(t), x(t) -x + ẋ(t) 2 .

From (ISIHD-Pert), we obtain

ḧ(t) + α t ḣ(t) = ẍ(t) + α t ẋ(t), x(t) -x + ẋ(t) 2 = -∇f (x(t) + β(t) ẋ(t)) + e(t), x(t) -x + ẋ(t) 2 = -∇f (x(t) + β(t) ẋ(t)), x(t) + β(t) ẋ(t) -x -e(t), x(t) -x + ẋ(t) 2 + β(t) ∇f (x(t) + β(t) ẋ(t)), ẋ(t) .
Convexity of f implies,

-∇f (x(t) + β(t) ẋ(t)), x(t) + β(t) ẋ(t) -x ≤ f -f (x(t) + β(t) ẋ(t)) ≤ 0,
and thus,

ḧ(t) + α t ḣ(t) ≤ x(t) -x e(t) + ẋ(t) 2 + β 0 ∇f (x(t) + β(t) ẋ(t)) ẋ(t) ≤ C e(t) + ẋ(t) 2 + β 0 ∇f (x(t) + β(t) ẋ(t)) ẋ(t)
where C = sup t≥t0 x(t) -x < +∞ thanks to Lemma 3.2, and we denoted β 0 = 1 + |β|/t 0 . Multiplying both sides by t, we arrive at

t ḧ(t) + α ḣ(t) ≤ Ct e(t) + t ẋ(t) 2 + β 0 √ t 0 (t ∇f (x(t) + β(t) ẋ(t)) )( √ t ẋ(t) ).
The right-hand side of this inequality belongs to L 1 (t 0 , +∞; R) by assumption on the error, and using the Cauchy-Schwarz inequality and Theorem 3.4(iv)-(v) for the last two terms. Since h ∈ C 2 ([t 0 , +∞[), it then follows from Lemma A.6 that lim t→+∞ x(t) -x exists. We have now shown that all conditions of Lemma A.1 are satisfied, hence the weak convergence of the trajectories.

Remark 3.7. For the unperturbed case, similar rates to ours in Theorem 3.4 and weak convergence of the trajectory were proved in [START_REF] Alecsa | An extension of the second order dynamical system that models Nesterov's convex gradient method[END_REF]. Again, handling errors necessitates new estimates and bounds, for instance those established in Lemma 3.2.

3.3. Discussion. We now discuss the main differences between the two systems in terms of their stability to perturbations and the corresponding assumptions.

Recall from Remark 3.6, the integrability assumption m(•)e(•) ∈ L 1 (t 0 , +∞; H) required to ensure stability for system (ISIHD-Pert) is equivalent to ensuring that the second-order moment of the error e(•) is finite. One may wonder whether this is more stringent than the integrability assumptions for the explicit Hessian system (ISEHD-Pert) involving the control of the first-order moments of the error and its derivative (see Section 3. 1). The answer is clearly affirmative in the scalar case with a simple integration by parts argument. Indeed, supposing without loss of generality that e(•) is a non-increasing and non-negative function, one has

+∞ t 0 t| ė(t)|dt = - +∞ t 0 t ė(t)dt ≤ t0e(t0) + +∞ t 0 e(t)dt ≤ t0e(t0) + t -2 0 +∞ t 0 t 2 |e(t)|dt.
Another intuitive way to understand this is to look at what happens if the system is discretized with finite differences. In this case, the integrability assumptions on the errors for system (ISEHD-Pert) boil down to controlling only the first-order moment of the (discretized) error. Indeed, temporal discretization with fixed step size of t ẋ(t) gives k x k+1 -x k whose summability is clearly implied by the summability of k x k . We conclude this discussion by noting that Lipschitz continuity of the gradient is not needed for the estimates and convergence analysis of (ISEHD-Pert) while it is used extensively to analyze (ISIHD-Pert). This is a distinctive avantage of (ISEHD-Pert) compared to (ISIHD-Pert). This will be even more notable when extending to the non-smooth case; see Section 5.

4. Smooth strongly convex case. We will successively examine the Explicit Hessian Damping, then the Implicit Hessian Damping. 4.1. Explicit Hessian damping. In this section we consider the explicit Hessian system under the assumption of strong convexity of f . Following Polyak's heavy ball system [START_REF] Polyak | Introduction to Optimization[END_REF], consider the second-order perturbed system

ẍ(t) + 2 √ µ ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + β ė(t) + ∇f (x(t)) + e(t) = 0, ( 55
)
which has a fixed positive damping coefficient that is adjusted to the modulus µ of strong convexity of f . To study (55), we define the function

E : [t 0 , +∞[→ R + t → E(t) := f (x(t)) -min H f + 1 2 v(t) 2 , ( 56
)
where

v(t) = √ µ(x(t) -x ) + ẋ(t) + β∇f (x(t)). ( 57
)
Theorem 4.1. Suppose that f : H → R is µ-strongly convex for some µ > 0, let x be the unique minimizer of f . Let x(•) : [t 0 , +∞[→ H be a solution trajectory of (55). Suppose that

a) 0 ≤ β ≤ 1 2 √ µ . b) +∞ t0 e(t) dt < +∞ and +∞ t0 ė(t) dt < +∞.
Then the following properties are satisfied: (i) Minimizing properties: for all t ≥ t 0 (ii) Convergence rates: suppose moreover that for some p > 0,

E(t) ≤ E(t 0 )e - √ µ 2 (t-t0) + M e -
e(t) + β ė(t) = O 1 t p , as t → +∞. Then E(t) = O 1 t p , i.e
., E(t) inherits the decay rate of the error terms. As a consequence, as t → +∞

f (x(t)) -min H f = O 1 t p ; x(t) -x 2 = O 1 t p ; ẋ(t) 2 = O 1 t p .
In addition, when β > 0

e -√ µt t t0 e √ µs ∇f (x(s)) 2 ds = O 1 t p .
Proof. Recall f := min H f = f (x ). Define g(t) := e(t) + β ė(t), so that the constitutive equation is written in the compact form

ẍ(t) + 2 √ µ ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇f (x(t)) + g(t) = 0. ( 58
)
Derivation of E(•) gives

Ė(t) = ∇f (x(t)), ẋ(t) + v(t), v(t) = ∇f (x(t)), ẋ(t) + v(t), √ µ ẋ(t) + ẍ(t) + β∇ 2 f (x(t)) ẋ(t) .
Using the definition of v(t) and (58), we get

Ė(t) = ∇f (x(t)), ẋ(t) + √ µ(x(t) -x ) + ẋ(t) + β∇f (x(t)), - √ µ ẋ(t) -∇f (x(t)) -v(t), g(t) .
After developing and simplifying, we obtain

Ė(t) + √ µ ∇f (x(t)), x(t) -x + µ x(t) -x , ẋ(t) + √ µ ẋ(t) 2 +β √ µ ∇f (x(t)), ẋ(t) + β ∇f (x(t)) 2 = -v(t), g(t) .
According to strong convexity of f , we have

