Asymptotic behavior of Newton-like inertial dynamics involving the sum of potential and nonpotential terms

In a Hilbert space H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}, we study a dynamic inertial Newton method which aims to solve additively structured monotone equations involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator A=∇f+B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A= \nabla f +B $\end{document}, where ∇f is the gradient of a continuously differentiable convex function f and B is a nonpotential monotone and cocoercive operator. Besides a viscous friction term, the dynamic involves geometric damping terms which are controlled respectively by the Hessian of the potential f and by a Newton-type correction term attached to B. Based on a fixed point argument, we show the well-posedness of the Cauchy problem. Then we show the weak convergence as t→+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\to +\infty $\end{document} of the generated trajectories towards the zeros of ∇f+B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\nabla f +B$\end{document}. The convergence analysis is based on the appropriate setting of the viscous and geometric damping parameters. The introduction of these geometric dampings makes it possible to control and attenuate the known oscillations for the viscous damping of inertial methods. Rewriting the second-order evolution equation as a first-order dynamical system enables us to extend the convergence analysis to nonsmooth convex potentials. These results open the door to the design of new first-order accelerated algorithms in optimization taking into account the specific properties of potential and nonpotential terms. The proofs and techniques are original and differ from the classical ones due to the presence of the nonpotential term.


Introduction and preliminary results
Let H be a real Hilbert space endowed with the scalar product ·, · and the associated norm · . Many situations coming from physics, biology, human sciences involve equations containing both potential and nonpotential terms. In human sciences, this comes from the presence of both cooperative and noncooperative aspects. In physics, this comes from the joint presence of terms of diffusion and convection. To describe such situations we will focus on solving additively structured monotone equations of the type Find x ∈ H : ∇f (x) + B(x) = 0. (1.1) In the above equation, ∇f is the gradient of a convex continuously differentiable function f : H → R (that's the potential part), and B : H → H is a nonpotential operator 1 which is supposed to be monotone and cocoercive. To this end, we will consider continuous inertial dynamics whose solution trajectories converge as t → +∞ to solutions of (1.1). Our study is part of the active research stream that studies the close relationship between continuous dissipative dynamical systems and optimization algorithms which are obtained by their temporal discretization. To avoid lengthening the paper, we limit our study to the analysis of the continuous dynamic. The analysis of the algorithmic part and its link with first-order numerical optimization will be carried out in a second companion paper. From this perspective, damped inertial dynamics offer a natural way to accelerate these systems. As the main feature of our study, we will introduce the dynamic geometric dampings which are respectively driven by the Hessian for the potential part and by the corresponding Newton term for the nonpotential part. In addition to improving the convergence rate, this will considerably reduce the oscillatory behavior of the trajectories. We will pay particular attention to the minimal assumptions which guarantee convergence of the trajectories, and which highlight the asymmetric role played by the two operators involved in the dynamic. We will see that many results can be extended to the case where f : H → R ∪ {+∞} is a convex lower semicontinuous proper function, which makes it possible to broaden the field of applications.

