Asymptotic for the perturbed heavy ball system with vanishing damping term

We investigate the long time behavior of solutions to the differential equation $\ddot{x}(t)+\frac{c}{\left( t+1\right) ^{\alpha}}\dot{x}(t)+\nabla \Phi\left( x(t)\right) =g(t),~t\geq0, $ where $c$ is nonnegative constant, $\alpha\in\lbrack0,1[,$ $\Phi$ is a $C^{1}$ convex function on a Hilbert space $\mathcal{H}$ and $g\in L^{1} (0,+\infty;\mathcal{H}).$ We obtain sufficient conditions on the source term $g(t)$ ensuring the weak or the strong convergence of any trajectory $x(t)$ as $t\rightarrow+\infty$ to a minimizer of the function $\Phi$ if one exists.

Using classical arguments (see for instance [7]), one can easily prove that if the function ∇Φ : H → H is Lipschitz on bounded subset of H, then for any initial data (x 0 , x 1 ) ∈ H × H, the equation (1.1) has a unique global solution x ∈ W 2,1 loc (0, +∞; H) satisfying (x(0), ẋ(0)) = (x 0 , x 1 ).Moreover, the associated energy function is nonincreasing and converges to 0 as t → +∞.Hence hereafter, we will assume that x ∈ W 2,1 loc (0, +∞; H) is a solution to (1.1) and we will focus our attention on the study of the asymptotic behavior of x(t) as t → +∞ and on the rate of convergence of the energy function W.
Before setting the main results of our present paper, let us first recall some previous results: In the pioneer paper [1], Alvarez considered the case where α = 0 and g = 0.He proved that x(t) converges weakly in H as t → +∞ to a minimizer of the function Φ.Moreover, he showed that the convergence is strong if either the function Φ is even or the interior of the set arg min Φ is not empty.In [6], Haraux and Jendoubi extended the weak convergence result of Alvarez to the case where the source term g belongs to the space L 1 (0, +∞; H).Recently, Cabot and Frankel [5] studied (1.1) where g = 0 and α ∈]0, 1[.They proved that every bounded solution x(t) (i.e.x ∈ L ∞ (0, +∞; H)) converges weakly toward a critical point of Φ.In a very recent work [8], the second author of this paper improved the result of Cabot and Frankel by getting rid of the superfluous hypothesis on the boundedness of the solution.Moreover he established that W (t) = •( 1 t 2 ᾱ ) as t → +∞ for every ᾱ < α.In [7], Jendoubi and May proved that the main convergence result of Cabot and Frankel remains true if the source term g satisfies the condition +∞ 0 (1 + t) g(t) dt < ∞.Recently, this result was improved in [4].In fact, we proved that if the solution x(t) is bounded and the function g satisfies the optimal condition then x(t) converges weakly to some element of arg min Φ and W (t) = •( 1 t 2α ) as t → +∞.One of the main purpose of this paper is to prove that the sole assumption (1.3) guarantees the boundedness (and therefore the weak convergence) of the solution x(t).We notice that, in a very recent work [2], Attouch, Chbani, Peypouquet and Redont have considered the equation (1.1) in the case α = 1.They have proved that if c > 3 and +∞ 0 (1 + t) g(t) dt < ∞ then x(t) converges weakly to some element of arg min Φ and that W (t) = O( 1 t 2 ).Moreover, they have established the strong convergence of x(t) in the case where the function Φ is even or the interior of the subset arg min Φ is not empty.In this paper, we extend their results to the case α < 1.
Our main first result is the weak convergence of the trajectories of (1.1) under the optimal condition (1.3) on the source term g.Theorem 1.1.Assume that +∞ 0 (t + 1) α g(t) dt < ∞.Then x(t) converges weakly in H as t → +∞ to some x * ∈ arg min Φ.Moreover the energy function W satisfies the two following properties: Our second theorem improves the result on the convergence rate of the energy function W obtained in [8] in the case where g = 0 and it will be useful in the proof of the strong convergence of the solution x(t) when the convex function Φ is even.
The next result shows that, as in the limit case α = 1 (see [2, Theorem 3.1]), the strong convergence of x(t) as t → +∞ holds if the interior of arg min Φ is not empty Theorem 1.3.Suppose that +∞ 0 (t + 1) α g (t) dt < ∞ and int (arg min Φ) = ∅.Then there exists some x * ∈ arg min Φ such that x (t) → x * strongly in H as t → +∞.
In the last theorem, we prove, under an assumption on the source term g slightly stronger than the optimal condition (1.3), the strong convergence of the solution x(t) when the potential function Φ is even Theorem 1.4.Suppose that +∞ 0 g (t) dt < ∞ and Φ is even (i,e.Φ(−x) = Φ(x), ∀x ∈ H).Then x (t) converges strongly in H as t → +∞ to some x * ∈ arg min Φ.

