Asymptotic for a second order evolution equation with damping and regularizing terms

1.   Introduction

Let $\mathcal{H}$ be a real Hilbert space endowed with the inner product $\langle., .\rangle$ and the associated norm $\left\Vert.\right\Vert.$ Let $\Phi, U:\mathcal{H}\rightarrow\mathbb{R}$ be two convex continuously differentiable functions and $\gamma, \varepsilon$ two real positive functions defined on a fixed time interval $[t_{0}, +\infty)$ for some $t_{0} > 0.$ Motivated by the work ^[3] of Attouch, Chbani, and Riahi on the asymptotic behavior of the trajectories of the asymptotic vanishing damping dynamical system with regularizing regularizing term

we investigate in this paper the long time behavior, as $t\rightarrow+\infty,$ of the trajectories of the following generalized version of the (AVD$_{\alpha, \varepsilon}$) dynamical system

For the importance and the applications of these two dynamical systems and many other related dynamical systems in Mechanics and Optimization, we refer the reader to ^[2,5,6,14] and references therein.

Throughout this paper, we assume the following general hypothesis:

(H$_{1}$) The functions $\Phi, U:\mathcal{H}\rightarrow\mathbb{R}$ are convex, differentiable, and bounded from below. We set $\Phi^{\ast} = \inf _{x\in\mathcal{H}}\Phi(x)$ and $U^{\ast} = \inf_{x\in\mathcal{H}}U(x).$

(H$_{2}$) The set $S_{\Phi}: = \mbox{argmin}\Phi = \{z\in\mathcal{H}:\; \Phi(z) = \Phi^{\ast}\}$ is nonempty.

(H$_{3}$) The gradient functions $\nabla\Phi$ and $\nabla U$ of $\Phi$ and $U$ are Lipschitz on bounded subsets of $\mathcal{H}.$

(H$_{4}$) The function $\gamma:[t_{0}, +\infty)\rightarrow (0, +\infty)$ is absolutely continuous and satisfies the following property: there exist $t_{1}\geq t_{0}$ and two real constants $K_{1}, K_{2} > 0$ such that

for almost every $t\geq t_{1}.$

(H$_{5}$) The function $\varepsilon:[t_{0}, +\infty)\rightarrow(0, +\infty)$ is absolutely continuous, nonincreasing and satisfies

Proceeding as in the proof of [Theorem 3.1, ^[6]] and using the classical Cauchy-Lipschitz theorem and the energy function

one can easily prove that for every initial data $(x_{0}, v_{0})\in \mathcal{H}\times\mathcal{H},$ the dynamical system (GAVD$_{\gamma, \varepsilon}$) has a unique solution $x(.)\in C^{2}(t_{0}, +\infty; \mathcal{H})$ which satisfies $x(t_{0}) = x_{0}$ and $x^{\prime}(t_{0}) = v_{0}.$ Therefore, we assume in what follows that $x(.)$ is a global solution of (GAVD$_{\gamma, \varepsilon}$) and focus our attention on the study of the long time behavior of $x(t)$ as $t$ goes to infinity. Before starting the presentation of the main contributions of this work in this direction, let us first recall some well known results on the asymptotic behavior of solutions of a variant dynamical systems related to (GAVD$_{\gamma, \varepsilon}$). In the pioneer work ^[1], Alvarez considered the case where $\gamma(.)$ is constant and $\varepsilon = 0.$ He established that the trajectory $x(t)$ converges weakly to some element $\bar{x}$ of $S_\Phi$. He also proved that the rate of convergence of $\Phi(x(t))$ to $\Phi^{\ast}$ is of order $\circ(1/t)$ (see ^[2]). To overcome the drawback of the weak convergence to a non identified minimizer of $\Phi$, Attouch and Cazerniki ^[5] proved that adding a regularizing term $\varepsilon(t)x(t)$ forces any trajectory $x(t)$ of the system

to converge strongly to the element $x^{\ast}$ of minimum norm of the set $S_\Phi$ provided that $\int_{t_{0}}^{+\infty} \varepsilon(t)dt = +\infty$. Using a different approach, Jendoubi and May ^[10] proved that this strong convergence result remains true even if a perturbation integrable term $g(t)$ is added to the equation (1.2)$.$ In an other direction, in order to improve the rate of convergence of $\Phi(x(t))$ to $\Phi^{\ast},$ Su, Boyd, and Candes ^[14] introduced the following dynamical system which is the continuous version of the Nestrov's accelerated minimization method ^[12]:

They proved that if $\alpha\geq3$ then

This result was later improved in ^[4] and ^[11]. In fact it was proved that if $\alpha > 3$ then $x(t)$ converges weakly to some element $\bar{x}$ of $S_\Phi$ and that

In order to benefit at the same time of the quick minimization property $\Phi(x(t))-\Phi^{\ast} = \circ(1/t^{2})$ due to the presence of the vanishing damping term $\gamma(t) = \frac{\alpha}{t}$ in (1.3) and the strong convergence of the trajectories of (1.2) to a particular minimizer of $\Phi$ which is a consequence of the regularizing term $\varepsilon(t)x(t),$ Attouch, Chbani, and Riahi ^[3] have considered the dynamical system (AVD$_{\alpha, \varepsilon}$) and have established some properties of the asymptotic behavior of its trajectories which are summarized in the following theorem.

Theorem 1.1 (Attouch, Chbani and Riahi). Let $x\in C^{2}(t_{0}, +\infty; \mathcal{H})$ be a solution of (AVD$_{\alpha, \varepsilon}$). The following assertions hold:

(A) If $\alpha > 1$ and $\int_{t_{0}}^{+\infty}\frac{\varepsilon(t)} {t}dt < +\infty,$ then $\int_{t_{0}}^{+\infty}\frac{\left\Vert x^{\prime }(t)\right\Vert ^{2}}{t}dt < +\infty,$ $\lim \limits_{t\rightarrow+\infty}x^{\prime}(t) = 0$ and $\lim\limits_{t\rightarrow+\infty} \Phi(x(t)) = \Phi^{\ast}.$

(B) If $\alpha > 3$ and $\int_{t_{0}}^{+\infty}t\varepsilon(t)dt < +\infty,$ then $x(t)$ converges weakly to some element of $S_\Phi$.

Furthermore, the associated energy function $E(t) = \frac{1}{2}\left\Vert x^{\prime }(t)\right\Vert ^{2}+\Phi(x(t))-\Phi^{\ast}$ satisfies $E(t) = \circ(1/t^{2})$ and $\int_{t_{0}}^{+\infty}t E(t)dt < +\infty.$

(C) If the function $\varepsilon$ satisfies moreover one of the following hypothesis

(H_5a) $\lim\limits_{t\rightarrow+\infty}t^{2}\varepsilon(t) = +\infty$ if $\alpha = 3$

(H_5b) $t^{2}\varepsilon(t)\geq c > \frac{4}{9}\alpha(\alpha-3)$ if $\alpha > 3$

(H_5c) $\int_{t_{0}}^{+\infty}\frac{\varepsilon(t)}{t}dt = +\infty$

then $\liminf\limits_{t\rightarrow+\infty}\left\Vert x(t)-x^{\ast}\right\Vert = 0$ where $x^{\ast}$ is the element of minimal norm of the set $S_\Phi.$

In our present paper, we improve and extend these results to the general dynamical system (GAVD$_{\gamma, \varepsilon}$). Moreover, we highlight some new asymptotic properties of the trajectories of (GAVD$_{\gamma, \varepsilon}$).

Our first main result is a general minimization property of (GAVD$_{\gamma, \varepsilon}$) which improves the assertion (A) in the previous theorem.