∇f (x(t)), x(t) -x ≥ f (x(t)) -f + µ 2 x(t) -x 2 .
Thus, by combining the last two relations, and by the Cauchy-Schwarz inequality, we obtain

Ė(t) + √ µA(t) ≤ v(t) g(t) , Set M := 2E(t 0 ) + +∞ t0
g(τ ) dτ . By assumption, +∞ t0 g(τ ) dτ < +∞, and thus sup t≥t0 v(t) ≤ M < +∞. Returning to (59) we deduce that

Ė(t) + √ µ 2 E(t) + √ µ 2 ẋ(t) 2 ≤ M g(t) . ( 60
) Therefore Ė(t) + √ µ 2 E(t) ≤ M g(t) . ( 61
)
By integrating the differential inequality above, we obtain

E(t) ≤ E(t 0 )e - √ µ 2 (t-t0) + M e - √ µ 2 t t t0 e √ µ 2 τ g(τ ) dτ. ( 62
)
We now use Lemma A.5, which is the continuous version of Kronecker's Theorem for series, with f (t) = g(t) and ϕ(t) = e Combining the above results with (64), we deduce that lim t→+∞ ẋ(t) = 0.

(ii) Let us make precise the argument developed above, and assume that, as t → +∞

g(t) = O 1 t p ,
where p > 0. Then, from (62) we get

E(t) ≤ E(t 0 )e - √ µ 2 (t-t0) + M e - √ µ 2 t t 2 t0 e √ µ 2 τ g(τ ) dτ + t t 2 e √ µ 2 τ g(τ ) dτ ≤ E(t 0 )e - √ µ 2 (t-t0) + M e - √ µ 2 t C 1 e √ µt 4
+ t t 2 e √ µ 2 τ C 2 τ p dτ ≤ E(t 0 )e - √ µ 2 (t-t0) + M e - √ µ 2 t C 1 e √ µt 4 + C 2 t p e √ µ 2 t ≤ E(t 0 )e - √ µ 2 (t-t0) + M C 1 e - √ µt 4 + C 2 t p = O 1 t p .
By definition of E(t) and strong convexity of f , we infer

µ 2 x(t) -x 2 ≤ f (x(t)) -min f (H) = O 1 t p and √ µ(x(t) -x ) + ẋ(t) + β∇f (x(t)) 2 = O 1 t p .
Developing the left-hand side of the last expression, we obtain

µ x(t) -x 2 + ẋ(t) 2 + β 2 ∇f (x(t)) 2 + 2β √ µ x(t) -x , ∇f (x(t)) + ẋ(t), 2β∇f (x(t)) + 2 √ µ(x(t) -x ) ≤ C t p . By convexity of f , we have x(t) -x , ∇f (x(t)) ≥ f (x(t)) -f . Moreover, ẋ(t), 2β∇f (x(t)) + 2 √ µ(x(t) -x ) = d dt 2β(f (x(t)) -f ) + √ µ x(t) -x 2 .
Combining the above results, we obtain

√ µ 2β(f (x(t)) -f ) + √ µ x(t) -x 2 + β 2 ∇f (x(t)) 2 + d dt 2β(f (x(t)) -f ) + √ µ x(t) -x 2 ≤ C t p . Set Z(t) := 2β(f (x(t)) -f ) + √ µ x(t) -x 2 . We have d dt Z(t) + √ µZ(t) + β 2 ∇f (x(t)) 2 ≤ C t p .

By integrating this differential inequality, elementary computation gives

e -√ µt t t0 e √ µs ∇f (x(s)) 2 ds ≤ C t p .
This completes the proof.

4.2. Implicit Hessian damping. We now turn to the implicit Hessian system, and take in the Polyak heavy ball system a fixed positive damping coefficient which is adjusted to the modulus of strong convexity of f . This gives the system

ẍ(t) + 2 √ µ ẋ(t) + ∇f (x(t) + β ẋ(t)) + e(t) = 0. (65)
To analyze (65), we define the function

E : [t 0 , +∞[→ R + t → E(t) := f (x(t) + β ẋ(t))) -min H f + 1 2 √ µ(x(t) -x ) + ẋ(t) 2 . ( 66
)
Theorem 4.2. Suppose that f : H → R is µ-strongly convex for some µ > 0, and let x be the unique minimizer of f . Let x(•) : [t 0 , +∞[→ H be a solution trajectory of (65). Suppose that

a) 0 ≤ β ≤ 1 2 √ µ . b) +∞ t0 e(t) dt < +∞.
Then the following properties are satisfied: (i) Minimizing properties: there exists a positive constant M such that for all t ≥ t 0

E(t) ≤ E(t 0 )e - √ µ 2 (t-t0) + M e - √ µ 2 t t t0 e √ µ
2 τ e(τ ) dτ.

More precisely,

M := E(t 0 ) c + 1 2c +∞ t0 e(τ ) dτ with c = min{µ, 1} 4 max{β 2 L 2 , 1}
and L is the Lipschitz constant of ∇f . Consequently,

lim t→+∞ E(t) = 0; lim t→+∞ f (x(t)) = min H f lim t→+∞ x(t) -x = lim t→+∞ ∇f (x(t)) = lim t→+∞ ẋ(t) = 0.
(ii) Convergence rates: suppose moreover that for some p > 0, e(t) = O 1 t p , as t → +∞. Then E(t) = O 1 t p , i.e., E(t) inherits the decay rate of the error terms. In turn, as t → +∞

f (x(t)) -min H f = O 1 t p ; x(t) -x 2 = O 1 t p ; ẋ(t) 2 = O 1 t p ; ∇f (x(t)) 2 = O 1 t p . Proof. Let us define v(t) = √ µ(x(t) -x ) + ẋ(t) (67)
and thus, E equivalently reads

E(t) = f (x(t) + β ẋ(t)) -min H f + 1 2 v(t) 2 . (68)
Taking the derivative in time of

E(•) gives Ė(t) = ∇f (x(t) + β ẋ(t)) , ẋ(t) + β ẍ(t) + v(t), v(t) = ∇f (x(t) + β ẋ(t)) , ẋ(t) + β ẍ(t) + √ µ(x(t) -x ) + ẋ(t), √ µ ẋ(t) + ẍ(t) .
Using the constitutive equation (58), we get

Ė(t) = ∇f (x(t) + β ẋ(t)) , (1 -2β √ µ) ẋ(t) -β∇f (x(t) + β ẋ(t)) -βe(t) + √ µ(x(t) -x ) + ẋ(t), - √ µ ẋ(t) -∇f (x(t) + β ẋ(t)) -e(t) .
After developing and simplifying, we obtain

