Hedy Attouch
email: hedy.attouch@umontpellier.fr

Aïcha Balhag
email: aichabalhag@gmail.com

Zaki Chbani •

Hassan Riahi
email: h-riahi@uca.ac.ma

Zaki Chbani
email: chbaniz@uca.ac.ma

Accelerated gradient methods combining Tikhonov regularization with geometric damping driven by the Hessian

Keywords: Accelerated gradient methods, convex optimization, damped inertial dynamics, Hessian-driven damping, hierarchical minimization, Nesterov accelerated gradient method, Tikhonov approximation Mathematics Subject Classification (2010) 37N40, 46N10, 49M30, 65K05, 65K10, 65K15, 65L08, 65L09, 90B50, 90C25

In a Hilbert setting, for convex differentiable optimization, we consider accelerated gradient dynamics combining Tikhonov regularization with Hessian-driven damping. The Tikhonov regularization parameter is assumed to tend to zero as time tends to infinity, which preserves equilibria. The presence of the Tikhonov regularization term induces a strong convexity property which vanishes asymptotically. To take advantage of the exponential convergence rates attached to the heavy ball method in the strongly convex case, we consider the inertial dynamic where the viscous damping coefficient is taken proportional to the square root of the Tikhonov regularization parameter, and therefore also converges towards zero. Moreover, the dynamic involves a geometric damping which is driven by the Hessian of the function to be minimized, which induces a significant attenuation of the oscillations. Under an appropriate tuning of the parameters, based on Lyapunov's analysis, we show that the trajectories have at the same time several remarkable properties: they provide fast convergence of values, fast convergence of gradients towards zero, and strong convergence to the minimum norm minimizer. This study extends a previous paper by the authors where similar issues were examined but without the presence of Hessian driven damping.

Introduction

Throughout the paper, H is a real Hilbert space which is endowed with the scalar product •, • , with x 2 = x, x for x ∈ H. Given f : H → R a general convex function, which is continuously differentiable, we will develop fast gradient methods for solving the minimization problem min {f (x) : x ∈ H} .

(

) 1
Our approach is based on the convergence properties as t → +∞ of the trajectories generated by the damped inertial dynamic (TRISH) ẍ(t) + δ ε(t) ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇f (x(t)) + ε(t)x(t) = 0, and on the link between dynamical systems and the algorithms that result from their temporal discretization. We use (TRISH) as shorthand for Tikhonov regularized inertial system with Hessian-driven damping. As a basic ingredient, this system involves a nonnegative function ε(•) which enters both in the viscous damping and the Tikhonov regularization terms. We assume that lim t→+∞ ε(t) = 0, which preserves the equilibria. According to the structure of (TRISH) this makes the damping coefficient asymptotically vanish, in coordination with the Tikhonov regularization coefficient. The other basic ingredient is the Hessian driven damping term which induces several favorable properties, notably a significant reduction of the oscillations. We will show that a judicious setting of ε(•) and of the positive parameter δ ensures that the trajectories generated by (TRISH) verify the following three properties at the same time:

• rapid convergence of values (one can approach arbitrarily close to the optimal convergence rate),

• rapid convergence of the gradients towards zero,

• strong convergence towards the minimum norm element of S = argmin f . Throughout the paper, we assume that the objective function f and the Tikhonov regularization parameter ε(•) satisfy the following hypothesis:

(A)        f : H → R is convex, of class C 2 ,
∇f is Lipschitz continuous on bounded sets; S := argmin H f = ∅. We denote by x * the element of minimum norm of S; ε : [t 0 , +∞[→ R + is a nonincreasing function, of class C 1 , such that lim t→∞ ε(t) = 0.

We will explain at the end of the article how our study can be extended to the case of a convex lower semicontinuous proper function f : H → R ∪ {+∞}, and give existence and uniqueness results for the Cauchy problem associated with our dynamics.

The role of the Tikhonov regularization

Initially designed for the regularization of ill-posed inverse problems [START_REF] Tikhonov | Solution of incorrectly formulated problems and the regularization method[END_REF][START_REF] Tikhonov | Solutions of Ill-Posed Problems[END_REF], the field of application of the Tikhonov regularization was then considerably widened. The coupling of first-order in time gradient systems with a Tikhonov approximation whose coefficient tends asymptotically towards zero has been highlighted in a series of papers [START_REF] Alvarez | Asymptotic selection of viscosity equilibria of semilinear evolution equations by the introduction of a slowly vanishing term[END_REF], [START_REF] Attouch | Viscosity solutions of minimization problems[END_REF], [START_REF] Attouch | A dynamical approach to convex minimization coupling approximation with the steepest descent method[END_REF], [START_REF] Attouch | Asymptotic behavior of coupled dynamical systems with multiscale aspects[END_REF], [START_REF] Baillon | A convergence result for non-autonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming[END_REF], [START_REF] Cabot | Proximal point algorithm controlled by a slowly vanishing term: Applications to hierarchical minimization[END_REF], [START_REF] Cominetti | Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization[END_REF], [START_REF] Hirstoaga | Approximation et résolution de problèmes d'équilibre, de point fixe et d'inclusion monotone[END_REF]. Our approach builds on several previous works that have paved the way concerning the coupling of damped second-order in time gradient systems with Tikhonov approximation. First studies concerned the heavy ball with friction system of Polyak [START_REF] Polyak | Introduction to Optimization[END_REF], where the damping coefficient γ > 0 is fixed. In [START_REF] Attouch | Asymptotic control and stabilization of nonlinear oscillators with nonisolated equilibria[END_REF] Attouch and Czarnecki considered the system ẍ(t) + γ ẋ(t) + ∇f (x(t)) + ε(t)x(t) = 0.

(2)

In the slow parametrization case +∞ 0 ε(t)dt = +∞, they proved that any solution x(•) of (2) converges strongly to the minimum norm element of argmin f , see also [START_REF] Attouch | Asymptotic behavior of gradient-like dynamical systems involving inertia and multiscale aspects[END_REF], [START_REF] Cabot | Inertial gradient-like dynamical system controlled by a stabilizing term[END_REF], [START_REF] Cabot | On the long time behavior of second order differential equations with asymptotically small dissipation[END_REF], [START_REF] Jendoubi | On an asymptotically autonomous system with Tikhonov type regularizing term[END_REF]. This hierarchical minimization result contrasts with the case without the Tikhonov regularization term, where the convergence holds only for weak convergence, and the limit depends on the initial data. In the quest for a faster convergence, the following system with asymptotically vanishing damping (AVD) α,ε ẍ(t) + α t ẋ(t) + ∇f (x(t)) + ε(t)x(t) = 0,

was studied by Attouch, Chbani, and Riahi in [START_REF] Attouch | Combining fast inertial dynamics for convex optimization with Tikhonov regularization[END_REF]. It is a Tikhonov regularization of the dynamic

(AVD) α ẍ(t) + α t ẋ(t) + ∇f (x(t)) = 0, (4)
which was introduced by Su, Boyd and Candès in [START_REF] Su | A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights[END_REF]. (AVD) α is a low resolution ODE of 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] and of the Ravine method [START_REF] Attouch | From the Ravine method to the Nesterov method and vice versa: a dynamic system perspective[END_REF], [START_REF] Su | A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights[END_REF]. (AVD) α has been the subject of many recent studies which have given an in-depth understanding of the Nesterov accelerated gradient method, see [START_REF] Apidopoulos | The differential inclusion modeling the FISTA algorithm and optimality of convergence rate in the case b ≤ 3[END_REF], [START_REF] Attouch | Damped inertial dynamics with vanishing Tikhonov regularization: Strong asymptotic convergence towards the minimum norm solution[END_REF], [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF], [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF], [START_REF] Attouch | The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1/k 2[END_REF], [START_REF] Chambolle | On the convergence of the iterates of Fista[END_REF], [START_REF] May | Asymptotic for a second order evolution equation with damping and regularizing terms[END_REF], [START_REF] Su | A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights[END_REF], [START_REF] Siegel | Accelerated first-order methods: Differential equations and Lyapunov functions[END_REF], [START_REF] Wilson | A Lyapunov analysis of momentum methods in optimization[END_REF].

As an original aspect of our approach, we rely on the properties of the heavy ball with friction method of Polyak in the strongly convex case, which provides exponential convergence rates. To take advantage of this remarkable property, and adapt it to our situation, we consider the nonautonomous dynamic version of the heavy ball method which at time t is governed by the gradient of the regularized function x → f (x) + ε(t)

2

x 2 , where the Tikhonov regularization parameter satisfies ε(t) → 0 as t → +∞. This idea was first developed in [START_REF] Attouch | Damped inertial dynamics with vanishing Tikhonov regularization: Strong asymptotic convergence towards the minimum norm solution[END_REF], [START_REF] Attouch | Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution[END_REF]. Let us make this precise.

Recall that a function f :

H → R is µ-strongly convex for some µ > 0 if f -µ 2 • 2 is convex.
In this setting, we have the following exponential convergence result for the heavy ball with friction dynamic where the viscous damping coefficient is twice the square root of the modulus of strong convexity of f , see [START_REF] Polyak | Some methods of speeding up the convergence of iteration methods[END_REF]:

Theorem 1 Suppose that f : H → R is a function of class C 1 which is µ-strongly convex for some µ > 0. Let x(•) : [t 0 , +∞[→ H be a solution trajectory of ẍ(t) + 2 √ µ ẋ(t) + ∇f (x(t)) = 0. (5)
Then, the following property holds:

f (x(t)) -min H f = O e -√ µt as t → +∞.
To adapt this result to the case of a general convex differentiable function f : H → R, a natural idea is to use Tikhonov's method of regularization. This leads to consider the non-autonomous dynamic which at time t is governed by the gradient of the strongly convex function

ϕ t : H → R, ϕ t (x) := f (x) + ε(t) 2 x 2 .