Dynamical inertial Newton method for additively structured monotone problems
For t ≥ t 0 , let us introduce the following second-order differential equation which will form the basis of our analysis: We use (DINAM) as an abbreviation for dynamical inertial Newton method for additively structured monotone problems. We call t 0 ∈ R the origin of time. Since we are considering autonomous systems, we can take any arbitrary real number for t 0 . For simplicity, we set t 0 = 0. When considering the corresponding Cauchy problem, we add the initial conditions: x(0) = x 0 ∈ H andẋ(0) = x 1 ∈ H. The term B (x(t))ẋ(t) is interpreted as d dt (B(x(t))) taken in the distribution sense. Likewise the term ∇ 2 f (x(t))ẋ(t) is interpreted as d dt (∇f (x(t))) taken also in the distribution sense. Because of the assumptions made below, these terms are indeed measurable functions which are bounded on the bounded time intervals. So, we will consider strong solutions of the above equation (DINAM).
We emphasize the fact that we do not assume the gradient of f to be globally Lipschitz continuous. Developing our analysis without using any bound on the gradient of f is a key to further extend the theory to the nonsmooth case. As a specific property, the inertial system (DINAM) combines two different types of driving forces associated respectively with the potential operator ∇f and the nonpotential operator B. It also involves three different types of friction: (a) The term γẋ(t) models viscous damping with a positive coefficient γ > 0.
is the so-called Hessian driven damping, which allows to attenuate the oscillations that naturally occur with the inertial gradient dynamics. (c) The term β b B (x(t))ẋ(t) is the nonpotential version of the Hessian driven damping.
It can be interpreted as a Newton-type correction term. Note that each driving force term enters (DINAM) with its temporal derivative. In fact, we have This is a crucial observation which makes (DINAM) equivalent to a first-order system in time and space, and makes the corresponding Cauchy problem well posed. This will be proved later (see Sect. 2.1 for more details). The cocoercivity assumption on the operator B plays an important role in the analysis of (DINAM), not only to ensure the existence of solutions, but also to analyze their asymptotic behavior as time t → +∞.
Recall that the operator B : H → H is said to be λ-cocoercive for some λ > 0 if Note that B is λ-cocoercive is equivalent to B -1 is λ-strongly monotone, i.e., cocoercivity is a dual notion of strong monotonicity. It is easy to check that B is λ-cocoercive implies that B is 1/λ-Lipschitz continuous. The reverse implication holds true in the case where the operator is the gradient of a convex and differentiable function. Indeed, according to Baillon-Haddad's theorem [17], ∇f is L-Lipschitz continuous implies that ∇f is a 1/Lcocoercive operator (we refer to [18,Corollary 18.16] for more details).

Historical aspects of the inertial systems with Hessian-driven damping
The following inertial system with Hessian-driven damping was first considered by Alvarez, Attouch, Peypouquet, and Redont in [6]. Then, according to the continuous interpretation by Su, Boyd, and Candès [28] of the accelerated gradient method of Nesterov, Attouch, Peypouquet, and Redont [14] replaced the fixed viscous damping parameter γ with an asymptotic vanishing damping parameter α t , with α > 0. At first glance, the presence of the Hessian may seem to entail numerical difficulties. However, this is not the case as the Hessian intervenes in the above ODE in the form ∇ 2 f (x(t))ẋ(t), which is nothing but the derivative with respect to time of ∇f (x(t)). So, the temporal discretization of these dynamics provides first-order algorithms of the form As a specific feature, and by comparison with the classical accelerated gradient methods, these algorithms contain a correction term which is equal to the difference of the gradients at two consecutive steps. While preserving the convergence properties of the accelerated gradient method, they provide fast convergence to zero of the gradients and reduce the oscillatory aspects. Several recent studies have been devoted to this subject, see Attouch, Chbani, Fadili, and Riahi [7], Boţ, Csetnek, and László [20], Kim [24], Lin and Jordan [25], Shi, Du, Jordan, and Su [27], and Alesca, Lazlo, and Pinta [4] for an implicit version of the Hessian driven damping. Application to deep learning has been recently developed by Castera, Bolte, Févotte, and Pauwels [23]. In [3], Adly and Attouch studied the finite convergence of proximal-gradient inertial algorithms combining dry friction with Hessiandriven damping.

Inertial dynamics involving cocoercive operators
Let us come to the transposition of these techniques to the case of maximally monotone operators. Álvarez and Attouch [5] and Attouch and Maingé [10] studied the equation when A : H → H is a cocoercive (and hence maximally monotone) operator (see also [19]). The cocoercivity assumption plays an important role in the study of (1.2), not only to ensure the existence of solutions, but also to analyze their long-term behavior. Assuming that the cocoercivity parameter λ and the damping coefficient γ satisfy the inequality λγ 2 > 1, Attouch and Maingé [10] showed that each trajectory of (1.2) converges weakly to a zero of A, i.e., x(t) x ∞ ∈ A -1 (0) as t → +∞. Moreover, the condition λγ 2 > 1 is sharp. For general maximally monotone operators, this property has been further exploited by Attouch and Peypouquet [13] and by Attouch and Laszlo [8,9]. The key property is that, for λ > 0, the Yosida approximation A λ of A is λ-cocoercive and A -1 λ (0) = A -1 (0). So the idea is to replace the operator A with its Yosida approximation and adjust the Yosida regularization parameter. Another related work has been done by Attouch and Maingé [10] who first considered the asymptotic behavior of the second-order dissipative evolution equation with f : H → R convex and B : H → H cocoercivë combining potential with nonpotential effects. Our study will therefore consist initially in introducing the Hessian term and the Newton-type correcting term into this dynamic.