Ė(t) + 2β √ µ ∇f (x(t) + β ẋ(t)) , ẋ(t) + √ µ ∇f (x(t) + β ẋ(t)) , x(t) -x + β ∇f (x(t) + β ẋ(t)) 2 + √ µ ẋ(t) 2 + µ x(t) -x , ẋ(t) = - √ µ(x(t) -x ) + ẋ(t) + β∇f (x(t) + β ẋ(t)) , e(t) .
In view of strong convexity of f , we have

∇f (x(t) + β ẋ(t)) , x(t) -x = ∇f (x(t) + β ẋ(t)) , x(t) + β ẋ(t) -x -∇f (x(t) + β ẋ(t)) , β ẋ(t) ≥ f (x(t) + β ẋ(t)) -f + µ 2 x(t) -x + β ẋ(t) 2 -∇f (x(t) + β ẋ(t)) , β ẋ(t) .
Thus, by combining the last two relations, we obtain

Ė(t) + β √ µ ∇f (x(t) + β ẋ(t)) , ẋ(t) + √ µ f (x(t) + β ẋ(t)) -f + µ 2 x(t) -x + β ẋ(t) 2 + β ∇f (x(t) + β ẋ(t)) 2 + √ µ ẋ(t) 2 + µ x(t) -x , ẋ(t) ≤ w(t) e(t) , (69)
Thus under this condition we get

Ė(t) + √ µ 2 E(t) ≤ w(t) e(t) . (70)
From (70), we first deduce that Ė(t) ≤ w(t) e(t) , which, after integration, gives

E(t) ≤ E(t 0 ) + t t0 w(τ ) e(τ ) dτ.
By definition of w we have

w(t) ≤ v(t) + β ∇f (x(t) + β ẋ(t)) -∇f (x ) ≤ v(t) + βL x(t) -x + β ẋ(t) ,
where L is the Lipschitz constant of ∇f . On the other hand, strong convexity of f entails

E(t) ≥ µ 2 x(t) -x + β ẋ(t) 2 + 1 2 v(t) 2 .
Hence, there exists a positive constant c such that 1

E(t) ≥ c w(t) 2 .
This in turn gives

c w(t) 2 ≤ E(t 0 ) + t t0 w(τ ) e(τ ) dτ.
According to Lemma A. Returning to (70) we deduce that

Ė(t) + √ µ 2 E(t) ≤ M e(t) . (71)
By integrating the differential inequality above, we obtain

E(t) ≤ E(t 0 )e - √ µ 2 (t-t0) + M e - √ µ 2 t t t0 e √ µ 2 τ e(τ ) dτ. (72)
(i) We first deduce from (72) that E(t) tends to zero as t → +∞. This implies that

lim t→+∞ f (x(t) + β ẋ(t))) = min H f, (73)
lim t→+∞ √ µ(x(t) -x ) + ẋ(t) = 0. (74)
From (73) and strong convexity of f we deduce that

lim t→+∞ (x(t) -x ) + β ẋ(t) = 0. (75)
From ( 74) and (75), and

β = 1 √ µ (a consequence of the assumption β ≤ 1 2 √ µ ), elementary algebra gives lim t→+∞ x(t) -x = lim t→+∞ ẋ(t) = 0. 1 One can take c = min{µ,1} 4 max{β 2 L 2 ,1} .
In turn, continuity of f and ∇f imply lim (ii) Let us now assume that, as t → +∞, we have e(t) = O 1 t p , where p > 0. Based on (72), a similar argument as in the explicit case (see the proof of Theorem 4.1) gives E(t) = O 1 t p . By definition of E(t), we infer that

f (x(t) + β ẋ(t)) -min H f = O 1 t p (76)
and

√ µ(x(t) -x ) + ẋ(t) 2 = O 1 t p . (77)
From ( 76) and strong convexity of f we deduce that

(x(t) -x ) + β ẋ(t) 2 = O 1 t p . (78)
Combining ( 77) and ( 78), and recalling that β √ µ = 1 we immediately obtain

x(t) -x 2 ≤ C t p and ẋ(t) 2 = O 1 t p . (79)
According to the Lipschitz continuity of ∇f , and ∇f (x ) = 0 we deduce that

∇f (x(t)) 2 ≤ L 2 x(t) -x 2 = O 1 t p .
Now, combining the descent lemma with (76), ( 78) and (79) shows that

f (x(t)) -min H f ≤ f (x(t) + β ẋ(t)) -min H f -β ∇f (x(t) + β ẋ(t)), ẋ(t) + Lβ 2 2 ẋ(t) 2 ≤ f (x(t) + β ẋ(t)) -inf H f + Lβ x(t) -x + β ẋ(t) ẋ(t) + Lβ 2 2 ẋ(t) 2 = O 1 t p ,
which completes the proof.

Remark 4.1. The results of Theorem 4.2 appear new. Even for the unperturbed case of system (65), where e ≡ 0, we are not aware of any guarantees for these dynamics in the literature.

5. The non-smooth case.

5.1. Explicit Hessian damping. In the sequel, we will show that most properties obtained in the smooth case still hold for the global strong solution of (4) (and in particular, all properties that do not require x(t) to be twice differentiable).

for almost all t > t 0 . Using (82), we obtain

ü(t) = 1 β - α t ẋ(t) - 1 β u(t) -(e(t) + β ė(t)) (84)
for almost all t > t 0 . We will need the following energy function of the system, defined for all T ≥ t ≥ t 0 (recall (81) for the definition of u(t)):

W T (t) := 1 2 u(t) 2 + f (x(t)) - T t u(τ ), e(τ ) + β ė(τ ) dτ, (85)
and when the following expression is well-defined (we will prove it later)

W (t) := 1 2 u(t) 2 + f (x(t)) - +∞ t u(τ ), e(τ ) + β ė(τ ) dτ. ( 86
)
Theorem 5.1. Let α > 0. Suppose that inf H f > -∞. Suppose that e(•) ∈ W According to (89), we have u(t) 2 ≤ -2t α Ẇ (t). Therefore,

+ β ė(t) = u(t), 1 β - α t ẋ(t) - 1 β u(t) + ξ(t), ẋ(t) = - 1 β u(t) 2 + ẋ(t), 1 β - α t u(t) + ξ(t) = - 1 β u(t) 2 + ẋ(t), 1 β - α t u(t) + 1 β ( u(t) -ẋ(t)) = - 1 β u(t) 2 + ẋ(t), 2 β - α t u(t) - 1 β ẋ(t) = - 1 β u(t) 2 - 1 β ẋ(t) 2 + 2 β - α t ẋ(t), u(t) ≤ - α 2t ẋ(t) 2 - α 2t u(t)
ḧ(t) + α t ḣ(t) + 1 - αβ t (W (t) -f (z)) + 1 - αβ t İ(t) ≤ - 3t α -β Ẇ (t) + e(t) + β ė(t) u(t) -z -1 - αβ t +∞ t u(τ ), e(τ ) + β ė(τ ) dτ.
Dividing by t and rearranging the terms, we have with g(t) := e(t) + β ė(t)