The viscosity curve

ε → xε := argmin H f (•) + ε 2
• 2 will play a key role in our analysis. By definition of ϕ t , we have x ε(t) = argmin H ϕ t . The first-order optimality condition gives

∇f (x ε(t) ) + ε(t)x ε(t) = 0. ( 6
)
We call t → x ε(t) the parametrized viscosity curve. Then, replacing f by ϕ t in [START_REF] Apidopoulos | The differential inclusion modeling the FISTA algorithm and optimality of convergence rate in the case b ≤ 3[END_REF], and noticing that ϕ t is ε(t)-strongly convex, this gives the following dynamic which was introduced in [START_REF] Attouch | Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution[END_REF] and [START_REF] Attouch | Damped inertial dynamics with vanishing Tikhonov regularization: Strong asymptotic convergence towards the minimum norm solution[END_REF] (δ is a positive parameter)

(TRIGS) ẍ(t) + δ ε(t) ẋ(t) + ∇f (x(t)) + ε(t)x(t) = 0.
(TRIGS) stands shortly for Tikhonov regularization of inertial gradient systems. In order not to asymptotically modify the equilibria, it is supposed that ε(t) → 0 as t → +∞1 . This condition implies that (TRIGS) falls within the framework of the inertial gradient systems with asymptotically vanishing damping. It has been shown in [START_REF] Attouch | Damped inertial dynamics with vanishing Tikhonov regularization: Strong asymptotic convergence towards the minimum norm solution[END_REF], [START_REF] Attouch | Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution[END_REF] that a judicious tuning of ε(t) in (TRIGS) ensures both rapid convergence of values, and strong convergence of the trajectories towards the minimum norm element of S = argmin H f (which is reminiscent of the Tikhonov method).

The role of the Hessian-driven damping

As is the case with inertial dynamics which are only damped by viscous damping, the system (TRIGS) may exhibit oscillations which are undesirable from an optimization point of view. To remedy this situation, we introduce into the dynamic a geometric damping which is driven by the Hessian of the function f to be minimized. So doing, we obtain the system (TRISH). The presence of the Hessian does not entail numerical difficulties, since the Hessian intervenes in the above ODE in the form ∇ 2 f (x(t)) ẋ(t), which is nothing but the derivative wrt time of ∇f (x(t)). This explains why the time discretization of this dynamic provides first-order algorithms. The importance of the Hessian driven damping has been demonstrated in several areas. We list some of them below.

• In the field of PDEs for mechanics and physics, it is called strong damping, or geometric damping because it takes into account the geometry of the function to be minimized. In the PDE's framework, when f is quadratic, and ∇f = A is a linear elliptic operator, the strong damping involves the action of a fractional power A θ of A on the velocity vector. When θ ≥ 1 2 , this induces notably reduced oscillations. The Hessian-driven damping corresponds to θ = 1. It can be combined with various other types of damping, such as the dry friction [START_REF] Adly | Newton-type inertial algorithms for solving monotone equations governed by sums of potential and nonpotential operators[END_REF]. It also makes it possible to model shocks which are completely damped in unilateral mechanics [START_REF] Attouch | A second-order differential system with Hessian-driven damping; Application to non-elastic shock laws[END_REF].

• It has been shown in [START_REF] Attouch | From the Ravine method to the Nesterov method and vice versa: a dynamic system perspective[END_REF] and [START_REF] Shi | Understanding the acceleration phenomenon via high-resolution differential equations[END_REF] that the high resolution ODE of the Ravine and Nesterov methods exhibits the Hessian driven damping. This explains the rapid convergence of the gradients towards zero which is verified by these dynamics and algorithms [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], [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF], [START_REF] Bot | Tikhonov regularization of a second order dynamical system with Hessian damping[END_REF], [START_REF] Shi | Understanding the acceleration phenomenon via high-resolution differential equations[END_REF]. Our approach is in accordance with Nesterov [START_REF] Nesterov | How to Make the Gradients Small[END_REF], where it is conjectured that the introduction of an adapted Tikhonov regularization term helps to make the gradients small.

• The Hessian driven damping comes into the study of Newton's method in optimization. Given a general maximally monotone operator A : H → 2 H , to overcome the ill-posedness of Newton's continuous method for solving 0 ∈ A(x), the following first-order evolution system was considered by Attouch and Svaiter [START_REF] Attouch | A continuous dynamical Newton-Like approach to solving monotone inclusions[END_REF] and studied further in [START_REF] Abbas | Newton-like dynamics and forward-backward methods for structured monotone inclusions in Hilbert spaces[END_REF], [START_REF] Attouch | A dynamic approach to a proximal-Newton method for monotone inclusions in Hilbert Spaces, with complexity O(1/n 2 )[END_REF]. Formally, this system is written as

γ(t) ẋ(t) + β d dt (A(x(t))) + A(x(t)) = 0.
It can be considered as a continuous version of the Levenberg-Marquardt method, which acts as a regularization of the Newton method. Under a fairly general assumption on the regularization parameter γ(•), this system is well posed and generates trajectories that converge weakly to equilibria. Thus, (TRISH) and its nonsmooth extension can be considered as an inertial and regularized version of this system when A is the subdifferential of a convex lower semicontinuous proper function.

A model result

In section 3, we will prove the following result in the case ε(t) = 1 t r . It is expressed with the help of the parametrized viscosity curve which converges strongly to the minimum norm solution.

Theorem 2 Take 0 < r < 2, δ > 2, β > 0. Let x : [t 0 , +∞[→ H be a solution trajectory of ẍ(t) + δ t r 2 ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇f (x(t)) + 1 t r x(t) = 0. ( 7
)
Then, we have fast convergence of the values, fast convergence of the gradients towards zero, and strong convergence of the trajectory to the minimum norm solution, with the following rates:

• f (x(t)) -min H f = O 1 t r as t → +∞ ; • +∞ t0 t 3r-2 2 ∇f (x(t)) 2 dt < +∞; • x(t) -x ε(t) 2 = O 1 t 2-r 2 as t → +∞.
This is the first time that these three properties have been obtained within the same dynamic. Let us note that by taking r close to 2, one obtains convergence rates comparable to the most recent results concerning the introduction of the Hessian driven damping in the dynamic associated with the accelerated gradient of Nesterov. Precisely, letting r → 2 in the formulas above gives f (x(t))-min H f = O(1/t 2 ), and +∞ t0 t 2 ∇f (x(t)) 2 dt < +∞. Thus, by taking r sufficiently close to 2, we can obtain convergence rates arbitrarily close to these rates. The case r = 2, which corresponds to the Nesterov accelerated gradient method, is critical: in this case, the strong convergence towards the minimum norm solution is an open question. The above results show the balance between fast convergence of values and strong convergence to the minimum norm solution.

Contents

The paper is organized as follows. In section 2, for a general Tikhonov regularization parameter ε(•), we study the asymptotic convergence properties of the solution trajectories of (TRISH). Based on Lyapunov analysis, we show their strong convergence to the minimum norm element of S, and establish the convergence rates of the values and integral estimates of the gradients. In section 3, we apply these results to the particular case ε(t) = 1 t r , 0 < r < 2, and obtain fast convergence results. Section 4 contains numerical illustrations. Section 5 gives indications concerning the extension of our study to the nonsmooth case, and provides existence and uniqueness results for the Cauchy problem associated with the considered dynamics. We conclude with a perspective and open questions.

Convergence results via Lyapunov analysis

Given a general regularization parameter ε(•), we successively present the idea guiding the Lyapunov analysis, then some preparatory lemmas, and finally the detailed proof. In the next section, we will particularize our results to the case ε(t) = 1 t r , 0 < r < 2, and obtain fast convergence results.

General idea of the proof

As we already mentioned, the function

ϕ t : H → R, ϕ t (x) := f (x) + ε(t) 2 x 2 (8)
plays a central role in the Lyapunov analysis, via its strong convexity property. Thus, it is convenient to reformulate (TRISH) with the help of the function ϕ t , which gives

(TRISH) ẍ(t) + δ ε(t) ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇ϕ t (x(t)) = 0, (9)
where δ, β are positive parameters. We recall that ε : [t 0 , +∞[→ R + is a nonincreasing function of class C 1 , such that lim t→+∞ ε(t) = 0. In the mathematical analysis of inertial gradient dynamics and algorithms with Hessian-driven damping, the basic equality

d dt ∇f (x(t)) = ∇ 2 f (x(t)) ẋ(t) (10)
makes these systems relevant to first-order methods, a crucial property for numerical purposes.

In the presence of the Tikhonov term, to keep the structural property attached to [START_REF] Attouch | First order optimization algorithms via inertial systems with Hessian driven damping[END_REF], let us introduce the following variant of (TRISH) where the above relation comes with ϕ t instead of f :

(TRISHE) ẍ(t) + δ ε(t) ẋ(t) + β d dt (∇ϕ t (x(t))) + ∇ϕ t (x(t)) = 0. ( 11
)
Adding the suffix E after TRISH recalls that the dynamic has been adapted to take advantage of the Equality in [START_REF] Attouch | First order optimization algorithms via inertial systems with Hessian driven damping[END_REF], with ϕ t instead of f . To encompass these two dynamic systems, we consider

ẍ(t) + δ ε(t) ẋ(t) + β d dt [∇ϕ t (x(t)) + (p -1)ε(t)x(t)] + ∇ϕ t (x(t)) = 0, (12)
where the parameter p ∈ [0, 1]. When p = 0 we get (TRISH), and for p = 1 we get (TRISHE). Given p ∈ [0, 1], let us introduce the real-valued function t ∈ [t 0 , +∞[ → Ep(t) ∈ R + that plays a key role in our Lyapunov analysis. It is defined by

Ep(t) := ϕ t (x(t)) -ϕ t (x ε(t) ) + 1 2 vp(t) 2 (13)
where ϕ t has been defined in ( 8), x ε(t) in ( 6), and

vp(t) := λ ε(t) x(t) -x ε(t) + ẋ(t) + β [∇ϕ t (x(t)) + (p -1)ε(t)x(t)] , (14)
with 0 ≤ λ < δ. We will show that under a judicious setting of parameters, Ep(•) satisfies the first-order differential inequality

Ėp(t) + µ(t)Ep(t) + β 2δ (δ -λ) ∇ϕ t (x(t)) 2 ≤ x * 2 2 G(t), (15)
where

µ(t) := - ε(t) 2ε(t) + (δ -λ) ε(t), (16)
and

G(t) := (λc + 2a)λ ε2 (t) ε 3 2 (t) -ε(t) + (1 -p)βλ(δ -λ)ε 2 (t).
Since µ(t) > 0, this will allow us to estimate the rate of convergence of Ep(t) towards zero. In turn, this provides convergence rates of values and trajectories, as the following lemma shows.