Link with Newton-like methods for solving monotone inclusions
Let us specify the link between our study and Newton's method for solving (1.1). To overcome the ill-posed character of the continuous Newton method for a general maximally

Contents
The paper is organized as follows. Section 1 introduces (DINAM) with some historical perspective. In Sect. 2, based on the first-order equivalent formulation of (DINAM), we show that the Cauchy problem is well-posed (in the sense of existence and uniqueness of solutions). In Sect. 3, we analyze the asymptotic convergence properties of the trajectories generated by (DINAM). Using appropriate Lyapunov functions, we show that any trajectory of (DINAM) converges weakly as t → +∞, and that its limit belongs to S = (∇f + B) -1 (0). The interplay between the damping parameters β f , β b , γ and the cocoercivity parameter λ will play an important role in our Lyapunov analysis. In Sect. 4, we perform numerical experiments showing that the well-known oscillations in the case of the heavy ball with friction are damped with the introduction of the geometric (Hessianlike) damping terms. An application to the LASSO problem with a nonpotential operator as well as a coupled system in dynamical games are considered. Section 5 deals with the extension of the study to the nonsmooth and convex case. Section 6 contains some concluding remarks and perspectives.

Well-posedness of the Cauchy-Lipschitz problem
We first show the existence and the uniqueness of the solution trajectory for the Cauchy problem associated with (DINAM) for any given initial condition data (x 0 , x 1 ) ∈ H × H.

First-order in time and space equivalent formulation
The following first-order equivalent formulation of (DINAM) was first considered by Alvarez, Attouch, Bolte, and Redont [6] and Attouch, Peypouquet, and Redont [14] in the framework of convex minimization. Specifically, in our context, we have the following equivalence, which follows from a simple differential and algebraic calculation. Suppose that β f > 0. Then the following problems are equivalent: which gives the first equation of (ii). By differentiating y(·) and using (i), we geṫ By combining (2.1) and (2.2), we obtaiṅ This gives the second equation of (ii).
(ii) ⇒ (i). By differentiating the first equation of (ii), we obtain Let us eliminate y from this equation to obtain an equation involving only x. For this, we successively use the second equation in (ii), then the first equation in (ii) to obtaiṅ Therefore, From (2.4) and (2.5), we obtain (i).

Well-posedness of the evolution equation (DINAM)
In the following theorem, we show the well-posedness of the Cauchy problem for the evolution equation (DINAM).
Proof System (ii) in Proposition 2.1 can be written equivalently aṡ is a Lipschitz continuous map. Indeed, the Lipschitz continuity of G is a direct consequence of the Lipschitz continuity of B. The existence of a classical solution tȯ follows from Brézis [21,Proposition 3.12]. In fact, the proof of this result relies on a fixed point argument. It consists in finding a fixed point of the mapping It is proved that the sequence of iterates (w n ) generated by the corresponding Picard iterationẇ continuous on the bounded time intervals, andẍ taken in the distribution sense is locally essentially bounded.
Remark 2.1 Note that when ∇f is supposed to be globally Lipschitz continuous, the above proof can be notably simplified by just applying the classical Cauchy-Lipschitz theorem.