1 t ḧ(t) + 1 t - αβ t 2 (W (t) -f (z)) ≤ - 3 α - β t Ẇ (t) - α t 2 ḣ(t) + 1 t - αβ t 2 İ(t) + 1 t g(t) u(t) -z - 1 t - αβ t 2 +∞ t u(τ ), g(τ ) dτ.
After integration, and using Lemma A.2, we get

1 t ḣ(t) + t t 1 1 s - αβ s 2 W (s) -f (z) ds ≤ - t t 1 3 α - β s Ẇ (s) ds + C + K1(t) + K2(t), (91)
where

K1(t) = t t 1 1 s g(s) u(s) -z ds and K2(t) = t t 1 1 s - αθ s 2 ∞ s u(τ ) g(τ ) dτ ds.
Let us majorize K 1 (t) and K 2 (t). The relation

u(s) -z ≤ u(t 1 ) -z + s t1 u(τ ) dτ,
and u(•) bounded (see (87)) give

K 1 (t) ≤ t t 1 1 s g(s) u(s) -z ds ≤ u(t 1 ) -z t 1 + sup t≥t 1 u(τ ) +∞ t 1 g(s) ds ≤ C < +∞.
For K 2 (t), we use again u(•) bounded (see (87)) and integration by parts to obtain

K2(t) ≤ C t t 1 1 s ∞ s g(τ ) dτ ds ≤ C log t ∞ t g(τ ) dτ + t t 1 g(τ ) log τ dτ + 1 .
Let us examine the integral terms that enter (91). Since W (•) is non-increasing

t t 1 1 s - αβ s 2 W (s) -f (z) ds ≥ (W (t) -f (z)) t t 1 1 s - αβ s 2 ds = (W (t) -f (z)) log t -log t1 + αβ t - αβ t1 . (92)
In turn, integration by parts gives 93) with z = x ∈ S, and using nonnegativity of h, we infer that for some positive constant C

- t t 1 3 α - β s Ẇ (s) ds = 3 α - β t 1 W (t 1 ) -f (z) - 3 α - β t (W (t) -f (z)) + β t t 1 W (s) -f (z) s 2 ds ≤ 3 α - β t 1 W (t 1 ) -f (z) - 3 α - β t (W (t) -f (z)) + β W (t 1 ) -f (z) 1 t 1 - 1 t , ≤ 3 α W (t 1 ) -f (z) - 3 α - β t (W (t) -f (z)) (93) since t → W (t) -f (
1 - αβ t 1 t t1 1 s (W (s) -f (z)) ds ≤ C + K 2 (t).
Arguing similarly as for proving part (v)-(a), we can show that K 2 (•) is bounded. Thus

t t 1 1 s f (x(s)) -f (z) + u(s) 2 ds ≤ t t 1 1 s (W (s) -f (z)) ds + 1 t 1 t t 1 u(s) g(s) ds < +∞,
which completes the proof.

(i) f (x(t)) -min H f = O t -2 . (ii) +∞ t 0 t(f (x(t)) -min H f )dt < +∞, +∞ t 0 t 2 ξ(t) 2 dt < +∞, +∞ t 0 t ẋ(t) 2 dt < +∞. (iii) ẋ(t) + βξ(t) = O(t -1 ).
Proof. Let (x, y) : [t 0 , +∞[→ H × H be a global strong solution of (4). Take α ≥ 3 and x ∈ S. Recall f := min H f and g(t) = e(t) + β ė(t). Our analysis relies on the non-smooth version of the Lyapunov function in [START_REF] Su | A differential equation for modeling nesterov's accelerated gradient method: Theory and insights[END_REF], which is defined for

λ ∈ [2, α -1], as E λ,T : [t 0 , T ] → R by E λ,T (t) =t(t -β(λ + 2 -α))(f (x(t)) -f ) + 1 2 v λ (t) 2 + λ(α -λ -1) 1 2 x(t) -x 2 - T t τ v λ (τ ), g(τ ) dτ, (95)
where v λ (t) := λ(x(τ ) -x ) + τ u(t), and u is defined on [t 0 , +∞[ by (80) and u is given by (81). E λ,T (•) is the sum of four terms, each of which is absolutely continuous on [t 0 , T ] for all T > t 0 . Hence E λ,T is differentiable almost everywhere. We first differentiate each term of E λ,T :

d dt t(t -β(λ + 2 -α))(f (x(t)) -f ) =(2t -β(λ + 2 -α))(f (x(t)) -f ) + t(t -β(λ + 2 -α)) ξ(t), ẋ(t) .
Using (83), we have d dt

1 2 v λ (t) 2 = λ(x(t) -x ) + t u(t), λ ẋ(t) + u(t) + tü(t) = λ(x(t) -x ) + t u(t), (λ + 1 -α) ẋ(t) -(t -β)ξ(t) -tg(t) = λ(λ + 1 -α) x(t) -x , ẋ(t) -t(α -λ -1) ẋ(t) 2 -βt(t -β) ξ(t) 2 -λ(t -β) x(t) -x , ξ(t) -t(t -β(λ + 2 -α)) ξ(t), ẋ(t) -t v λ (t), g(t) , d dt λ(α -λ -1) 1 2 x(t) -x 2 = λ(α -λ -1) x(t) -x , ẋ(t) , and
d dt - T t τ v λ (τ ), g(τ ) dτ = t v λ (t), g(t) .
By collecting these results, the perturbation terms cancel each other out. We get

d dt E λ,T (t) = (2t -β(λ + 2 -α))(f (x(t)) -f ) -λ(t -β) x(t) -x , ξ(t) -t(α -λ -1) ẋ(t) 2 -βt(t -β) ξ(t) 2 , (96)
for almost all t > t 0 . Since ξ(t) ∈ ∂f (x(t)) for all t > t 0 , we have ξ(t), x(t) -x ≥ f (x(t) -f (x ), and we deduce from (96), that

d dt E λ,T (t) ≤ -((λ-2)t-β(α-2))(f (x(t))-f )-t(α-λ-1) ẋ(t) 2 -βt(t-β) ξ(t) 2 , (97)
for almost all t ≥ t 1 = max (t 0 , β). It follows that E λ,T is non-increasing on [t 1 , T ]. In particular, E λ,T (t) ≤ E λ,T (t 1 ) for t 1 ≤ t ≤ T . This gives the existence of a constant C such that

1 2 v λ (t) 2 ≤ C + t t0 v λ (t) τ g(τ ) dτ. (98)
Applying Lemma A.3 to (98), and using the integrability of t → tg(t), it follows that

sup t≥t0 v λ (t) ≤ √ 2C + ∞ t0 τ g(τ ) dτ < +∞. (99)
As a consequence, we can define the energy function

E λ (t) := t(t -β(λ + 2 -α))(f (x(t)) -f ) + 1 2 v λ (t) 2 + λ(α -λ -1) 1 2 x(t) -x 2 - ∞ t τ v λ (t), g(τ ) dτ,
which has the same derivative as E λ,T . Hence E λ (t) ≤ E λ (t 0 ). Combined with (99), this gives

t(t -β(λ + 2 -α))(f (x(t)) -f ) ≤ C + sup t≥t0 v λ (t) ∞ t0 τ g(τ ) dτ < +∞,
whence statement (i). Claim (iii) is obtained by letting λ = 0 in (99). Integration of (97) gives the integral estimates of (ii), which completes the proof.