Lemma 1 Let x(•) : [t 0 , +∞[→ H be a solution trajectory of the damped inertial dynamic [START_REF] Attouch | Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity[END_REF], and t ∈ [t 0 , +∞[ → Ep(t) ∈ R + be the energy function defined in [START_REF] Attouch | Combining fast inertial dynamics for convex optimization with Tikhonov regularization[END_REF]. Then, the following estimates are satisfied: for any t ≥ t 0 ,

f (x(t)) -min H f ≤ Ep(t) + ε(t) 2 x * 2 ; ( 17
) x(t) -x ε(t) 2 ≤ 2Ep(t) ε(t) . (18)
Therefore, x(t) converges strongly to x * as soon as lim t→+∞

Ep(t) ε(t) = 0. Proof i) According to the definition of ϕ t , we have f (x(t)) -min H f = ϕ t (x(t)) -ϕ t (x * ) + ε(t) 2 x * 2 -x(t) 2 = ϕ t (x(t)) -ϕ t (x ε(t) ) +   ϕt(xε(t)) -ϕ t (x * ) ≤0    + ε(t) 2 x * 2 -x(t) 2 ≤ ϕ t (x(t)) -ϕ t (x ε(t) ) + ε(t) 2 x * 2 .
By definition of Ep(t) we have

ϕ t (x(t)) -ϕ t (x ε(t) ) ≤ Ep(t) (19)
which, combined with the above inequality, gives [START_REF] Attouch | Asymptotic behavior of gradient-like dynamical systems involving inertia and multiscale aspects[END_REF].

ii) By the strong convexity of ϕ t , and x ε(t) := argmin H ϕ t , we have

ϕ t (x(t)) -ϕ t (x ε(t) ) ≥ ε(t) 2 x(t) -x ε(t) 2 .
By combining the inequality above with [START_REF] Attouch | From the Ravine method to the Nesterov method and vice versa: a dynamic system perspective[END_REF], we get

Ep(t) ≥ ε(t) 2 x(t) -x ε(t) 2 ,
which gives (18).

Preparatory results for Lyapunov analysis

The parametrized viscosity curve t → x ε(t) plays a central role in the definition of Ep(•), and therefore in the Lyapunov analysis. We review below some its topological and differential properties.

Topological properties

The following properties are immediate consequences of the classical properties of the Tikhonov regularization (see [START_REF] Attouch | Viscosity solutions of minimization problems[END_REF] for a general overview of viscosity methods), and of lim t→+∞ ε(t) = 0:

• ∀t ≥ t 0 x ε(t) ≤ x * (20)
• lim

t→+∞ x ε(t) -x * = 0 where x * = proj argmin f 0. (21)

Differential properties

To evaluate the terms According to [START_REF] Attouch | A dynamical approach to convex minimization coupling approximation with the steepest descent method[END_REF], [START_REF] Hirstoaga | Approximation et résolution de problèmes d'équilibre, de point fixe et d'inclusion monotone[END_REF], [START_REF] Torralba | Convergence epigraphique et changements d'échelle en analyse variationnelle et optimisation[END_REF], the viscosity curve is Lipschitz continuous on the compact intervals of ]0, +∞[. So it is absolutely continuous, and almost everywhere differentiable. Based on these properties we have the following lemma, which was established in [START_REF] Attouch | Damped inertial dynamics with vanishing Tikhonov regularization: Strong asymptotic convergence towards the minimum norm solution[END_REF], and which we reproduce here for ease of reading.

d dt ϕ t (x ε(t) ) and

Lemma 2

The following properties are satisfied:

i) For each t ≥ t 0 , d dt ϕ t (x ε(t) ) = 1 2 ε(t) x ε(t) 2 .
ii) The function t → x ε(t) is Lipschitz continuous on the compact intervals of ]t 0 , +∞[, hence almost everywhere differentiable, and the following inequality holds: for almost every

t ≥ t 0 d dt x ε(t) 2 ≤ - ε(t) ε(t) d dt x ε(t) , x ε(t) .
Therefore, for almost every

t ≥ t 0 d dt x ε(t) ≤ - ε(t) ε(t) x ε(t) .
Proof i) We use some classical differentiability properties of the Moreau envelope. We have

ϕ t (x ε(t) ) = inf ξ∈H f (ξ) + ε(t) 2 ξ -0 2 = f 1 ε(t) (0),
where, for any θ > 0, the Moreau envelope f θ : H → R is defined by

f θ (x) = min ξ∈H f (ξ) + 1 2θ
x -ξ 2 .

(

) 22
Recall that infimum in the above expression is achieved at the unique point prox θf (x), i.e.

f θ (x) = f (prox θf (x)) + 1 2θ x -prox θf (x) 2 . ( 23
)
One can consult [28, section 12.4] for more details on the Moreau envelope. Since

d dθ f θ (x) = -1 2 ∇f θ (x) 2 , (see [7, Appendix, Lemma 3]
), we have:

d dt f θ(t) (x) = - θ(t) 2 ∇f θ(t) (x) 2 .
Therefore,

d dt ϕ t (x ε(t) ) = d dt f 1 ε(t) (0) = 1 2 ε(t) ε 2 (t) ∇f 1 ε(t) (0) 2 . ( 24
)
On the other hand, we have

∇ϕ t (x ε(t) ) = 0 ⇐⇒ ∇f (x ε(t) ) + ε(t)x ε(t) = 0 ⇐⇒ x ε(t) = prox 1 ε(t) f (0). Since ∇f 1 ε(t) (0) = ε(t) 0 -prox 1 ε(t) f (0) , we get ∇f 1 ε(t) (0) = -ε(t)x ε(t)
. This combined with ( 24) gives

d dt ϕ t (x ε(t) ) = 1 2 ε(t) x ε(t) 2 . ii) We have -ε(t)x ε(t) = ∇f (x ε(t) ) and -ε(t + h)x ε(t+h) = ∇f (x ε(t+h) ).
According to the monotonicity of ∇f, we have

ε(t)x ε(t) -ε(t + h)x ε(t+h) , x ε(t+h) -x ε(t) ≥ 0, which implies -ε(t) x ε(t+h) -x ε(t) 2 + (ε(t) -ε(t + h)) x ε(t+h) , x ε(t+h) -x ε(t) ≥ 0.
After division by h 2 , we obtain

(ε(t) -ε(t + h)) h x ε(t+h) , x ε(t+h) -x ε(t) h ≥ ε(t) x ε(t+h) -x ε(t) h 2 .
We now rely on the differentiability properties of the viscosity curve → x , which have been recalled above. It is Lipschitz continuous on the compact intervals of ]0, +∞[, so almost everywhere differentiable. Therefore, the mapping t → x (t) satisfies the same differentiability properties. By letting h → 0, we obtain that, for almost every

t ≥ t 0 -ε(t) x ε(t) , d dt x ε(t) ≥ ε(t) d dt x ε(t) 2 ,
which gives the claim. The last statement follows from Cauchy-Schwarz inequality.

Lyapunov analysis of (TRISH): main theorem and its proof

Take p ∈ [0, 1]. In the Lyapunov analysis of

ẍ(t) + δ ε(t) ẋ(t) + β d dt [∇ϕ t (x(t)) + (p -1)ε(t)x(t)] + ∇ϕ t (x(t)) = 0, (25)
we assume that the Tikhonov regularization parameter ε(•) satisfies the following growth condition.

Definition 1 The Tikhonov regularization parameter t → ε(t) satisfies the condition (Hp) if there exists a > 1, c > 2, λ > 0 and t 1 ≥ t 0 such that for all t ≥ t 1 , (Hp)

d dt 1 ε(t) ≤ min 2λ -δ , 1 2 δ - a + 1 a λ
, and δβ ≤ 1

ε(t) ,
where, in the above inequality, it is supposed that the parameter λ is such that δ 2 < λ < δ and satisfies

• F or p / ∈ [0, 1 2 ] and c > max 2, 1+ √ 2(1-p) 2p-1 δ 2 < λ < min a a+1 δ, δ+ √ δ 2 -4(1-p) 2 when 2 √ 1 -p < δ ≤ √ 2 -1 c , 1 2 δ + 1 c + (δ + 1 c ) 2 -2 < λ < min a a+1 δ, δ+ √ δ 2 -4(1-p) 2 when δ > √ 2 -1 c . • For p ∈ [0, 1 2 ] 1 2 δ + 1 c + (δ + 1 c ) 2 -2 < λ < min a a+1 δ, δ+ √ δ 2 -4(1-p) 2 when δ > 2 (1 -p).
Remark 1 Integrating the differential inequality (Hp) shows that the damping coefficient in [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF] (which is proportional to ε(t)) must be greater than or equal to C/t for some positive constant C. This is consistent with the theory of inertial gradient systems with time-dependent viscosity coefficient, which shows that the asymptotic optimization property is valid provided that the integral of the viscous damping coefficient over [t 0 , +∞[ be infinite, [START_REF] Attouch | Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity[END_REF], [START_REF] Cabot | On the long time behavior of second order differential equations with asymptotically small dissipation[END_REF]. See also [START_REF] Attouch | Damped inertial dynamics with vanishing Tikhonov regularization: Strong asymptotic convergence towards the minimum norm solution[END_REF], [START_REF] Attouch | Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution[END_REF] where a similar growth condition on the Tikhonov parameter is considered.

For ease of reading, let us recall the functions that enter the Lyapunov analysis:

Ep(t) := ϕ t (x(t)) -ϕ t (x ε(t) ) + 1 2 vp(t) 2 (26)
vp(t) := λ ε(t) x(t) -x ε(t) + ẋ(t) + β [∇ϕ t (x(t)) + (p -1)ε(t)x(t)] (27)
µ(t) := - ε(t) 2ε(t) + (δ -λ) ε(t) (28)
γ(t) := exp t t1 µ(s)ds . (29)
We can now state our main convergence result.