Theorem 1.2 (A general minimization property of (GAVD$_{\gamma, \varepsilon}$)). Let $x(.)$ be a solution of (GAVD$_{\gamma, \varepsilon}$). Then $\int_{t_{0}}^{+\infty}\gamma(t)\left\Vert x^{\prime }(t)\right\Vert ^{2}dt < +\infty,$ and the energy function $W(t),$ defined by (1.1) decreases and converges to $0$ as $t\rightarrow+\infty$. In particular $\lim\limits_{t\rightarrow+\infty}x^{\prime}(t) = 0$ and $\lim\limits_{t\rightarrow+\infty }\Phi(x(t)) = \Phi^{\ast}.$

The second result concerns the weak convergence properties of the trajectories of (GAVD$_{\gamma, \varepsilon}$). The first part of this result is similar to the assertion (B) in Theorem 1.1. Our proof, which is different from the arguments given by Attouch, Chbani, and Riahi [Theorem 3.1, ^[3]], provides an other confirmation of the fact, noticed recently in many works as ^[2,3,11] and ^[14], that the value $\alpha = 3$ in the the system (1.3) is critical and somehow mysterious. The second part of the theorem is a simple result on the weak convergence to a common minimizer of the two convex functions $\Phi$ and $U$. At our knowledge, this result was not known even in the case where the damping term $\gamma$ is constant. A comparable result has been proved by Cabot(see [Proposition 2.5, ^[7]]) for the first order system $x^{\prime}(t)+\nabla\Phi(x(t))+\varepsilon(t)\nabla U(x(t)) = 0$.

Theorem 1.3 (Weak convergence properties of (GAVD$_{\gamma, \varepsilon}$)). Assume that there exist $t_{1}\geq t_{0}, \; 0\leq\theta\leq1, \alpha > 0$ with $\alpha > 3$ if $\theta = 1$ such that

where $\left[ \left(t^{\theta}\gamma(t)\right) ^{\prime}\right] ^{+} = \max\{0, \left(t^{\theta}\gamma(t)\right) ^{\prime}\}.$ Let $x(.)$ be a solution of (GAVD$_{\gamma, \varepsilon}$). Then the two following properties hold:

(P$_{1}$) If $\int_{t_{0}}^{+\infty}t^{\theta}\varepsilon(t)dt < +\infty$ then $x(t)$ converges weakly to some element of $S_\Phi.$

(P$_{2}$) If $S_\Phi\cap S_U\neq\varnothing$ and $\liminf\limits_{t\rightarrow+\infty}t^{1+\theta}\varepsilon(t) > 0\;$then $x(t)$ converges weakly to some element of $S_\Phi\cap S_U.$

Moreover, in both case, the energy function $W$ satisfies the following asymptotical behavior

The last result deals with the strong convergence of the trajectories of (GAVD$_{\gamma, \varepsilon}$) to a minimizer of the function $U$ on the set of minimizers of $\Phi$.

Theorem 1.4 (Strong convergence properties of (GAVD$_{\gamma, \varepsilon})$). Assume that $U$ is strongly convex and $\gamma (t) = \frac{\alpha}{t^{\theta}}$ such that $\alpha > 0$ and $0\leq\theta < 1$ or $\alpha > 3$ if $\theta = 1.$ Suppose in addition that $\int_{t_{0}}^{+\infty }t^{\theta}\varepsilon(t)dt = +\infty.$ Let $x(.)$ be a solution of (GAVD$_{\gamma, \varepsilon}$). Then the two following assertions hold true:

(Q$_{1}$) If $x^{\prime}(t) = \circ(1/t^{\theta})$ and $\int_{t_{0} }^{+\infty}t^{\theta}\left\Vert x^{\prime}(t)\right\Vert ^{2}dt < +\infty$ then $x(t)$ converges strongly to the unique minimizer $p^{\ast}$ of $U$ on $S_\Phi.$

(Q$_{2}$) If the unique minimizer $x^{\ast}$ of $U$ on $\mathcal{H}$ belongs to $S_\Phi$ then $x(t)$ converges strongly to $x^{\ast}$ and the energy function $W$ satisfies the asymptotic properties (1.5).

Combining Theorem 1.2 and Theorem 1.4 provides a new proof of following important result due to Attouch and Cazernicki (see [Theorem 2.3, ^[5]]).