5.1.3. Convergence of the trajectories and faster asymptotic rates. Similar argument as in the smooth case (see Theorem 3.3), but now using the Lyapunov function (95), gives weak convergence of the trajectories of (4). Moreover, in the same vein as Theorem 3.2, o(•) rates can also be obtained. We leave the details to the readers for the sake of brevity. Then, for any global strong solution (x, y) of (4) (i) x(t) converges weakly, as t → +∞ to a point in S;

(ii) f (x(t)) -min H f = o t -2 and ẋ(t) + βξ(t) = o t -1 as t → +∞.

Remark 5.1. In the special case where e ≡ 0 in (4), i.e., unperturbed case, Theorem 4.2, Theorem 5.2 and Theorem 5.3 recover the result of [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF]Section 4]. In a nutshell, our result demonstrates that the properties of the unperturbed system are preserved under reasonable integrability conditions on the errors.

5.2. Implicit Hessian damping. As we have already discussed in the smooth case (see Section 3.3), the analysis of the convergence properties of the system with implicit Hessian driven damping heavily relies on Lipschitz continuity of the gradient. As shown above, such a property was not needed to analyze system (4). Therefore, the study of the convergence properties for the non-smooth system (13) (even without perturbations) is an open challenging topic.

6. Numerical experiments. To support our theoretical claims, we consider numerical examples in H = R 2 with two real-valued functions:

• The first one is given by f (x 1 , x 2 )) = (x 1 -1) 4 + (x 2 -5) 2 . This function is obviously convex (but not strongly so) and smooth, and has a unique minimizer at [START_REF] Adly | Finite convergence of proximal-gradient inertial algorithms combining dry friction with Hessian-driven damping[END_REF][START_REF] Apidopoulos | Convergence rate of inertial Forward-Backward algorithm beyond Nesterov's rule[END_REF]. For this function, we consider the continuous time dynamical system (ISEHD-Pert) with parameters (α, β) = (3. Although we have no theoretical guarantee for system (13), we do report the corresponding numerical results with parameters (α, γ, β) = (3.1, 1, 1).

For both examples, we take as an exogenous perturbation e(t) = cos(2πt) t δ with δ ∈ {0.1, 1.1, 3.1}.

All systems are solved numerically with a Runge-Kutta adaptive method in MAT-LAB on the time interval [START_REF] Adly | Finite convergence of proximal-gradient inertial algorithms combining dry friction with Hessian-driven damping[END_REF]50] with initial data (x 0 , ẋ0 ) = (-10, 20, 5, -5). The results are displayed in Figure 1 and Figure 2.

Let us first comment on the results for the smooth function (Figure 1). For δ = 3.1, all required moment assumptions on the errors are fulfilled (for the explicit Hessian, the term ė is dominated by e and can then be discarded). Hence the fast rates predicted by Theorem 3.2(i) and Theorem 3.4(i) as well as convergence of the trajectories (see Theorem 3.3 and Theorem 3.5) hold true. For the value δ = 0.1, since the error is not even integrable, neither the convergence of the objective value nor that of the trajectories is ensured, with large oscillations appearing. The implicit Hessian damping seems also less stable as anticipated from our discussion in Section 3.3. For δ = 1.1, though there is no convergence guarantee for the trajectory, the objective value for (ISEHD-Pert) decreases but at a rate which is dominated by the error decrease. This can be explained in light of the proof of Theorem 3.2(i), where a close inspection of ( 29) and [START_REF] Nesterov | Introductory Lectures on Convex Optimization: A Basic Course[END_REF] shows that the bound on the objective error decomposes as

f (x(t)) -f ≤ O 1 t 2 + C t t0
τ e(τ ) dτ For δ ∈]1, 2], the second term indeed dominates the first one and decreases at the slower rate t -2(δ-1) . This confirms the known rule that there is a trade-off between fast convergence of the methods and their robustness to perturbations. Similar observations remain true for the non-smooth function (Figure 2) with system (4) where we now invoke Theorem 5.2 and Theorem 5.3. As for system [START_REF] Attouch | Fast proximal methods via time scaling of damped inertial dynamics[END_REF], it seems that it has a behaviour similar to what we observed in the smooth case for system (ISIHD-Pert). As we argued in Section 5.2, supplementing the numerical observations for system [START_REF] Attouch | Fast proximal methods via time scaling of damped inertial dynamics[END_REF] with theoretical guarantees is an open problem that we leave to a future work.

7. Conclusion and perspectives. The introduction of the correction term attached to the damping driven by the Hessian in first-order accelerated optimization algorithms makes it possible to considerably dampen the oscillations in the trajectory. The study of the robustness of these algorithms with respect to error perturbations is crucial for their further development in a stochastic framework. Our systematic study of these questions for the dynamics underlying these algorithms is a fundamental first step in this direction. We paid particular attention to the explicit and the implicit forms of the Hessian driven damping, showing several advantages of the explicit form. Our study concerns the dynamics with damping driven by the Hessian within the framework of the Nesterov acceleration gradient method. It shows that the convergence of the values still holds when the error terms satisfy an appropriate integrability condition, and fast convergence is satisfied when the (first or second-order) moment of the error is finite. Indeed, as a general rule, there is a balance between the rate of convergence of the methods and their robustness with respect to error disturbances. An interesting technique studied in [START_REF] Adly | Finite convergence of proximal-gradient inertial algorithms combining dry friction with Hessian-driven damping[END_REF][START_REF] Adly | Finite time stabilization of continuous inertial dynamics combining dry friction with Hessian-driven damping[END_REF] is the introduction of a dry friction term. This makes it possible to have errors which do not necessarily go to zero, they must not exceed a certain threshold, but on Introduce a primitive of β, p(t) = t t0 β(r)dr Therefore, (102) can be equivalently written Let t → +∞ to deduce that 0 ≤ lim sup t→+∞ 1 ϕ(t) t t0 ϕ(s)f (s)ds ≤ . Since this is true for any > 0, the result follows.