Theorem 3 Let x(•) : [t 0 , +∞[→ H be a solution trajectory of the system [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF]. Take δ > 2 (1 -p). Suppose that ε(•) satisfies the condition (Hp). Then, the following properties are satisfied: for all t ≥ t 1

Ep(t) ≤ x * 2 2γ(t) t t1 G(s)γ(s)ds + γ(t 1 )Ep(t 1 ) γ(t) , (30)
t t1 ∇ϕs(x(s)) 2 ds ≤ 2δ β(δ -λ) Ep(t 1 ) + δ x * 2 β(δ -λ) t t1 G(s)ds, (31)
where

G(t) = (λc + 2a)λ ε2 (t) ε 3 2 (t) -ε(t) + (1 -p)βλ(δ -λ)ε 2 (t). ( 32
)
Proof Since the mapping

Ėp(t) = ∇ϕ t (x(t)), ẋ(t) + 1 2 ε(t) x(t) 2 - 1 2 ε(t) x ε(t) 2 + vp(t), vp(t) . ( 33
)
According to the definition of vp (see [START_REF] Baillon | A convergence result for non-autonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming[END_REF]), and the equation ( 25), the derivation of vp gives

vp(t) = λ 2 ε(t) ε(t) x(t) -x ε(t) + λ ε(t) ẋ(t) -λ ε(t) d dt x ε(t) + ẍ(t) + β d dt (∇ϕ t (x(t)) + (p -1)ε(t)x(t)) = λ 2 ε(t) ε(t) x(t) -x ε(t) + (λ -δ) ε(t) ẋ(t) -λ ε(t) d dt x ε(t) -∇ϕ t (x(t)).
Let us write shortly Ap(t) := ∇ϕ t (x(t)) + (p -1)ε(t)x(t). We get

vp(t), vp(t) = λ 2 ε(t) ε(t) x(t) -x ε(t) + (λ -δ) ε(t) ẋ(t) -λ ε(t) d dt x ε(t) -∇ϕ t (x(t)), vp(t) = λ 2 2 ε(t) x(t) -x ε(t) 2 + λ ε(t) 2 ε(t) + (λ -δ) ε(t) x(t) -x ε(t) , ẋ(t) + (λ -δ) ε(t) ẋ(t) 2 + β (λ -δ) ε(t) Ap(t), ẋ(t) C0 +λ β ε(t) 2 ε(t) -ε(t) ∇ϕ t (x(t)), x(t) -x ε(t) ) =D0 -∇ϕ t (x(t)), ẋ(t) , -β ∇ϕ t (x(t)), Ap(t) -βλ ε(t) Ap(t), d dt x ε(t) -λ 2 ε(t) d dt x ε(t) , x(t) -x ε(t) -λ ε(t) d dt x ε(t) , ẋ(t) + (p -1)λβ 2 ε(t) ε(t) x(t) -x ε(t) , x(t) .
(34) Since ϕ t is ε(t)-strongly convex, we have

ϕ t (x ε(t) ) -ϕ t (x(t)) ≥ ∇ϕ t (x(t)), x ε(t) -x(t) + ε(t) 2 x(t) -x ε(t) 2 .
Recall that ε(t) is nonincreasing, i.e. ε(t) ≤ 0. Therefore, by using the above estimation, we get

D 0 ≤ β ε(t) 2 ε(t) -ε(t) ϕ t (x(t)) -ϕ t (x ε(t) ) + 1 2 β ε(t) 2 ε(t) -ε(t) ε(t) x(t) -x ε(t) 2 . ( 35
)
For all a > 1 we have the following elementary inequalities

-λ ε(t) d dt x ε(t) , ẋ(t) ≤ λ ε(t) 2a ẋ(t) 2 + aλ ε(t) 2 d dt x ε(t) 2 ( 36
) -βλ ε(t) Ap(t), d dt x ε(t) ≤ λβ 2 ε(t) 2a Ap(t) 2 + aλ ε(t) 2 d dt x ε(t) 2 . ( 37
)
Similarly, for all b > 0,

-λ 2 ε(t) d dt x ε(t) , x(t) -x ε(t) ≤ bλ ε(t) 2 d dt x ε(t) 2 + λ 3 ε 3 2 (t) 2b x(t) -x ε(t) 2 .
(

) 38
The coefficients a and b will be adjusted later, conveniently. Note that

C 0 = (λ -δ) ε(t) ẋ(t) 2 + β Ap(t), ẋ(t) = (λ -δ) ε(t) 2 ẋ(t) 2 + ẋ(t) + βAp(t) 2 -β 2 Ap(t) 2 . ( 39
) -β ∇ϕ t (x(t)), Ap(t) = - β 2 ∇ϕ t (x(t)) 2 + Ap(t) 2 -∇ϕ t (x(t)) -Ap(t) 2 = - β 2 ∇ϕ t (x(t)) 2 + Ap(t) 2 -(p -1) 2 ε 2 (t) x(t) 2 . ( 40
)
By combining the inequalities ( 35)-( 36)-( 37)-( 38)-( 39)-( 40) with [START_REF] Cabot | Proximal point algorithm controlled by a slowly vanishing term: Applications to hierarchical minimization[END_REF], we obtain

vp(t), vp(t) ≤ λ β ε(t) 2 ε(t) -ε(t) ϕ t (x(t)) -ϕ t (x ε(t) ) +λ ε(t) 2 ε(t) + (λ -δ) ε(t) x(t) -x ε(t) , ẋ(t) + λ 2 2 ε(t) + λε(t) 2 β ε(t) 2 ε(t) -ε(t) + λ 3 ε 3 2 (t) 2b x(t) -x ε(t) 2 + 1 2 1 + 1 a λ -δ ε(t) ẋ(t) 2 + 1 2 (λ -δ) ε(t) ẋ(t) + βAp(t) 2 + 1 2 (2a + b) λ ε(t) d dt x ε(t) 2 + (p -1)λβ 2 ε(t) ε(t) x(t) -x ε(t) , x(t) + β 2 βλ ε(t) a -1 -β(λ -δ) ε(t)) Ap(t) 2 -∇ϕ t (x(t)), ẋ(t) + β 2 (p -1) 2 ε 2 (t) x(t) 2 - β 2 ∇ϕ t (x(t)) 2 . (41)
Combining ( 41) with [START_REF] Cabot | Inertial gradient-like dynamical system controlled by a stabilizing term[END_REF], the terms ∇ϕ t (x(t)), ẋ(t) cancel each other out, which gives

Ėp(t) ≤ λ β ε(t) 2 ε(t) -ε(t) ϕ t (x(t)) -ϕ t (x ε(t) ) + 1 2 ε(t) + β(p -1) 2 ε 2 (t) x(t) 2 - 1 2 ε(t) x ε(t) 2 + λ ε(t) 2 ε(t) + (λ -δ) ε(t) x(t) -x ε(t) , ẋ(t) + λ 2 2 ε(t) + λε(t) 2 β ε(t) 2 ε(t) -ε(t) + λ 3 ε 3 2 (t) 2b x(t) -x ε(t) 2 (42)
+ 1 2 1 + 1 a λ -δ ε(t) ẋ(t) 2 + 1 2 (λ -δ) ε(t) ẋ(t) + βAp(t) 2 + 1 2 (2a + b) λ ε(t) d dt x ε(t) 2 + (p -1)λβ 2 ε(t) ε(t) x(t) -x ε(t) , x(t) + β 2 βλ ε(t) a -1 -β(λ -δ) ε(t)) Ap(t) 2 - β 2 ∇ϕ t (x(t)) 2 .
To build the differential inequality satisfied by Ep(•), let us majorize

µ(t)Ep(t) = µ(t) ϕ t (x(t)) -ϕ t (x ε(t) ) + µ(t) 2 vp(t) 2 = µ(t) ϕ t (x(t)) -ϕ t (x ε(t) ) + µ(t)λ 2 ε(t) 2 x(t) -x ε(t) 2 + µ(t) 2 ẋ(t) + βAp(t) 2 +µ(t)λ ε(t) x(t) -x ε(t) , ẋ(t) + βAp(t) ≤ µ(t) ϕ t (x(t)) -ϕ t (x ε(t) ) + µ(t)λ 2 ε(t) 2 x(t) -x ε(t) 2 + µ(t) 2 ẋ(t) + βAp(t) 2 +µ(t)λ ε(t) x(t) -x ε(t) , ẋ(t) + βµ(t)λ ε(t) x(t) -x ε(t) , ∇ϕ t (x(t)) +(p -1)βµ(t)λε(t) ε(t) x(t) -x ε(t) , x(t) ≤ µ(t) ϕ t (x(t)) -ϕ t (x ε(t) ) + µ(t)λ 2 ε(t) x(t) -x ε(t) 2 + µ(t) 2 ẋ(t) + βAp(t) 2 +µ(t)λ ε(t) x(t) -x ε(t) , ẋ(t) + β 2 2 µ(t) ∇ϕ t (x(t)) 2 +(p -1)βµ(t)λε(t) ε(t) x(t) -x ε(t) , x(t) . ( 43
)
By adding ( 42) and ( 43), using µ(t) = -ε(t) 2ε(t) + (δ -λ) ε(t), and after simplification, we get