Asymptotic convergence properties of (DINAM)
In this section, we study the asymptotic behavior of the solution trajectories of (DINAM). For each solution trajectory t → x(t) of (DINAM), we show that the weak limit w- Before stating our main result, notice that B(p) is uniquely defined for p ∈ S.
By the monotonicity of ∇f , we have Replacing ∇f (p 1 ) with -B(p 1 ) and ∇f (p 2 ) with -B(p 2 ), we get

General case
The general line of the proof is close to that given by Attouch and Laszlo in [8,9]. The first major difference with the approach developed in [8,9] is that in our context, thanks to the hypothesis of cocoercivity on the nonpotential part, we do not need to go through the Yosida regularization of the operators. The second difference is that we treat the potential and nonpotential operators in a differentiated way. These points are crucial for applications to numerical algorithms, because the computation of the Yosida regularization of the sum of the two operators is often out of reach numerically.
The following theorem states the asymptotic convergence properties of (DINAM).
Then, for any solution trajectory x : [0, +∞[ → H of (DINAM), the following properties are satisfied: Proof Lyapunov analysis. Set A := B + ∇f and A β : where c and δ are coefficients to adjust. Using the differentiation chain rule for absolutely continuous functions (see [22,Corollary VIII.10]) and (DINAM), we geṫ We have Using the fact that p ∈ S, ∇f is monotone, and B is λ-cocoercive, we have and E p : [0, +∞[ → R be the energy function given by Since f is convex, we have (t) ≥ 0 for all t ≥ 0. This implies E p (t) ≥ 0 for all t ≥ 0 as well. We havė (3.9) By using (3.8) and (3.9), equation (3.7) can be rewritten aṡ Let us eliminate the term ∇f (x(t)) -∇f (p) from this relation by using the elementary algebraic inequality We obtaiṅ . According to Lemma A.3, and since a = δ = cγ -1 > 0, we have that q is positive definite if and only if 4agb 2 > 0. Equivalently Our aim is to find c such that cγ -1 > 0 and such that (3.12) is satisfied. Take δ := cγ -1 > 0 as a new variable. Equivalently, we must find δ > 0 such that After development and simplification we obtain Therefore, we just need to assume that Elementary optimization argument gives that Therefore we end up with the condition When β b = β f = β, we recover the condition λγ > β + 1 γ .
Combining this with +∞ 0 According to (3.16) the trajectory x(·) is bounded. Set R := sup t≥0 x(t) . By assumption, ∇f is Lipschitz continuous on the bounded sets. Let L R < +∞ be the Lipschitz constant of ∇f on B(0, R). Since B is λ-cocoercive, it is 1 λ -Lipschitz continuous. Therefore A is L-Lipschitz continuous on the trajectory with L := L R + 1 λ . Therefore for all t ≥ 0.   By using (DINAM), we havë Since the second member of the above equality belongs to L 2 (0, +∞; H), we finally get +∞ 0 ẍ(t) 2 dt < +∞.
Note that the left member of (3.28) can be rewritten as a derivative of a function, preciselÿ Let us prove that the function h given in (3.29) is bounded from below by some constant. Indeed, since the terms q p (t) and A β (x(t)) -A β (p), x(t)p are nonnegative, we have According to the boundedness of x(·) andẋ(·) (see (3.16) and (3.26)), we deduce that there exists m ∈ R such that Let us introduce the real-valued function ϕ : We have ϕ (t) =ḣ(t)g(t) ≤ 0. Hence, the function ϕ is nonincreasing on [0, +∞[. This classically implies that the limit of ϕ exists as t → +∞. Since g ∈ L 1 (0, +∞), we deduce that lim t→+∞ h(t) exists. Using the fact that A β (x(t)) -A β (p), x(t)p tends to zero as t → +∞ (a consequence of (3.25) and x(·) bounded), we obtaiṅ with limit of θ (t) exists as t → +∞. The existence of the limit of q p then follows from a classical general result concerning the convergence of evolution equations governed by strongly monotone operators (here γ Id, see Theorem 3.9, p. 88 in [21]). This means that, for all p ∈ S, To complete the proof via Opial's lemma, we need to show that every weak sequential cluster point of x(t) belongs to S. Let t n → +∞ such that x(t n ) x * , n → +∞. We have A x(t n ) → 0 strongly in H and x(t n ) x * weakly in H.