Lemma A.6 ([11, Lemma 5.9]). Let t 0 > 0, and let w : [t 0 , +∞[→ R be a twice differentiable2 function which is bounded from below. Assume that t ẅ(t) + α ẇ(t) ≤ g(t), for some α > 1, almost every t > t 0 , and some non-negative function g ∈ L 1 (t 0 , +∞; R). Then, the positive part [ ẇ] + of ẇ belongs to L 1 (t 0 , +∞; R) and lim t→+∞ w(t) exists.

1 β

 1 ( u(t) -ẋ(t)), we obtain the convergence of ∇f (x(t)) to zero. Remark 3.1. The result in [11, Lemma 4.1] is a particular case of Theorem 3.1 when β = 0. Theorem 3.1 is also a generalization of [19, Theorem 1.3 and Proposition 1.5] to the perturbed case. Performing this generalization necessitates a different Lyapunov function together with particular novel estimates in the course of derivation.

Remark 3 . 2 .

 32 The first three claims (resp. fourth claim) of Theorem 3.2 are a nontrivial generalization of [9, Theorem 3] (resp. [10, Theorem 2.1]) to the perturbed case. The presence of perturbations necessitates a careful analysis of several bounds and new estimates to handle the presence of errors and eventually preserve the convergence rates.

2α 3 isTheorem 3 . 3 .

 333 achieved by taking f (x) = x[START_REF] Apidopoulos | Convergence rate of inertial Forward-Backward algorithm beyond Nesterov's rule[END_REF]. Theorem 3.2 is consistent with these optimality results. The condition α > 3 is important to get the asymptotic rate o(1/t 2 ). 3.1.3. Convergence of the trajectories. We complete our analysis by showing weak convergence of the trajectories. Assume that e ∈ C 1 ([t 0 , +∞[; H) with +∞ t0 t e(t) dt < +∞ and+∞ t0 t ė(t) dt < +∞. Let x(t) be a solution trajectory to (ISEHD-Pert) for α > 3 and β > 0. Then x(t) converges weakly to a minimizer of f .

2 τ

 2 e(τ ) + β ė(τ ) dτ, where M := 2E(t 0 ) + +∞ t0 e(τ ) + β ė(τ ) dτ . As a consequence, lim t→+∞ E(t) = 0; lim t→+∞ f (x(t)) = min H f lim t→+∞ x(t) -x = lim t→+∞ ∇f (x(t)) = lim t→+∞ ẋ(t) = 0.

√ µ 2 t

 2 . By assumption we have+∞ t0 g(τ ) dτ < +∞. We deduce that lim x(t) -x ) + ẋ(t) + β∇f (x(t)) = 0. (64)Acoording to (63) and the strong convexity of f we deduce that lim t→+∞ x(t) -x = 0 By continuity of ∇f , and since ∇f (x ) = 0, we deduce that lim t→+∞ ∇f (x(t)) = 0.

  ) dτ < +∞.

  t→+∞ ∇f (x(t)) = 0 and lim t→+∞ f (x(t)) = min H f.

Theorem 5 . 3 .

 53 Let α > 3. Let f ∈ Γ 0 (H) and assume that S = argmin f = ∅. Suppose that e(•) ∈ W

1 , 1 )

 11 , and (ISIHD-Pert) with parameters (α, γ, β) = (3.1, 1, 1).• The second example we consider is with the convex non-smooth functionf (x 1 , x 2 ) = (x 1 -1) 4 + (x 2 -5) 2 + 0.1(|x 1 | + |x 2 |).For this function, we use the continuous time non-smooth system (4) with parameters (α, β) = (3.1, 1).

Figure 1 .

 1 Figure 1. Example on a smooth function: Evolution of the objective error and distance to the minimizer as a function of t for different error decay exponents.

  Implicit Hessian damping: non-smooth function

Figure 2 .

 2 Figure 2. Example on a non-smooth function: Evolution of the objective error and distance to the minimizer as a function of t for different error decay exponents.

d

  ds p(τ (s)) = 1 ⇐⇒ p(τ (s)) = τ (s) t0 β(r)dr = s + C, for any constant C. Thus, τ defines a change of variable if and only if +∞ t0 β(r)dr = +∞, hence our assumption on β. Lemma A.5. Take t 0 > 0, and let f ∈ L 1 (t 0 , +∞; R) be continuous. Consider a nondecreasing function ϕ : [t 0 , +∞[→ R + such that lim t→+∞ ϕ(t) = +∞. Then, s)f (s)ds = 0. Proof. Given > 0, fix t so that ∞ t |f (s)|ds ≤ . Then, for t ≥ t , split the integral t t0 ϕ(s)f (s)ds into two parts to obtain 1

  ) with α > 3, γ > 0. Define, for t ≥ t 0 , Assume that ∇f is L-Lipschitz continuous, e ∈ C([t 0 , +∞[) and m(•)e(•) ∈ L 1 (t 0 , +∞; H). Then, we have sup t≥t0 E(t) < +∞, sup t≥t0 t ẋ(t) < +∞ and sup t≥t0 x(t) -x < +∞.

	m(t) := max t, L|a(t)β(t)|, L|a(t)|β(t) 2 .	(49)
	Proof. Consider the function E(t) in (40) with the choices (47)-(48) for c(t), d(t),

b(t) and a(t), with b ∈]2, α -1[. For such a choice, there exists t 1 ≥ t 0 such that for all t ≥ t 1 , a(t) > 0, β(t) > 0 (and in turn, m(t) > 0), and all conditions of (41) are satisfied. Thus, E(t) is monotonically decreasing on [t 1 , +∞[ according to Lemma 3.1. Since the solution x(t) is continuous, it is bounded on [t 0 , t 1 ] and so without loss of generality we can assume that t 1 = t 0 and proceed to show,

  Recall our discussion on the parameters in (47)-(48). Notice that we have t 2 ≤ a(t) ≤ t 2 + κ 1 and γ/2 ≤ β(t) ≤ β 0 for t large enough, where β 0 = γ + |β|/t 0 and κ is a non-negative constant. In turn, for t large enough, we have max (1, γ/2) Lγ/2t 2 ≤ m(t) ≤ max (1, β 0 ) Lβ 0 (t + κ 1 ) 2 .

	whence we get boundedness of x(t)-x and b(x(t)-x )+t ẋ(t) . The triangle in-
	equality then shows that t ẋ(t) is also bounded. Using this into (50) together with
	Cauchy-Schwarz inequality and our integrability assumption, we deduce bounded-
	ness of E(t).
	Remark 3.6.
	(51)

3, we conclude that, for all t ≥ t 0 b(x(t)-x )+t ẋ(t) + √ d x(t)-x ≤ 2 |E(t 0 )|+2C +∞ t0 |m(τ )| e(τ ) dτ < +∞,