Ėp(t) + µ(t)Ep(t) ≤ ε(t) - ε(t) 2ε 3 2 (t) + (δ -2λ) + λβ ε(t) 2ε(t) ϕ t (x(t)) -ϕ t (x ε(t) ) + 1 2 ε(t) + β(p -1) 2 ε 2 (t) x(t) 2 - 1 2 ε(t) x ε(t) 2 λ 4 β ε(t) ε(t) + 2 2δλ -2λ 2 -1 ε 3 2 (t) + 2 λ 2 b ε 3 2 (t) x(t) -x ε(t) 2 + 1 2 (1 + 1 a )λ -δ ε(t) ẋ(t) 2 - ε(t) 4ε(t) ẋ(t) + βAp(t) 2 + 1 2 (2a + b) λ ε(t) d dt x ε(t) 2 + (p -1)βλ(δ -λ)ε 2 (t) x(t) -x ε(t) , x(t) + β 2 βλ ε(t) a -1 -β(λ -δ) ε(t)) Ap(t) 2 + β 2 -1 -β ε(t) 2ε(t) + β (δ -λ) ε(t) ∇ϕ t (x(t)) 2 . ( 44
) Since ε(•) is nonincreasing, - ε(t) 4ε(t) ẋ(t) + βAp(t) 2 ≤ - ε(t) 2ε(t) ẋ(t) 2 - β 2 ε(t) 2ε(t) Ap(t) 2 .
Combining this inequality with x(t) -x ε(t) , x(t) = 1 2

x(t) -x ε(t) 2 + x(t) 2 -x ε(t) 2 , we get Ėp(t) + µ(t)Ep(t) ≤ ε(t) - ε(t) 2ε 3 2 (t) + (δ -2λ) + λβ ε(t) 2ε(t) ϕ t (x(t)) -ϕ t (x ε(t) ) + 1 2 ε(t) + β(p -1) (p -1 + λ(δ -λ)) ε 2 (t) x(t) 2 - 1 2 ε(t) + (p -1)βλ(δ -λ)ε 2 (t) x ε(t) 2 + λ 4 β ε(t) ε(t) + 2 2δλ -2λ 2 -1 ε 3 2 (t) + 2 λ 2 b ε 3 2 (t) + 2(p -1)β(δ -λ)ε 2 (t) x(t) -x ε(t) 2 + 1 2 (1 + 1 a )λ -δ ε(t) - ε(t) 2ε(t) ẋ(t) 2 + 1 2 (2a + b) λ ε(t) d dt x ε(t) 2 + β 2 βλ ε(t) a -1 -β(λ -δ) ε(t)) - β ε(t) ε(t) Ap(t) 2 + β 2 -1 -β ε(t) 2ε(t) + β (δ -λ) ε(t) ∇ϕ t (x(t)) 2 . ( 45
)
By using Lemma 2, we have

d dt x ε(t) 2 ≤ ε2 (t) ε 2 (t) x ε(t) 2 ≤ ε2 (t) ε 2 (t) x * 2 ,
which gives

Ėp(t) + µ(t)Ep(t) ≤ ε(t) - ε(t) 2ε 3 2 (t) + (δ -2λ) + λβ ε(t) 2ε(t) ϕ t (x(t)) -ϕ t (x ε(t) ) + 1 2 ε(t) + β(p -1) (p -1 + λ(δ -λ)) ε 2 (t) x(t) 2 + λ 4 β ε(t) ε(t) + 2 2δλ -2λ 2 -1 ε 3 2 (t) + 2 λ 2 b ε 3 2 (t) +2(p -1)β(δ -λ)ε 2 (t) x(t) -x ε(t) 2 + 1 2 (1 + 1 a )λ -δ ε(t) - ε(t) 2ε(t) ẋ(t) 2 + 1 2 (2a + b) λ ε2 (t) ε 3 2 (t) -ε(t) + (1 -p)βλ(δ -λ)ε 2 (t) x ε(t) 2 + β 2 βλ ε(t) a -1 -β(λ -δ) ε(t) - β ε(t) ε(t) Ap(t) 2 + β 2 -1 -β ε(t) 2ε(t) + β (δ -λ) ε(t) ∇ϕ t (x(t)) 2 . ( 46
)
Let us make precise the choice of the parameter b and take

b := c 2 λ with c > 2.
Let us analyze the sign of the coefficients involved in [START_REF] Siegel | Accelerated first-order methods: Differential equations and Lyapunov functions[END_REF]. Since ε(t) is nonincreasing and p ≤ 1,

Ėp(t) + µ(t)Ep(t) ≤ ε(t)      - ε(t) 2ε 3 2 (t) + (δ -2λ) =A +λβ ε(t) 2ε(t) ≤0      ϕ t (x(t)) -ϕ t (x ε(t) ) + 1 2    ε(t) + β(p -1) (p -1 + λ(δ -λ)) ε 2 (t) =B    x(t) 2 + λ 4      β ε(t) ε(t) ≤0 +2      2 δ + 1 c λ -2λ 2 -1 =C      ε 3 2 (t) + 2 (p -1)β(δ -λ) ≤0 since δ≥λ ε 2 (t)      x(t) -x ε(t) 2 + 1 2 (1 + 1 a )λ -δ ε(t) - ε(t) 2ε(t) =D ẋ(t) 2 + β 2 β λ a + δ -λ ε(t) -1 - β ε(t) ε(t) ∆1 Ap(t) 2 + 1 2 (2a + cλ) λ ε2 (t) ε 3 2 (t) -ε(t) + (1 -p)βλ(δ -λ)ε 2 (t) x ε(t) 2 + β 2 -1 -β ε(t) 2ε(t) + β (δ -λ) ε(t) ∆2 ∇ϕ t (x(t)) 2 . ( 47
)
• Condition (Hp) implies that

d dt 1 ε(t) ≤ (2λ -δ), d dt 1 ε(t) ≤ 1 2 δ -(1 + 1 a )λ and δβ ≤ 1 ε(t)
.

So, we have

A = - ε(t) 2ε 3 2 (t) + (δ -2λ) = d dt 1 ε(t) + (δ -2λ) ≤ 0. • According to δ > 2 (1 -p) and λ < δ+ √ δ 2 -4(1-p) 2 , we have p -1 + δλ -λ 2 ≥ 0. Therefore, B = ε(t) ≤0 +β(p -1) (p -1 + λ(δ -λ)) ε 2 (t) ≤ β (p -1) ≤0 p -1 + δλ -λ 2 ε 2 (t) ≤ 0. • When δ ≤ √ 2 -1 c we have 2 δ + 1 c λ -2λ 2 -1 ≤ 2 √ 2λ -2λ 2 -1 = -( √ 2λ -1) 2 ≤ 0. When δ > √ 2 -1 c , we have (δ + 1 c )λ -λ 2 - 1 2 ≤ 0, because λ ≥ 1 2 δ + 1 c + (δ + 1 c ) 2 -2 . Therefore (δ + 1 c )λ -λ 2 -1 2 ≤ 0, which implies that C = 2 δ + 1 c λ -2λ 2 -1 ≤ 0.
• Condition (Hp) implies that

D = 1 2 (1 + 1 a )λ -δ - 1 2 ε(t) ε 3 2 (t) = d dt 1 ε(t) + 1 2 (1 + 1 a )λ -δ ≤ 0.
• For nonpositivity of ∆ 1 , we use δβ ≤ 1

ε(t)
and

d dt 1 √ ε(t) ≤ 1 2 δ -(1 + 1 a )λ to conclude ∆ 1 = β λ a + δ -λ ε(t) -1 - β ε(t) ε(t) = β λ a + δ -λ ε(t) -1 + 2β ε(t) - ε(t) 2ε 3 2 (t) - 1 2 δ -(1 + 1 a )λ ≤0 +β δ -(1 + 1 a )λ ε(t) ≤ 2β (δ -λ) ε(t) -1 = β (δ -2λ) ≤0 ε(t) + βδ ε(t) -1 ≤0 ≤ 0.
• Finally,

∆ 2 = λβ ε(t) -1 + β ε(t) (δ -2λ) + d dt 1 ε(t) ≤0 ≤ λ β ε(t) - 1 δ + λ δ -1 ≤ λ δ
It follows from the estimate (47) that

Ėp(t) + µ(t)Ep(t) ≤ 1 2 (2a + cλ) λ ε2 (t) ε 3 2 (t) -ε(t) + (1 -p)βλ(δ -λ)ε 2 (t) x ε(t) 2 + β 2 λ δ -1 ∇ϕ t (x(t)) 2 . Since x ε(t) ≤ x * , we get Ėp(t) + µ(t)Ep(t) ≤ x * 2 2 G(t) + β 2 λ δ -1 ∇ϕ t (x(t)) 2 , ( 48
)
where

G(t) = (λc + 2a)λ ε2 (t) ε 3 2 (t) -ε(t) + (1 -p)βλ(δ -λ)ε 2 (t).
We have

By integrating ( 50) on [t 1 , t], and dividing by γ(t), we obtain our claim ( 30)

Ep(t) ≤ x * 2 2γ(t) t t1 G(s)γ(s)ds + γ(t 1 )Ep(t 1 ) γ(t) . (51)
Coming back to [START_REF] Su | A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights[END_REF], we get by integration

Ep(t) -Ep(t 1 ) + t t1 µ(s)Ep(s)ds + β 2 1 - λ δ t t1 ∇ϕs(x(s)) 2 ds ≤ x * 2 2 t t1
G(s)ds.

Our assertion [START_REF] Brézis | Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution[END_REF]

(∇ϕ t (x(t)))) + ∇ϕ t (x(t)) = 0. ( 52
)
Let us assume that there exists a, c > 1 and t 1 ≥ t 0 such that for all t ≥ t 1 , (H 1 ) holds. Then,

E 1 (t) ≤ x * 2 2 t t1 (λc + 2a)λ ε2 (s) ε 3 2 (s) -ε(s) γ(s) ds γ(t) + γ(t 1 )E 1 (t 1 ) γ(t) (53)
where γ(t) = exp t t1 µ(s)ds and µ(t) = -ε(t) 2ε(t) + (δ -λ) ε(t).

Corollary 2 Let x(•) : [t 0 , +∞[→ H be a solution trajectory of the system (TRISH) with δ > 2

ẍ(t) + δ ε(t) ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇ϕ t (x(t)) = 0. ( 54
)
Let us assume that there exists a, c > 1 and t 1 ≥ t 0 such that for all t ≥ t 1 , condition (H 0 ) holds. Then,

E 0 (t) ≤ x * 2 2 t t1 (λc + 2a)λ ε2 (s) ε 3 2 (s) -ε(s) + βλ(δ -λ)ε 2 (t) γ(s) ds γ(t) + γ(t 1 )E 0 (t 1 ) γ(t) (55)
where γ(t) = exp t t1 µ(s)ds and µ(t) = -ε(t) 2ε(t) + (δ -λ) ε(t).