Theorem 1.5 (Attouch and Cazernicki). Let $\alpha > 0$. If $\int_{t_{0}}^{+\infty }\varepsilon(t)dt = +\infty$, then any trajectory $x(.)$ to the dynamical system

converges strongly to the projection of zero on the closed and convex subset $S_{\Phi}$.

2.   A general minimization property of (GAVD$_{\gamma, \varepsilon}$)

This section is devoted to the proof of Theorem 1.2. The main idea of the proof is inspired by the paper ^[9].

Proof. Differentiating the energy function $W$ defined by (1.1) and using the equation (GAVD$_{\gamma, \varepsilon}$), we obtain

Hence

and the function $W(t)$ decreases and converges to some positive real number $W_{\infty}$ as $t\rightarrow+\infty$. Therefore, to conclude, we just have to show that $W_{\infty}\leq0.$ Let $v$ be an arbitrarily element of $\mathcal{H}$. Consider the function

Using the equation (GAVD$_{\gamma, \varepsilon}$) and the convexity of $\Phi$ and $U,$ one can easily check that

Recalling that $W(t)\geq W_{\infty},$ we get

where $A_{\infty} = W_{\infty}+\Phi^{\ast}-\Phi(v).$

Integrating the last inequality from $t_{0}$ to $t > t_0$ and using the fact that $\gamma h_{v}\geq0$ and the assumption $\gamma^{\prime}(t)\leq\frac{K_{2}}{t^{2}},$ we find

where

Since $\gamma(t)\geq \frac{K_1}{t}$, we deduce from (2.2) that $\int_{t_{0}}^{+\infty}\frac{\left\Vert x^{\prime}(s)\right\Vert ^{2}}{s}ds < \infty$ which, thanks to [Lemma 3.2, ^[9]], implies that

Using now the Cauchy-Schwarz inequality, we infer

which implies that $\lim\limits_{t\rightarrow+\infty}f_{v}(t) = 0$. Hence, we deduce that

Recalling that since $W\;$ is bounded, $x^{\prime}$ is also bounded. Thus, from (2.7), we infer that

Finally, dividing the inequality (2.4) by $t$, using the estimates (2.6), (2.8), (2.9) and letting $t\rightarrow+\infty$, we obtain $A_{\infty}\leq0,$ which implies that $W_{\infty}\leq\Phi (v)-\Phi^{\ast}.$ Since this holds for every $v\in\mathcal{H},$ the required result $W_{\infty}\leq0$ follows.

Remark 2.1. Let us notice that in the proof of Theorem 1.2, we did not use the hypothesis ($H_2$). Moreover, we can prove that if $S_\Phi$ is empty, then any solution $x(.)$ of the (GAVD$_{\gamma, \varepsilon}$) system satisfies $\|x(t)\|\rightarrow+\infty$ as $t\rightarrow+\infty$. Indeed, otherwise there exists a sequence $(t_n)_n$ tending to $+\infty$ so that $(x(t_n))_n$ converges weakly to an element $\bar{x}\in\mathcal{H}$. From the lower semi-continuity property it follows that

which, thanks to the fact $\lim_{t\rightarrow+\infty }\Phi(x(t)) = \Phi^{\ast}$, implies that $\Phi(\bar{x})\leq \Phi^*$ and contradicts the assumption $S_\Phi = \emptyset$.

3.   Weak convergence properties of (GAVD$_{\gamma, \varepsilon}$)

In this section, we prove Theorem 1.3. The proof relies on the classical Opial's lemma and the following technical lemma which will be also useful in the study of the strong convergence properties of the trajectories of (GAVD$_{\gamma, \varepsilon}$) in the next section.

Lemma 3.1. Assume that the function $\gamma(.)$ satisfies the assumption (1.4) in Theorem 1.3. Let $x(.)$ be a solution of (GAVD$_{\gamma, \varepsilon}$) and let $v\in S_\Phi$ such that the positive function $t^{\theta}r_{v}(t)$ belongs to $L^{1}(t_{0}, +\infty; \mathbb{R})$ where $r_{v}(t) = \varepsilon(t)(U(v)-U^{\ast})$. Then the function $h_{v}(t) = \frac{1}{2}\left\Vert x(t)-v\right\Vert ^{2}$ converges as $t\rightarrow+\infty$ and the energy function $W$ satisfies the asymptotic property (1.5).