  5.1.1. Minimizing properties. From now on, we assume that, for all T > t 0 , e(•) ∈ W 1,1 (t 0 , T ; H). Let (x, y) : [t 0 , +∞[→ H × H be the global strong solution to[START_REF] Alvarez | A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics[END_REF] with Cauchy data (x(t 0 ), y(t 0 )) = (x 0 , y 0 ) ∈ dom(f ) × H. For t ≥ t 0 define

	u(t) =	t t0	-βe(s) +	1 β	-	α s		x(s) -	1 β	y(s) ds.	(80)
	Thus u is continuously differentiable, with derivative satisfying
	u(t) = -βe(t) +	1 β	-	α t	x(t) -	1 β	y(t), ∀t ≥ t 0 ,	(81)
	= ẋ(t) + βξ(t),					for almost all t > t 0 ,	(82)
	where ξ(t) ∈ ∂f (x(t)), and the last equality follows from Theorem 2.2(vi)-(b).
	Therefore, u can be also written equivalently as
											t
			u(t) = x(t) -x 0 + β		ξ(s)ds.
										t0
	With parts (i) and (ii) of Theorem 2.2, equality (81) shows that u is absolutely
	continuous on any compact subinterval of [t 0 , +∞[, hence differentiable almost ev-
	erywhere on [t 0 , +∞[. Therefore,
	ü(t) = -β ė(t) +	α t 2 x(t) +	1 β	-	α t	ẋ(t) -	1 β	ẏ(t).
	The equality above, combined with ẏ(t) = αβ t 2 x(t) + ẋ(t) + β(ξ(t) + e(t)) (which is
	obtained by taking the difference of the two equations in (4)), yields
		ü(t) = -	α t	ẋ(t) -ξ(t) -(e(t) + β ė(t)),

  1,1 (t 0 , T ; H) for all T > t 0 , with W is well-defined and non-increasing on [t 1 , +∞[ for some t 1 ≥ t 0 . iii) lim t→+∞ W (t) = lim t→+∞ f (x(t)) = inf H f , lim t→+∞ ẋ(t) + βξ(t) = 0. (iv) As t → +∞, every sequential weak cluster point of x(t) belongs to S. (v) If, moreover, the solution set S = ∅ and

								+∞
								log t e(t) < +∞ and
								t0
		+∞
			log t ė(t) < +∞, then
		t0
		(a) f (x(t)) -inf H f = O	1 log t	and u(t) = O	√	1 log t	as t → +∞.
		(b)	+∞ t0	1 t	(f (x(t)) -inf H	f )dt < +∞.
	Proof. Since we are interested in asymptotic analysis, we can assume t ≥ t 1 =
	max (t 0 , 2αβ).
	Claim (i). According to Theorem 2.2, W T is absolutely continuous. Taking the
	derivative and using the chain rule we get
				ẆT (t) = u(t), ü(t) + ξ(t), ẋ(t) + u(t), e(t) + β ė(t) ,
	for almost every T > t > t 0 . Now use (82) and (84) to obtain
	ẆT (t) = u(t),		1 β	-	α t	ẋ(t) -	1 β	u(t) -(e(t) + β ė(t)) + ξ(t), ẋ(t) + u(t), e(t)
								+∞	+∞
								e(t) < +∞ and	ė(t) < +∞.
								t0	t0
	Then for any global strong solution of (4), (x, y) : [t 0 , +∞[→ H × H
	(i) (ii)	+∞ t0	1 t	ẋ(t) 2 dt < +∞,	+∞ t0	1 t	ξ(t) 2 dt < +∞.

(

  2 , for almost every t ≥ t 1 . So W T is non-increasing on [t 1 , +∞[, because it is absolutely continuous and its derivative is non-positive therein. Therefore W T (t) ≤ W T (t 1 ) for all t ∈ [t 1 , T ]. Equivalently Claim (iii). Given z ∈ H, let us define h : [t 0 , +∞[→ R + by h(t) = 1 2 u(t) -z 2 .

	1 2 1 2 After simplification, and setting C = 1 u(t) 2 + f (x(t)) -≤ u(t 1 ) 2 + f (x(t 1 )) -t 2 u(t 1 ) 2 + f (x(t 1 )) -inf f (H), we obtain T u(τ ), e(τ ) + β ė(τ ) dτ T t1 u(τ ), e(τ ) + β ė(τ ) dτ. 1 2 u(t) 2 ≤ C -t t1 u(τ ), e(τ ) + β ė(τ ) dτ. By Cauchy-Schwarz inequality we get 1 2 u(t) 2 ≤ C + t t0 u(τ ) e(τ ) + β ė(τ ) dτ. According to Gronwall's Lemma A.3 u(t) ≤ √ 2C + t t 0 e(τ ) + β ė(τ ) dτ ≤ M := √ 2C + +∞ t 0 e(τ ) + β ė(τ ) dτ. (87) So, u(t) is bounded on [t 0 , +∞[, which allows us to define W (t) = 1 2 u(t) 2 + f (x(t)) -+∞ t u(τ ), e(τ ) + β ė(τ ) dτ. (88) Noticing that W and W T have the same derivative we conclude that Ẇ (t) + α 2t ẋ(t) 2 + α 2t u(t) 2 ≤ 0, (89) and thus W is non-increasing on [t 1 , +∞[. Claim (ii). Integrating (89), and using that f , and hence W , is bounded from below, we obtain, +∞ t0 1 t ẋ(t) 2 dt < +∞, and +∞ t0 1 t u(t) 2 dt < +∞. (90) Using Jensen's inequality, we get the integrability claim on ξ(t). The function h is continuously differentiable with ḣ(t) = u(t) -z, u(t) , and ḣ is absolutely continuous on compact subintervals of [t 0 , +∞[ (since u is) and satisfies ḧ(t) = u(t) -z, ü(t) + u(t) 2 for almost every t > t 0 . Using (82) and (83) we get ü(t) + α t u(t) = -1 -αβ t ξ(t) -(e(t) + β ė(t)). Therefore, for almost every t > t 0 ḧ(t) + α t ḣ(t) = u(t) 2 -u(t) -z, 1 -αβ t ξ(t) -u(t) -z, e(t) + β ė(t) = u(t) 2 -1 -αβ t x(t) -z -x0 + β t t 0 ξ(s)ds, ξ(t) -u(t) -z, e(t) + β ė(t) ≤ u(t) 2 -1 -αβ t x(t) -z, ξ(t) -1 -αβ t -x0 + β t t 0 ξ(s)ds, ξ(t) + e(t) + β ė(t) u(t) -z . To interpret -x 0 + β t t0 ξ(s)ds , ξ(t) as a temporal derivative, let us introduce I(t) = 1 2β -x 0 + β t t0 ξ(s)ds 2 . Then I(•) is locally absolutely continuous and İ(t) = -x 0 + t t0 ξ(s)ds, ξ(t) al-most everywhere, because ξ ∈ L 2 (t 0 , T ; H) ⊆ L 1 (t 0 , T ; H) for all T > t 0 ; see part (vi)-(c) of Theorem 2.2. So, ḧ(t) + α t ḣ(t) ≤ u(t) 2 -1 -αβ t x(t) -z, ξ(t) -1 -αβ t İ(t) + e(t) + β ė(t) u(t) -z , for almost every t > t 0 . On the other hand, by convexity of f and ξ(t) ∈ ∂f (x(t)) x(t) -z, ξ(t) ≥ f (x(t) -f (z). Therefore ḧ(t) + α t ḣ(t) + 1 -αβ t (f (x(t)) -f (z)) + 1 -αβ t ḧ(t) + α t ḣ(t) + 1 -αβ t (W (t) -f (z)) + 1 -αβ t İ(t) ≤ 3 2 -αβ 2t u(t) 2 İ(t) ≤ u(t) Using the definition (88) of W , we get + e(t) + β ė(t) u(t) -z -1 -+∞ αβ t t u(τ ), e(τ ) + β ė(τ ) dτ.