Particular cases

Take ε(t) = 1 t r , 0 < r < 2, t 0 > 0, and consider the systems (TRISHE) and (TRISH). The convergence rate of the values and the strong convergence to the minimum norm solution will be obtained by particularizing Theorem 3 to these situations. In these cases, the integrals that enter into the formulation of Theorem 3 can be calculated explicitly. In fact, obtaining sharp convergence rate of the gradients requires another Lyapunov analysis based on the function Ep defined by

Ep(t) := ϕ t (x(t)) -ϕ t (x ε(t) ) + 1 2 ẋ(t) + β [∇ϕ t (x(t)) + (p -1)ε(t)x(t)] 2 . ( 56
)
We can notice that when λ = 0, we have Ep(t) = Ep(t). So, with λ = 0, the estimation [START_REF] Nesterov | Introductory lectures on convex optimization: A basic course[END_REF] becomes

Ėp(t) = Ėp(t) ≤ 1 2 ε(t) + β(p -1) 2 ε 2 (t) x(t) 2 - 1 2 ε(t) x ε(t) 2 - δ 2 ε(t) ẋ(t) 2 - δ 2 ε(t) ẋ(t) + βAp(t) 2 + β 2 -1 + βδ ε(t)) Ap(t) 2 - β 2 ∇ϕ t (x(t)) 2 .
By supposing δβ ≤ 1

ε(t)
, we conclude that

Ėp(t) ≤ 1 2 ε(t) + β(p -1) 2 ε 2 (t) x(t) 2 - 1 2 ε(t) x * 2 - δ 2 ε(t) ẋ(t) 2 - δ 2 ε(t) ẋ(t) + βAp(t) 2 - β 2 ∇ϕ t (x(t)) 2 .
(57)

System (TRISHE)

Theorem 4 Take ε(t) = 1 t r and 0 < r < 2. Let x : [t 0 , +∞[→ H be a solution trajectory of

ẍ(t) + δ t r 2 + β t r ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇f (x(t)) + 1 t r - rβ t r+1 x(t) = 0. ( 58
)
Then, we have convergence of values, strong convergence to the minimum norm solution, and

f (x(t)) -min H f = O 1 t r as t → +∞; (59) x(t) -x ε(t) 2 = O 1 t 2-r 2 as t → +∞. ( 60
) ẋ(t) + β∇f (x(t)) = O 1 t min(( 2+r 4 ,r) as t → +∞. ( 61
)
In addition, we have the following integral estimates

+∞ t1 t r-1 ẋ(t) 2 dt < +∞, +∞ t1 t ( 3r 2 -1) ∇f (x(t)) 2 dt < +∞.
Proof a) By taking ε(t) = 1 t r in Corollary 1, we get (58). So if (H 1 ) is satisfied, we get

E 1 (t) ≤ x * 2 2γ(t) t t1 (λc + 2a)λ ε2 (s) ε 3 2 (s) -ε(s) γ(s)ds + γ(t 1 )E 1 (t 1 ) γ(t) . ( 62
)
Using the same technique as before, we start by choosing the parameters a > 1, c > 2, λ > 0 such that

δ 2 < λ < a a+1 δ for 0 < δ ≤ √ 2 -1 c and δ 2 < 1 2 δ + 1 c + (δ + 1 c ) 2 -2 < λ < a a+1 δ for δ > √ 2 -1 c .
We can easily check that for r < 2 and for t ≥ t 1 large enough,

d dt 1 ε(t) = r 2 t r-2 2 ≤ min 2λ -δ , 1 2 δ - a + 1 a λ
, and βδ ≤ t This expresses that the condition (H 1 ) holds. With the notations of Theorem 3, we have

µ(t) = - ε(t) 2ε(t) + (δ -λ) ε(t) = r 2t + δ -λ t r 2 (63) γ(t) = exp t t1 µ(s)ds = t t 1 r 2 exp 2(δ -λ) 2 -r t 2-r 2 -t 2-r 2 1 = C 1 t r 2 exp 2(δ -λ) 2 -r t 2-r 2
where

C 1 = t r 2 1 exp 2(δ -λ) 2 -r t 2-r 2 1 -1 . ( 64
)
Setting λ 0 := (λc + 2a)λ, δ 0 := 2(δ -λ) 2 -r , and replacing ε(t) and γ(t) by their values in (62), we get

E 1 (t) ≤ r x * 2 2t r 2 exp δ 0 t 2-r 2 t t1 λ 0 r s 2 + 1 s r+2 2 exp δ 0 s 2-r 2 ds + γ(t 1 )E 1 (t 1 ) γ(t) (65)
Then notice that

d ds 1 ρs exp δ 0 s 2-r 2 = - 1 ρs 2 + δ 0 (2 -r) 2ρs r+2 2 exp δ 0 s 2-r 2 .
For s large enough, by taking 0 < ρ < 1 a+1 δ, we have

λ 0 r s 2 + 1 s r+2 2 ≤ - 1 ρs 2 + δ 0 (2 -r) 2ρs r+2 2
, which gives

E 1 (t) ≤ r 2t r 2 exp δ 0 t 2-r 2 t t1 - 1 ρs 2 + δ 0 (2 -r) 2ρs r+2 2 exp δ 0 s 2-r 2 ds + γ(t 1 )E 1 (t 1 ) γ(t) = 1 2t r 2 exp δ 0 t 2-r 2 t t1 d ds 1 ρs exp δ 0 s 2-r 2 ds + γ(t 1 )E 1 (t 1 ) γ(t) = r 2ρt r+2 2 - r t r 2 exp δ 0 t 2-r 2 1 2ρt 1 exp δ 0 t 2-r 2 1 + γ(t 1 )E 1 (t 1 ) γ(t) ≤ r 2ρt r+2 2 + γ(t 1 )E 1 (t 1 ) γ(t) .
We have

γ(t 1 )E 1 (t 1 ) γ(t) ≤ Ct -r 2 exp -δ 0 t 2-r 2
. Since 0 < r < 2 and δ 0 > 0, we deduce that γ(t 1 )E 1 (t 1 )

γ(t)
tends to zero at an exponential rate, as t → +∞. Therefore, there exists a positive constant C such that for t large enough

E 1 (t) ≤ C t r+2 2 . ( 66
)
By Lemma 1, we deduce that there exists positive constants C and M such that, for t large enough,

f (x(t)) -min H f ≤ C 1 t r+2 2 + 1 t r , x(t) -x ε(t) 2 ≤ 2Ep(t) ε(t) ≤ 2C t 2-r 2
, and

ẋ(t) + β∇ϕ t (x(t)) + (p -1)ε(t)x(t) 2 ≤ M Ep(t) ≤ M C t 2+r 2 (67)
Since 0 < r < 2, we conclude that

f (x(t)) -min H f = O 1 t r , x(t) -x ε(t) 2 = O 1 t 2-r 2
, as t → +∞.

By (67), we have

ẋ(t) + β∇f (x(t)) ≤ ẋ(t) + β∇ϕ t (x(t)) + ε(t) x(t) ≤ √ M C t 2+r 4 + 1 t r x(t) .
Since x is bounded, we conclude that, as t → +∞,

ẋ(t) + β∇f (x(t)) =        O 1 t 2+r 4 if r ∈ [ 2 3 , 2[ O 1 t r if r ∈]0, 2 3 [. b)
We now come to the integral estimates of the velocities and gradient terms. For this, we use the pointwise estimates already established, and proceed with the Lyapunov function Ep defined in (56). The system (TRISHE) corresponds to p = 1, so we consider

E 1 (t) := ϕ t (x(t)) -ϕ t (x ε(t) ) + 1 2 ẋ(t) + β∇ϕ t (x(t)) 2 .
Since for t > t 1 , δβ ≤

, then according to (57), we have

Ė1 (t) ≤ 1 2 ε(t) x(t) 2 - 1 2 ε(t) x * 2 - δ 2 ε(t) ẋ(t) 2 - δ 2 ε(t) ẋ(t) + β∇ϕ t (x(t)) 2 - β 2 ∇ϕ t (x(t)) 2 ≤ - 1 2 ε(t) x * 2 - δ 2 ε(t) ẋ(t) 2 - β 2 ∇ϕ t (x(t)) 2 (68)
Equivalently,

δ 2 ε(t) ẋ(t) 2 + β 2 ∇ϕ t (x(t)) 2 ≤ -Ė1 (t) - 1 2 ε(t) x * 2 .
By multiplying this last equality by t 3r 2 -1 and integrating on [t 1 , T ], we get

δ 2 T t1 t r-1 ẋ(t) 2 dt + β 2 T t1 t 3r 2 -1 ∇ϕ t (x(t)) 2 dt ≤ - T t1 t 3r 2 -1 Ė1 (t)dt + x * 2 2 T t1 t r 2 -2 dt. (69)
We have

- T t1 t 3r 2 -1 Ė1 (t)dt = t 3r 2 -1 1 E 1 (t 1 ) -t 3r 2 -1 E 1 (t) + 3r -2 2 T t1 t 3r 2 -2 E 1 (t)dt ≤ t 3r 2 -1 1 E 1 (t 1 ) + 3r -2 2 T t1 t 3r 2 -2 ϕ t (x(t)) -ϕ t (x ε(t) ) dt + 3r -2 4 T t1 t 3r 2 -2 ẋ(t) + β∇ϕ t (x(t)) 2 dt ≤ t 3r 2 -1 1 E 1 (t 1 ) + 3r -2 2 T t1 t 3r 2 -2 E 1 (t)dt + 3r -2 4 T t1 t 3r 2 -2 ẋ(t) + β∇ϕ t (x(t)) 2 dt (70)
According to (66) and (67), we deduce that there exists C > 0 such that

- T t1 t 3r 2 -1 Ė1 (t)dt ≤ t 3r 2 -1 1 E 1 (t 1 ) + C T t1 t r-3 dt. (71)
From this, we deduce that -

+∞ t1 t 3r 2 -1 Ė1 (t)dt < +∞. By (69), we conclude that δ 2 T t1 t r-1 ẋ(t) 2 dt + β 2 T t1 t 3r 2 -1 ∇ϕ t (x(t)) 2 dt < +∞. Therefore T t1 t r-1 ẋ(t) 2 dt < +∞ and T t1 t 3r 2 -1 ∇ϕ t (x(t)) 2 dt < +∞.
We also have

∇f (x(t)) 2 ≤ ∇ϕ t (x(t)) + 1 t r x(t) 2 ≤ 2 ∇ϕ t (x(t)) 2 + 2 t 2r x(t) 2 .
Since x is bounded, we obtain

+∞ t1 t 3r 2 -1 ∇f (x(t)) 2 dt < +∞.
This completes the proof.

System (TRISH)

We now come to the corresponding result for (TRISH), stated as a model result in the introduction.