Proof. First, we notice that up to take $t_{1}$ large enough we can assume that

with $K > 3$ and $K = \alpha$ if $\theta = 1.$

Let $\lambda(t) = t^{1+\theta}$. Using (2.1) and the above inequality, we find

Therefore,

Multiplying (2.3) by $\lambda^{\prime}(t)$ (we recall that, since $v\in S_\Phi,$ $\Phi(v) = \Phi^{\ast}$) and using the above inequality, we obtain

Integrating this inequality from $t_{1}$ to $t > t_1$ leads after some simple computations to the following inequality

where

and

which is a finite real constant thanks to the assumption on the function $r_{v}$. Let $A(\theta)$ and $\mu(\theta)$ be two strictly positive constants such that $A(\theta)+\mu(\theta) < \alpha(\theta+1)$ if $\theta < 1$ and $A(\theta)+\mu(\theta) = 2(\alpha-1)$ if $\theta = 1.$ Since $\gamma(t)\geq\frac{\alpha}{t^{\theta}}$, we have

Therefore, up to take $t_{1}$ large enough in the case $\theta < 1$, we can assume that

Using now the fact that

it follows, from the estimate (3.5), that

Applying now the elementary inequality

with $x = \sqrt{h_{v}(t)}$, we deduce that for every $t\geq t_{1}$

where

Inserting (3.6) in the inequality (3.2), we obtain

Let us notice that if $0\leq \theta < 1$ then $\lim\limits_{t\to+\infty}B(\theta, t) = 0$ and in the case where $\theta = 1$, since $\alpha > 3$, one can choose $0 < \mu(1) < \frac{2}{3}(\alpha-3)$ to get

Hence, up to take $t_{1}$ large enough, we can assume that, for every $0 \leq\theta\leq 1$, there exists a constant $\nu(\theta) > 0$ such that

In particular, the inequality (3.7) implies

Recalling that the function $g_{\theta}$ is integrable over $[t_{1}, +\infty)$ and applying the Gronwall's lemma, we deduce that the function $h_{v}$ is bounded. Hence, by going back to the inequality (3.7), we infer that

and

Now, using the fact that the energy function $W$ is decreasing, we deduce from (3.8) that $t^{1+\theta}W(t)\rightarrow0$ as $t\rightarrow+\infty$; in fact, for every $t\geq 2t_{1}$, we have

To conclude, it remains to prove that $\lim\limits_{t\rightarrow+\infty}h_{v}(t)$ exists. From (2.3), the function $h_{v}$ satisfies the differential inequality

where

The assumption on the function $r_{v}$ and the estimate (3.8) imply that $t^{\theta}\zeta(t)\in L^{1}(t_0, +\infty; \mathbb{R}^{+}),$ thus the existence of $\lim\limits_{t\rightarrow+\infty}h_{v}(t)$ follows from the following lemma.

Lemma 3.2. Let $a > 0$ and $w:[a, +\infty)\rightarrow\mathbb{R}^{+}$ be a continuous function satisfying

where $\alpha$ and $\theta$ are nonnegative constants with $0\leq\theta\leq1$ and $\alpha > 1$ if $\theta = 1.$ Let $\varphi\in C^{2}(a, +\infty; \mathbb{R}^{+})$ satisfy a differential inequality

with $t^{\theta}\psi(t)\in L^{1}(a, +\infty; \mathbb{R}^{+}).$ Then $\lim\limits_{t\rightarrow+\infty}\varphi(t)$ exists.

Proof. Multiplying each member of the differential inequality (3.9) by $e^{\Gamma(t, a)}$, where

and integrating on $[t_0, t]$, we get

Applying Fubini's Theorem and using [Lemma 3.14, ^[8]], we deduce that there is a real constant $M > 0$ such that

We therefore infer from (2.2) that the positive part $[\varphi^{\prime}]^{+}$ of $\varphi^{\prime}$ belongs to $L^{1}(a, +\infty; \mathbb{R}^{+})$ which implies that $\lim\limits_{t\rightarrow+\infty}\varphi(t)$ exists.