2

+ e(t) + β ė(t) u(t) -z .

  z) is non-increasing and t ≥ t 1 ≥ αβ.for appropriate constants C, D, E ∈ R. Now, take t 2 ≥ t 1 such that log s+D+ E s ≥ 0 for all s ≥ t 2 . Integrate from t 2 to t and use again that W is non-increasing to obtain

	Claim (v)-(b). Putting together (91) and (
											Combining (91) with (92)
	and (93), we obtain
	1 t	ḣ(t) + (W (t) -f (z)) log t + D +	E t	≤ C log t	t	∞	g(τ ) dτ +	t t 1	g(τ ) log τ dτ + 1
	h(t) t	-	h(t 2 ) t 2	+	t t2	h(s) s 2 ds + (W (t) -f (z))	t t2	log s + D +	E s	ds
											t	∞
								≤ C	log s	g(τ ) dτ +
											t2	s
											+∞
	bounded (see (87)) yield lim t→+∞	t	u(τ ), g(τ ) dτ = 0. As a consequence,
								lim t→+∞	f (x(t)) +	1 2	ẋ(t) + βξ(t) 2 ≤ f (z)
	for each z ∈ H. Thus
	inf H	f ≤ lim inf t→+∞	f (x(t)) ≤ lim sup t→+∞	f (x(t)) ≤ lim t→+∞	f (x(t)) +	1 2	ẋ(t) + βξ(t) 2 ≤ inf
											To
	conclude, we note that
										∞	∞
							log t			g(τ ) dτ ≤	log τ g(τ ) dτ < +∞,
									t		t
							t t2	g(τ )	τ t	log τ dτ ≤	t t2	g(τ ) log τ dτ < +∞ and
							log t

s t1 g(τ ) log τ dτ + 1 ds.

Since h is non-negative, this implies

(W (t) -f (z)) (t log t + (D -1)t + E log t + F ) ≤ C t + t log t ∞ t g(τ ) dτ + t t2 g(τ ) τ log τ dτ + t t t1 g(τ ) log τ dτ +G,

(94)

for some appropriate constants D, E, F, G ∈ R. Divide by t log t, let t → +∞, and use Lemma A.5, to obtain lim t→+∞ W (t) ≤ f (z). The integrability of g and u(•) H f, whence we get lim t→+∞ f (x(t)) = inf H f , and thus lim t→+∞ ẋ(t) + βξ(t) = 0. Claim (iv). This follows from claim (iii) and lower semicontinuity of f . Claim (v)-(a). Let x ∈ S. We start from (94) with z = x and divide by t. +∞ t u(τ ), g(τ ) dτ ≤ C +∞ t log τ g(τ ) dτ, where C = sup t≥t0 u(t) < +∞ (see (87)).

  5.1.2. Fast convergence rates. When α ≥ 3, under a reinforced integrability assumption on the perturbation term, we will show fast convergence results. The following theorem is the non-smooth counterpart of Theorem 3.2.

Theorem 5.2. Suppose that α ≥ 3. Let f ∈ Γ 0 (H) such that S = ∅. Suppose that e(•) ∈ W 1,1 (t 0 , T ; H) for all T > t 0 , with +∞ t0 t e(t) + β ė(t) dt < +∞. Then, for any global strong solution (x, y) of (4)

  1,1 (t 0 , T ; H) for all T > t 0 , with

	+∞
	t e(t) + β ė(t) dt < +∞.
	t0

In[START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF] Lemma 5.9], twice differentiability was not stated, but is actually needed for the statement to make sense.

supported by the OP VVV project CZ.02.

where

Let us make appear E(t) in A(t),

After developing and simplifying, we obtain

≤ v(t) g(t) .

Since 0

Let us use again the strong convexity of f to write

By combining the two inequalities above, we obtain

where

Elementary algebraic computation gives that, under the condition 0

(i) From (59), we first deduce that

which by integration gives

By definition of E(t), we have

According to Lemma A.3, we obtain

where we have used Cauchy-Schwarz inequality, and we set

Let us make E(t) appear on the left-hand side of (69). We get

where

Let us use again the strong convexity of f to write

By combining the inequalities above, we obtain

where

and we set y(t) := x(t) + β ẋ(t). Let us show that, for an adequate choice of the parameters, C(t) is non-negative. Let us reformulate C(t) as follows: Young's inequality gives the following minorization for the two first terms of C(t)

By using this inequality in C(t), and after simplification, we arrive at

Elementary algebra gives that -

. According to the classical rule for the sign of a quadratic function of a real variable, we get that C(t) ≥ 0 under the condition

Setting Z = β √ µ, the latter inequality is equivalent to ensuring

, we get as a final condition

the other hand we only obtain an approximate solution. Finding the right balance between the convergence rate and robustness is an important issue that should be the subject of further study. Another important aspect of our study is the fact that several results are valid in the case of a non-smooth function. This opens the door to the study of similar topics with respect to structured composite optimization problems involving a non-smooth term. These are some of the many facets of these flexible dynamics and algorithms which, in the unperturbed case, have been applied in various fields including PDE's and mechanical shocks [START_REF] Attouch | A second-order differential system with Hessiandriven damping; Application to non-elastic shock laws[END_REF], deep learning [START_REF] Castera | An inertial newton algorithm for deep learning[END_REF], non-convex optimization [START_REF] Attouch | Fast optimization via inertial dynamics with closed-loop damping[END_REF], monotone inclusions [START_REF] Attouch | Newton-like inertial dynamics and proximal algorithms governed by maximally monotone operators[END_REF][START_REF] Attouch | Continuous Newton-like inertial dynamics for monotone inclusions[END_REF] to mention a few important applications.

Appendix A. Auxiliary results. Let us first recall the continuous form of the Opial's Lemma [START_REF]Weak convergence of the sequence of successive approximations for nonexpansive mappings[END_REF], a key ingredient to establish convergence of the trajectories.

Lemma A.