Theorem 5 Take δ > 2, ε(t) = 1 t r , with 1 ≤ r < 2. Let x : [t 0 , +∞[→ H be a solution trajectory of ẍ(t) + δ t r 2 ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇f (x(t)) + 1 t r x(t) = 0. (72)
Then, we have the following estimates

f (x(t)) -min H f = O 1 t r as t → +∞; (73) x(t) -x ε(t) 2 = O 1 t 2-r 2 as t → +∞. (74) ẋ
(t) + β∇f (x(t)) = O 1 t r+2 4 as t → +∞. ( 75
) +∞ t1 t r-1 ẋ(t) 2 dt < +∞, +∞ t1 t 3r-2 2 ∇f (x(t)) 2 dt < +∞. ( 76
) Proof a) Taking ε(t) = 1 t r in Corollary 2 gives (72). So if the condition (H 0 ) is satisfied, we get E 0 (t) ≤ x * 2 2γ(t) t t1 (λc + 2a)λ ε2 (s) ε 3 2 (s) -ε(s) + βλ(δ -λ)ε 2 (t) γ(s) ds + γ(t 1 )E 0 (t 1 ) γ(t) . (77)
As in the proof of Theorem 4, since r < 2, take the parameters a > 1, c > 2, λ > 0 such that

1 2 δ + 1 c + (δ + 1 c ) 2 -2 < λ < min a a+1 δ, δ+ √ δ 2 -4 2 .
For t ≥ t 1 large enough, one can prove that

d dt 1 ε(t) = r 2 t r-2 2 ≤ min a a + 1 δ, δ + √ δ 2 -4 2 , and βδ ≤ t r 2 .
This means that the condition (H 0 ) is satisfied. By setting λ 0 := (λc + 2a)λ, δ 0 := 2(δ -λ) 2 -r and

β 0 = βλ (δ -λ)
and combining the equations ( 63) and ( 64) with (77), we get

E 0 (t) ≤ r 2t r 2 exp δ 0 t 2-r 2 t t1 λ 0 r s 2 + 1 s r+2 2 + β 0 rs 3r 2 exp δ 0 s 2-r 2 ds + γ(t 1 )E 0 (t 1 ) γ(t) . (78)
Let us estimate the integral

t t1 λ 0 r s 2 + 1 s r+2 2 + β 0 rs 3r 2 exp δ 0 s 2-r 2 ds. For ρ > 0 d ds 1 ρs exp δ 0 s 2-r 2 = - 1 ρs 2 + δ 0 (2 -r) 2ρs r+2 2 exp δ 0 s 2-r 2
.

So, we need to show that

λ 0 r s 2 + 1 s r+2 2 + β 0 rs 3r 2
≤ -

1 ρs 2 + δ 0 (2 -r) 2ρs r+2 2 . ( 79
)
Since r ≥ 1, we have for s large enough,

λ 0 r s 2 + 1 s r+2 2 + β 0 rs 3r 2 ≤ λ 0 r s 2 + 1 + β0 r s r+2 2 .
By taking ρ < r (a+1)(r+β0) δ, we have

λ 0 r s 2 + 1 + β0 r s r+2 2 ≤ - 1 ρs 2 + δ 0 (2 -r) 2ρs r+2 2 ⇐⇒ λ 0 r + 1 ρ s 2 ≤ δ 0 (2 -r) 2ρ -1 -β0 r 1 s r+2 2 = δ-λ ρ -1 -β0 r s r+2 2 ⇐⇒ λ 0 r + 1 ρ s 2-r 2 ≤ δ -λ ρ -1 - β 0 r .
We have 0 < r < 2, which means that lim s→+∞ 1 s 2-r 2 = 0. Combining the fact that λ < a a+1 δ, with the choice of ρ, one can check that

δ -λ -ρ(1 + β 0 r ) > a a + 1 δ -λ >0 + 1 a + 1 δ -ρ(1 + β 0 r ) > -(1 + β 0 r ) ρ - r (a + 1)(r + β 0 ) δ > 0.
Therefore, for s large enough, the last above inequalities are satisfied, which implies that, for 1 ≤ r < 2, and t 1 large enough, we have

E 0 (t) ≤ r 2t r 2 exp δ 0 t 2-r 2 t t1 - 1 ρs 2 + δ 0 (2 -r) 2ρs r+2 2 + β 0 rs 3r 2 ' exp δ 0 s 2-r 2 ds + γ(t 1 )E 0 (t 1 ) γ(t) = 1 2t r 2 exp δ 0 t 2-r 2 t t1 d ds 1 ρs exp δ 0 s 2-r 2 ds + γ(t 1 )E 0 (t 1 ) γ(t) = r 2ρt r+2 2 - r t r 2 exp δ 0 t 2-r 2 1 2ρt 1 exp δ 0 t 2-r 2 1 + γ(t 1 )E 0 (t 1 ) γ(t) ≤ r 2ρt r+2 2 + γ(t 1 )E 0 (t 1 )
γ(t) .

We now proceed in the same way as in the proof of Theorem 4. Since γ(t 1 )E 0 (t 1 )

γ(t)
has also an exponential decay to zero, we deduce that for t large enough, (73), ( 74) and (75) are satisfied. b) We now come to the precise integral estimates of the velocities and gradient terms. Parallel to the study for (TRISHE) we proceed with the Lyapunov function Ep defined in (56). The system (TRISH) corresponds to p = 0, so we consider

E 0 (t) := ϕ t (x(t)) -ϕ t (x ε(t) ) + 1 2 ẋ(t) + β∇f (x(t)) 2 .
Using successively the derivation chain rule, the equation (TRISH), and ε(t) ≤ 0, we get

d dt E 0 (t) = ∇ϕ t (x(t)), ẋ(t) + ε(t) 2 x(t) 2 + ẋ(t) + β∇f (x(t)), ẍ(t) + β∇ 2 f (x(t)) ẋ(t) = -β ∇ϕ t (x(t)), ∇f (x(t)) -δ ε(t) ẋ(t) 2 + ε(t) 2 x(t) 2 -δβ ε(t) ∇f (x(t)), ẋ(t) ≤ -β ∇f (x(t)) 2 -βε(t) x(t), ∇f (x(t)) -δ ε(t) ẋ(t) 2 -δβ ε(t) ∇f (x(t)), ẋ(t) ≤ -β 2 ∇f (x(t)) 2 + βε 2 (t) 2 x(t) 2 -δ ε(t) ẋ(t) 2 -δβ ε(t) ∇f (x(t)), ẋ(t)
Equivalently,

β 2 ∇f (x(t)) 2 + δ ε(t) ẋ(t) 2 ≤ - d dt E 0 (t) + βε 2 (t) 2 x(t) 2 -δβ ε(t) ∇f (x(t)), ẋ(t) .
By multiplying this last equality by t 3r 2 -1 and integrating on [t 1 , T ], we get

β 2 T t1 t 3r 2 -1 ∇f (x(t)) 2 dt + δ T t1 t r-1 ẋ(t) 2 dt ≤ - T t1 t 3r 2 -1 d dt E 0 (t)dt + β T t1 t -r 2 -1 x(t) 2 dt -δβ T t1 t r-1 ∇f (x(t)), ẋ(t) dt. ( 80
)
Let us show that the second member of this last inequality converges when T goes to infinity.

• For the first term of second member of inequality (80), we have

- T t1 t 3r 2 -1 d dt E 0 (t)dt = -t 3r 2 -1 E 0 (t) T t1 + 3r 2 -1 T t1 t 3r 2 -2 E 0 (t)dt ≤ t 3r 2 -1 1 E 0 (t 1 ) C1 + 3r 2 -1 T t1 t 3r 2 -2 ϕ t (x(t)) -f (x * ) dt + 3r -2 4 T t1 t 3r 2 -2 ẋ(t) + β∇f (x(t)) 2 dt ≤ C 1 + 3r -2 2 T t1 t 3r 2 -2 f (x(t)) -f (x * ) dt + 3r -2 4 T t1 t r 2 -2 x(t) 2 dt + 3r -2 4 T t1 t 3r 2 -2 ẋ(t) + β∇f (x(t)) 2 dt. (81)
Using ( 75) and (73), and the fact that x(•) is bounded, we obtain

- +∞ t1 t 3r 2 -1 d dt E 0 (t)dt ≤ C 1 + C 2 +∞ t1 t r 2 -2 dt + C 3 +∞ t1 t r-3 dt < +∞, because r < 2. ( 82
)
• Consider now the third term of the second member of the inequality (80). According to the equality d dt (f (x(t)) -f (x * )) = ∇f (x(t)), ẋ(t) , we have

-δβ T t1 t r-1 ∇f (x(t)), ẋ(t) dt = -δβ t r-1 f (x(t)) -f (x * ) T t1 + δβ(r -1)β T t1 t r-2 f (x(t)) -f (x * ) dt ≤ δβt r-1 1 f (x( 1 )) -f (x * ) C4 +δβ(r -1)β T t1 t r-2 f (x(t)) -f (x * ) dt.
Using (73), we get

-δβ +∞ t1 t r-1 ∇f (x(t)), ẋ(t) dt ≤ C 4 + C 5 +∞ t1 t -2 dt < +∞. ( 83
)
Collecting ( 80), ( 82) and ( 83), we conclude that

β 2 +∞ t1 t 3r 2 -1 ∇f (x(t)) 2 dt + δ +∞ t1 t r-1 ẋ(t) 2 dt ≤ C + β +∞ t1 t -r 2 -1 x(t) 2 dt.
Using again that x is bounded, we deduce that

β 2 +∞ t1 t 3r 2 -1 ∇f (x(t)) 2 dt + δ +∞ t1 t r-1 ẋ(t) 2 dt < +∞.
This completes the proof.