Before starting the proof of Theorem 1.3, let us recall the classical Opial's lemma.

Lemma 3.3 (Opial's lemma).Let $x:[t_{0}, +\infty)\rightarrow\mathcal{H}.$ Assume that there exists a nonempty subset $S$ of $\mathcal{H}$ such that:

i) if $t_{n}\rightarrow+\infty$ and $x(t_{n})\rightharpoonup x$ weakly in $\mathcal{H}$, then $x\in S$,

ii) for every $z\in S,$ $\lim\limits_{t\rightarrow+\infty}\left\Vert x(t)-z\right\Vert$ exists.

Then there exists $z_{\infty}\in S$ such that $x(t)\rightharpoonup z_{\infty}$ weakly in $\mathcal{H}$ as $t\rightarrow+\infty.$

For a simple proof of Opial's lemma, we refer the reader to ^[13].

Proof of Theorem 1.3. :

Step 1: Proof of the property (P$_{1}$).

According to Lemma 3.1, $\lim\limits_{t\rightarrow+\infty}h_{v}(t)$ exists for every $v\in S_\Phi$ and the energy function $W$ satisfies (1.5)$.$ Let $t_{n}\rightarrow+\infty$ such that $x(t_{n})$ converges weakly in $\mathcal{H}$ to some $\bar{x}.$ Since $\Phi(x(t))\rightarrow\Phi^{\ast}$ as $t\rightarrow+\infty,$ the weak lower semi-continuity of the convex function $\Phi$ implies that $\Phi(\bar{x})\leq\Phi^{\ast}$ which means that $\bar{x}\in S_\Phi.$ By Opial's lemma, we deduce that $x(t)$ converges weakly in $\mathcal{H}$ as $t\rightarrow+\infty$ to some element of $S_\Phi.$

Step2: Proof of the property (P$_{2}$).

Let $v\in S = S_\Phi\cap S_U.$ Since $r_{v} = 0$, Lemma 3.1 implies that $\lim\limits_{t\rightarrow+\infty}h_{v}(t)$ exists and $W$ satisfies (1.5)$.$ Thus, in view of the assumption $\liminf\limits_{t\rightarrow+\infty }t^{\theta+1}\varepsilon(t) > 0$, we have $U(x(t))\rightarrow U^{\ast}$ as $t\rightarrow+\infty.$ Therefore, as in the above step, the weak lower semi-continuity of the convex functions $\Phi$ and $U$ yields that every sequential weak cluster point of $x(t), \;$as $t\rightarrow+\infty,$ belongs to the subset $S.$ This completes the proof of the property (P$_{2}$) thanks to Opial's lemma.

4.   Strong convergence properties of (GAVD$_{\gamma, \varepsilon}$)

This section is devoted to the proof of Theorem 1.4. Before proving separately the two properties (Q$_{1}$) and (Q$_{2}$), let us first recall some general facts about strongly convex functions and a Tikhonov approximation method ^[15]. Since the function $U$ is strongly convex, there exists a positive real $m$ such that $U(x)-\frac{m}{2}\left\Vert x\right\Vert ^{2}$ is convex (we say that $U$ is $m-$strongly convex). Moreover, for every nonempty, convex and closed subset $C$ of $\mathcal{H}$, the function $U$ has a unique minimizer $x_{C}^{\ast}$ on $C.$ Let $x^{\ast}$ be the minimizer of $U$ on $\mathcal{H}$ and $p^{\ast}$ its minimizer on $S_\Phi.$ For every $t\geq t_{0},$ we consider the function $\Phi_{t}$ defined on $\mathcal{H}$ by