Proof of Theorem 1.1
The proof makes use of a modified version of the method used Attouch, Chbani, Peypouquet and Redont in [2].It relies on the study of a suitable Lyapunov function E and uses the following two classical lemmas.
Then, for all t ∈ [a, b] , The proof of this lemma is easy and similar to the proof of the classical Gronwall's lemma.
Lemma 2.2 (Opial's lemma [9]).Let x : [0, +∞[→ H. Assume that there exists a nonempty subset S of H such that: The proof of Opial's lemma is easy, see for instance [3].Let us now start the proof our theorem.We first define on [0, +∞[ the function where Γ (t) = t 0 γ (s) ds.A simple calculation yields that h satisfies the differential equation .
Let x * ∈ arg min Φ and define the function By differentiating, we obtain Hence by sing (1.1), we get Since the function Φ is convex, we have Inserting this inequality in (2.5) yields From (2.2) and (2.3), 2h ′ (t) − 1 → −1 as t → +∞.Then there exists t 1 ≥ 0 such that Therefore E is a decreasing function on [t 1 , +∞[.Then for every t ≥ t 1 , E(t) ≤ E(t 1 ), which implies that where Using now the Cauchy-Schwarz inequality, we obtain Hence by applying the Gronwall-Bellman lemma we obtain Taking the inner product of (1.1) with ẋ (t) , we obtain Multiplying the last inequality by h 2 (t) and using the fact that we get after integration by parts on [0, t], Combining (2.10) and (2.12), we get (1.5).Let us now prove the weak convergence of x(t) as t → +∞.We first notice that since W (t) → 0 as t → +∞ and Φ is weak lower semicontinuous (in fact Φ is continuous and convex), then the first item i) of Opial's lemma is satisfied with S = arg min Φ.Hence, it remains to prove that for any x * in arg min Φ, the associated function z(t) := 1 2 x(t) − x * 2 converges as t → +∞.A simple calculation using (1.1) gives Hence by using the monotonicity property of the operator ∇Φ, the fact that ∇Φ(x * ) = 0, and the Cauchy-Schwarz inequality we get where h is the function defined by (2.1) at the beginning of the proof.Hence, by using (2.3) and (2.14) we deduce that the function K, and therefore the positive part [ ż] + (t) of ż(t) belongs to the space L 1 (0, +∞) .Then the limit of z (t) as t → +∞ exists.This proves the item ii) of the Opial's lemma and completes the proof of Theorem 1.1.

Proof of Theorem 1.3
We follow the same method used in the proof of [2,Theorem 3.1].The assumption int(arg min Φ) = ∅ implies the existence of z 0 ∈ H and r > 0 such that for any v ∈ H with v ≤ 1 we have ∇(z 0 + rv) = 0 which implies by the monotocity property of ∇Φ that for any z ∈ H we have ∇Φ(z), z − z 0 − rv ≥ 0. Thus by taking the supremum on v, we get ∇Φ(z), z − z 0 ≥ r ∇Φ .

Proof of Theorem 1.4
Let T > 0. We define on [0, T ] the function: By a classical calculus using (1.1) and the fact that Φ is convex and even, we obtain where M = 2 sup s≥0 x(s) (Recall that from Theorem 1.1 we have x ∈ L ∞ (0, +∞, H)).
Since Φ is even and convex, we have 0 ∈ arg min (Φ) .Hence by using the convergence of the function z (t) = 1 2 x (t) − x * 2 proved in Theorem 1.1 with x * = 0, we infer that the limit of x (t) 2 as t goes to +∞ exists which implies that (5.2) lim t,T →+∞ x (t) 2 − x (T ) 2 = 0.
On the other hand, in view of Fubini theorem and (2.3), we conclude that x (t) satisfies the Cauchy convergence criterion in the Hilbert space H as t → +∞, and hence converges strongly in H as t → +∞.