From the closedness property of the graph of the maximally monotone operator
Consequently, x(t) converges weakly to an element of S as t goes to +∞. The proof of Theorem 3.1 is thereby completed.
Remark 3.1 In the statement of Theorem 3.1, the parameters have to satisfy a certain condition. If the rest of parameters are fixed, then the set of λs that fulfill the inequality can easily be found. Likewise, the feasible set of γ s if the other parameters are fixed can be determined explicitly.
In fact, let us rewrite condition (3.1) as follows: Thanks to we immediately deduce that Therefore (3.30) is equivalent to This in turn is equivalent to (3.31) From the first inequation of (3.31), we deduce that From the second inequation of (3.31), we deduce that Therefore, γ > 1 4λ Since (3.33) implies (3.32), we obtain that the feasible set of γ s is defined by γ > 1 4λ

Case β b = β f
Let us specialize the previous results in the case β b = β f . We set β b = β f := β > 0 and A := ∇f + B. We thus consider the evolution system The existence of strong global solutions to this system is guaranteed by Theorem 2.1. The convergence properties as t → +∞ of the solution trajectories generated by this system is a consequence of Theorem 3.1 and are given below. introduced by Attouch and Maingé in [10] to study the second-order dynamic (1.3) without geometric damping. With respect to [10], the introduction of the geometric damping, i.e., taking β > 0, provides some useful additional estimates.

Numerical illustrations
In this section, we give some numerical illustrations of (DINAM).

From continuous dynamic to algorithms
Let us first give some indications concerning the algorithms obtained by temporal discretization of the continuous dynamic (DINAM). Their convergence analysis will be postponed to another research investigation. Let us recall the condensed formulation of (DINAM) where A := ∇f + B and A β := β b B + β f ∇f . Take a fixed time step h > 0, and consider the following finite-difference scheme for (DINAM): This scheme is implicit with respect to the nonpotential B and explicit with respect to the potential operator ∇f . The temporal discretization of the Hessian driven damping ). After expanding (4.1), we obtain Set s := h 1+γ h and α := 1 1+γ h . So we have where B h = (h + β b )B, and By combining (4.4) and (4.5), we obtain the following algorithm, called (DINAAM). It is a splitting algorithm which involves the operators ∇f and B separately. (DINAAM): Initialize: x k+1 = (I + sB h ) -1 (y k ). (4.6)