Numerical illustrations

Let us illustrate our results with the following examples where the function f is taken successively strictly convex, then convex with a continuum of solutions. In a third example, we compare the two systems (TRISH) and (TRISHE). The following numerical experiences describe in these three situations the behavior of the trajectories generated by the system (TRIGS) (without the Hessian driven damping) and by the systems (TRISH) and (TRISHE) (with the Hessian driven damping). All these systems take into account the effect of Tikhonov regularization. They are differentiated by the presence, or not, of the Hessian driven damping. According to the model situation described in Theorem 4 and Theorem 5, the Tikhonov regularization parameter is taken equal to ε(t) = t -r , with 0 < r ≤ 2. We consider different values of the parameter r which plays a key role in tuning the viscosity and Tikhonov parameters. We pay particular attention to the case r close to the value 2, which provides fast convergence results. The corresponding dynamical systems are given by:

(TRIGS) ẍ(t) + δt -r 2 ẋ(t) + ∇f (x(t)) + t -r x(t) = 0 (TRISH) ẍ(t) + δt -r 2 ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇f (x(t)) + t -r x(t) = 0 (TRISHE) ẍ(t) + δt -r 2 ẋ(t) + β∇ 2 f (x(t)) ẋ(t) -rt -r-1 x(t) + t -r ẋ(t) + ∇f (x(t)) + 1 t r x(t) = 0.
We choose δ = 3, β = 1.

To facilitate the comparison of the trajectories corresponding to different dynamics, for example (TRIGS) and (TRISH), they are represented respectively by continuous lines and dotted lines. All our numerical tests were implemented in Scilab version 6.1 as an open source software.

Fig. 1 Convergence rates of values f 1 (x(t)) -f (x * ), of trajectories x(t) -x * 2 , and gradients ∇f 1 (x(t) 2 .

Example 1 Take f

1 :] -1, +∞[ 2 → R which is defined by f 1 (x) = (x 1 + x 2 2 ) -2 ln(x 1 + 1)(x 2 + 1).
The function f 1 is strictly convex with

∇f 1 (x) = 1 -2 x1+1 2x 2 -2 x2+1 and ∇ 2 f 1 (x) = 2 (x1+1) 2 0 0 2 + 2 (x2+1) 2 .
The unique minimum of f 1 is x * = (1, ( √ 5 -1)/2). The corresponding trajectories to the systems (TRIGS), (TRISHE) are depicted in Figure 1.

Example 2 Consider the convex function f 2 : R 2 → R defined by

f 2 (x) = 1 2 (x 1 + x 2 -1) 2 .
We have

∇f 2 (x) = x 1 + x 2 -1 x 1 + x 2 -1 and ∇ 2 f 1 (x) = 1 1 1 1 . We have S 2 = argmin f 2 = {(x 1 , 1 -x 1 ) : x 1 ∈ R} and x * = ( 1 2 , 1
2 ) is the minimum norm solution. The corresponding trajectories to the systems (TRIGS), (TRISHE) are depicted in Figure 2.

Example 3 To compare the systems (TRISH) and (TRISHE), we take the same function f 2 as in the previous example. The corresponding trajectories are depicted in Figure 3.

As predicted by the theory, it is observed that the trajectories generated by the systems (TRISH) and (TRISHE) have at the same time several remarkable properties: they ensure fast convergence of the values, fast convergence of the gradients towards zero, and convergence to the minimum norm solution. The presence of the Hessian driven damping in these dynamics induces a significant attenuation of oscillations (by comparison with (TRIGS)). The third example shows that Fig. 2 Convergence rates of values f 2 (x(t)) -f (x * ), trajectories x(t) -x * 2 , and gradients ∇f 2 (x(t) 2 .

Fig. 3 Comparison of (TRISH) and (TRISHE)

the trajectories generated by (TRISH) and (TRISHE) share a very similar behaviour. We see the advantage of taking r close to 2 in the presence of the Hessian driven damping. Indeed, r = 2 gives viscous damping similar to that of the accelerated gradient method of Nesterov, in which case we know that the adjustment of the coefficient δ plays a crucial role. Note the criticality of the case r = 2, since for r < 2 the condition for δ is δ > 2, whereas for r = 2 we know that we must take δ ≥ 3 to get fast convergence. This is an interesting subject for further research.

Existence of solution trajectories for (TRISHE)

Let us start by establishing the equivalence between the inertial dynamic with Hessian driven damping (TRISHE) and a first order system in time and space in the case of a smooth function f . Similar result was first obtained in [START_REF] Alvarez | A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics[END_REF], with applications to mechanics [START_REF] Attouch | A second-order differential system with Hessian-driven damping; Application to non-elastic shock laws[END_REF] and deep learning [START_REF] Bolte | An Inertial Newton Algorithm for Deep Learning[END_REF].

Theorem 6 Let f : H → R be a convex C 2 function. 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 (TRISHE) ẍ(t) + δ ε(t) ẋ(t) + β d dt (∇ϕ t (x(t))) + ∇ϕ t (x(t)) = 0. ( 84
)
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∇ϕ t (x(t)) -1 β -δ ε(t) x(t) + 1 β y(t) = 0 ẏ(t) -1 β -δ ε(t) -βδ 2 ˙ (t) √ ε(t) x(t) + 1 β y(t) = 0, ( 85
) with initial conditions x(t 0 ) = x 0 , y(t 0 ) = -β( ẋ0 + β∇ϕ t0 (x 0 )) + (1 -δ ε(t 0 ))x 0 . Proof 2. ⇒ 1.
Differentiating the first equation of (85) gives

ẍ(t) + β d dt (∇ϕ t (x(t))) + δ ˙ (t) 2 ε(t) x(t) - 1 β -δ ε(t) ẋ(t) + 1 β ẏ(t) = 0. (86)
Replacing ẏ(t) by its expression as given by the second equation of (85) gives After simplification of the above expression, we obtain (84).

ẍ(t) + β d dt (∇ϕ t (x(t))) + δ ˙ (t) 2 ε(t) x(t) - 1 β -δ ε(t) ẋ(t) + 1 β 1 β -δ ε(t
1. ⇒ 2. Define y(t) by the first equation of (85). Differentiating y(t) and using equation (84) allows one to eliminate ẍ(t), which finally gives the second equation of (85).

Based on Theorem 6, the following first order formulation helps give meaning to the (TRISHE) system when f ∈ Γ 0 (H). It is obtained by substituting the subdifferential ∂ϕ t for the gradient ∇ϕ t in the first-order formulation (85).

Definition 2 Let δ > 0, β > 0 and f ∈ Γ 0 (H). Given (x 0 , y 0 ) ∈ dom(f ) × H, the Cauchy problem for the inertial system (TRISHE) with generalized Hessian driven damping is defined by x(t) + 1 β y(t) = 0

         ẋ(t
x(t 0 ) = x 0 , y(t 0 ) = y 0 .

( (90)

The differential inclusion (89) is governed by the sum of the time dependent maximally monotone operator ∂G(t, .•) (a convex subdifferential) and the time-dependent linear 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 [START_REF] Brézis | Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution[END_REF]Proposition 3.12], and of the fact that t → ϕ t (x) is a nonincreasing function, see [START_REF] Attouch | Strong solutions for parabolic variational inequalities[END_REF]. In this setting, the notion of classical solution is replaced by the notion of strong solution, see [START_REF] Brézis | Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution[END_REF]Definition 3.1], [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF]Theorem 4.4], [START_REF] Attouch | On the effect of perturbations, errors in first-order optimization methods with inertia and Hessian driven damping[END_REF]Theorem 2.4].

Conclusion, perspective

For convex optimization in Hilbert spaces, we have introduced a damped inertial dynamics which combines Hessian driven damping with Tikhonov regularization. The Hessian driven damping and the Tikhonov regularization term induce specific favorable geometric properties, related to curvature aspects. The Tikhonov term regulates the objective function. It makes the dynamic relevant of the heavy ball with friction method for a strongly convex function. The Hessian driven damping acts on the velocity vector in a similar way as continuous Newton's method. It has a corrective effect by damping the oscillations that arise with ill-conditioned optimization problems. It turns out that the two techniques combine well and provide a substantial improvement to Nesterov's accelerated gradient method. While preserving fast convergence of values, they ensure fast convergence of gradients to zero, they significantly reduce oscillations, and provide convergence to the minimum norm solution.

Our study provides a solid basis for the convergence analysis of algorithms obtained by temporal discretization, which is a subject of further work. Our approach calls for many developments. We showed that our approach can be naturally extended to the case of nonsmooth convex optimization, and the study of additively structured "smooth + nonsmooth" convex optimization problems. Our study naturally leads to applications in various fields such as inverse problems for which strong convergence of trajectories, and obtaining a solution close to a desired state are key properties.

It is likely that a parallel approach can be developed for multiobjective optimization for the dynamical approach to Pareto optima, and within the framework of potential games. The Lyapunov analysis developed in this paper could also be very useful to study the asymptotic stabilization of several classes of PDE's, for example nonlinear damped wave equations. One of the main challenges related to our study is whether similar convergence results can be obtained using autonomous systems, see [START_REF] Attouch | Fast optimization via inertial dynamics with closed-loop damping[END_REF] for a first systematic study of this question. Indeed the study of autonomous versions of the Tikhonov method, such as the Haugazeau method, in the context of dynamic systems, and rapid optimization, is a field largely to be explored.

  ), we use the differentiability properties of the viscosity curve → x = argmin f (ξ) + 2 ξ 2 .

  t → x (t) is absolutely continuous (indeed locally Lipschitz) the classical derivation chain rule can be applied to compute the derivative of the function Ep(•), see [32, section VIII.2]. According to Lemma 2 i), for almost all t ≥ t 0 the derivative of Ep(•) is given by:

  ) + β∂ϕ t (x(t)) -1 β -δ ε(t) x(t) + 1

		β y(t)	0
	ẏ(t) -1 β -δ ε(t) -βδ 2	˙ (t) √ ε(t)

  88)Let us formulate (88) in a condensed form as an evolution equation in the product space H × H. Setting Z(t) = (x(t), y(t)) ∈ H × H, (88) can be equivalently writtenŻ(t) + ∂G(t, Z(t)) + D(t, Z(t)) 0, Z(t 0 ) = (x 0 , y 0 ), (89)where G(, •) ∈ Γ 0 (H × H) is the function defined by G(t, Z) = βϕ t (x), and the time-dependent operator D : [t 0 , +∞[×H × H → H × H is given by

	D(t, Z) = -	1 β	-δ ε(t) x +	1 β	y, -	1 β	-δ ε(t) -	βδ 2	˙ (t) ε(t)	x +	1 β	y .

This is the key property of the asymptotic version (t → +∞) of the Browder-Tikhonov regularization method.

Acknowledgments: The research of Aïcha BALHAG was supported by the EIPHI Graduate School (contract ANR-17-EURE-0002).