Numerical experiments for the continuous dynamics (DINAM)
A general method to generate monotone cocoercive operators which are not gradients of convex functions is to start from a linear skew symmetric operator A and then take its Yosida approximation A λ . As a model situation, take H = R 2 and start from A equal to the rotation of angle π 2 . We have A = 0 -1 1 0 . An elementary computation gives that, for any λ > 0, A λ = 1 1+λ 2 λ -1 1 λ , which is therefore λ-cocoercive. As a consequence, for λ = 1, we obtain that the matrix B = 1 -1 1 1 is 1 2 -cocoercive. With these basic blocks, one can easily construct many other cocoercive operators which are not potential operators. For that, use Lemma A.1 which gives that the sum of two cocoercive operators is still cocoercive, and therefore the set of cocoercive operators is a convex cone. We take H = R 2 equipped with the usual Euclidean structure. Let us consider B as a linear operator whose matrix in the canonical basis of R 2 is defined by B = A λ for λ = 5. According to the above remark, we can check that B is λ-cocoercive with λ = 5 and that B is a nonpotential operator. To observe the classical oscillations, in the heavy ball with friction, we take f : We set γ = 0.9. It is clear that f is convex but not strongly convex. We study three cases: (1) β b = 1, β f = 0.5, (2) β b = 0.5, β f = 1, and (3) β b = β f = 0.5. As a straight application of Theorem 3.1, we obtain that the trajectory x(t) generated by (DINAM) converges to x ∞ , where x ∞ ∈ S = (B + ∇f ) -1 (0) = {0}. The trajectory obtained by using Matlab is depicted in Fig. 1, where we represent the components x 1 (t) and x 2 (t) in red and blue respectively. Now we study the behavior of the trajectories by considering more different values of β b and β f . We study four cases in Fig. 2. The plots of the second variable of the solutions have been depicted in Fig. 2(a), while in Fig. 2  The behavior of (DINAAM) for a high dimension problem Therefore f is strongly convex. Take Then B is cocoercive. Indeed, for any x, y ∈ R n , Bx -By, xy = x 1y 1 2 + x 2y 2 2 + · · · + x ny n 2 ≥ 1 2 2 x 1y 1 2 + x 2y 2 2 + x 3y 3 2 + · · · + x ny n 2 = 1 2 Bx -By 2 .
If the matrix M has not full column rank with M M + B nonsingular, then In our experiment, we pick M a random 10 × 100 matrix which has not full column rank. Set γ = 3, β b = 1, β f = 1 and the operator B as presented above. Thanks to Corollary 3.1, we conclude that the trajectory x(t) generated by the system (DINAM) converges to x ∞ = (M M + B) -1 M b. Implementing the algorithm (DINAAM) in Matlab, we obtain the plot of k versus the norm of B(x k ) + ∇f (x k ). Similarly, we study several cases by changing the parameters β b , β f . This is depicted in Fig. 3.
Before ending this part, we discuss an application of our model to dynamical games. The following example is taken from Attouch and Maingé [10] and adapted to our context. (i) H = X 1 × X 2 is the Cartesian product of two Hilbert spaces equipped with norms · X 1 and · X 2 respectively. In which, x = (x 1 , x 2 ), with x 1 ∈ X 1 and x 2 ∈ X 2 , stands for an element in H; (ii) f : X 1 × X 2 → R is a convex function whose gradient is Lipschitz continuous on bounded sets; (iii) B = (∇ x 1 L, -∇ x 2 L) is the maximally monotone operator which is attached to a smooth convex-concave function L : X 1 × X 2 → R. The operator B is assumed to be λ-cocoercive with λ > 0. In our setting, with x(t) = (x 1 (t), x 2 (t)) the system (DINAM) is written According to Theorem 3.1, Structured systems such as (4.8) contain both potential and nonpotential terms which are often present in decision sciences and physics. In game theory, (4.8) describes Nash equilibria of the normal form game with two players 1, 2 whose static loss functions are respectively given by F 2 : (x 1 , x 2 ) ∈ X 1 × X 2 → F 2 (x 1 , x 2 ) = f (x 1 , x 2 ) -L(x 1 , x 2 ). (4.9) f (·, ·) is their joint convex payoff, and L is a convex-concave payoff with zero-sum rule. For more details, we refer the reader to [10]. As an example, take X 1 = X 2 = R and L : R 2 → R given by L(x) = 1 2 (x 2 1 -2x 1 x 2x 2 2 ). Then B = (∇ x 1 L, -∇ x 2 L) = 1 -1 1 1 . Pick f (x) = 1 2 (3x 2 1 -2x 1 x 2 + x 2 2 )x 1 -2x 2 . The Nash equilibria described in (4.8) can be solved by using (DINAM). Take γ = 3, β b = 0.5, β f = 0.5 and x 0 = (1, -1),ẋ 0 = (-10, 10) as initial conditions, then the numerical solution for (DINAM) converges to x ∞ = ( 3 4 , 1) which is the solution of (4.8) as well. The numerical trajectories and phase portrait of our model applied to dynamical games are depicted in Fig. 4.

The nonsmooth case
The equivalence obtained in Proposition 2.1 between (DINAM) and a first-order evolution system in time and space allows a natural extension of both our theoretical and numerical results to the case of a convex, lower semicontinuous and proper function f : H → R ∪ {+∞}. It suffices to replace the gradient of f with the convex subdifferential ∂f . We recall that the subdifferential of f at x ∈ H is defined by (g-DINAM) The prefix g in front of (DINAM) stands for generalized. Note that the first equation of (g-DINAM) is now a differential inclusion, because of the possibility for ∂f (x(t)) to be multivalued. By taking f = f 0 + δ C , where δ C is the indicator function of a constraint set C, the system (g-DINAM) allows to model damped inelastic shocks in mechanics and decision sciences, see [11]. The original aspect comes from the fact that (g-DINAM) now involves both potential driven forces (attached to f 0 ) and nonpotential driven forces (attached to B). As we will see, taking into account shocks created by nonpotential driving forces is a source of difficulties. Let us first establish the existence and uniqueness of the solution trajectory of the Cauchy problem. Recall that strong solution means that x(·) and y(·) are locally absolutely continuous functions whose distributional derivativesẋ andẏ belong to L 2 (0, T, H) for any T > 0.