Clearly, $\Phi_{t}$ is $\varepsilon(t)m$-strongly convex. Therefore, $\Phi _{t}$ satisfies the convex inequality

and has a unique global minimizer which we denote by $x_{\varepsilon(t)}.$ Adopting the Tikhonov method, we can prove that $x_{\varepsilon(t)}$ converges strongly to $p^{\ast}$ as $t\rightarrow+\infty.$ Indeed, since

and

then

Furthermore, since $U$ is strongly convex, $U(x)\rightarrow+\infty$ as $\left\Vert x\right\Vert\rightarrow+\infty$; hence from the inequality 4.3 we deduce that $(x_{\varepsilon (t)})_{t\geq t_{0}}$ is bounded. So, let $\tilde{x}\in\mathcal{H}$ be a weak limit of a sequence $(x_{\varepsilon (t_{n})})$ where $t_{n}\rightarrow+\infty$. Using the weak lower semi-continuity of the two convex functions $\Phi$ and $U$ and letting $t = t_{n}\rightarrow+\infty$ in the inequalities (4.2) and (4.3), we deduce that $\Phi(\tilde{x})\leq\Phi(p^{\ast})$ and $U(\tilde{x})\leq U(p^{\ast})$ which is, from the definition of $p^{\ast},$ is equivalent to $\tilde{x} = p^{\ast}.$ Consequently, we infer that $x_{\varepsilon(t)}$ converges weakly to $p^{\ast}$ as $t_{n}\rightarrow+\infty$. Now, since $U$ is $m-$strongly convex, we have

Hence, by (4.3), we conclude that $\lim\limits_{t\rightarrow+\infty}\left\Vert x_{\varepsilon(t)}-p^{\ast}\right\Vert = 0$ which completes the proof of the claim.

Proof of Theorem 1.4. Let us first prove the assertion (Q$_{1}$). We consider the function $h(t) = h_{p^{\ast}}(t) = \frac{1}{2}\left\Vert x(t)-p^{\ast}\right\Vert ^{2}.$ Using the equation (GAVD$\gamma, \varepsilon$) and the convex inequality (4.1), we obtain

In the last inequality we have used the fact that $p^{\ast}$ is also a minimizer of $\Phi.$ Set

The inequality (4.4) becomes

Let us prove that $\liminf\limits_{t\rightarrow+\infty}h(t) = 0.$ We argue by contradiction. As consequence of

there exists $t_{2}\geq t_{0}$ large enough and $\mu > 0$ such that $\sigma(t)\geq\mu$ for every $t\geq t_{2}.$ Therefore the differential inequality (4.5) implies that, for every $t\geq t_{2}$, we have

where

Applying Fubini's theorem, we then infer that

Since $\gamma(t) = \frac{\alpha}{t^{\theta}}$, a simple integration by parts ensures the existence of two real constants $B_{\theta } > A_{\theta} > 0$ so that

Hence, by combining the inequality (4.6) and the assumption

we get

which contradicts our assumption on the function $\varepsilon(.)$. We therefore deduce that

Now let us suppose that

The continuity of the function $h$ combined with (4.7) and (4.8) ensures the existence of two real numbers $\lambda < \delta$ and two positive real sequences $(s_{n})_{n}$ and $(t_{n})_{n}$ such that for every $n\in\mathbb{N}$ we have

where $t_{\ast} > t_{2}$ is a fixed positive number such that for every $t\geq t_{\ast}$

We deduce from (4.5) that for every $n\in\mathbb{N}$ and for all $t\in\lbrack s_{n}, t_{n}]$

Multiplying the last differential inequality by $t^{\theta}$ and integrating over $[s_{n}, t_{n}],$ we obtain

Using now the three facts

and letting $n$ goes to $+\infty$ in the the inequality (4.9), we get

This contradicts the assumption $\delta > \lambda.$ We therefore conclude that $\lim\limits_{t\rightarrow+\infty}h(t) = 0,$ which completes the proof of the assertion (Q$_{1}$).

In order to prove the assertion (Q$_2$), we first apply 3.1 with $v = x^\star$ to deduce that the energy function $W$ satisfies the asymptotic property (1.5) which implies in particular the solution $x(.)$ fulfils the assumption in the previous assertion (Q$_1$). Therefore, we conclude that $x(t)$ converges strongly to $p^\star$ which is, in the present case, equal to $x^\star$. This ends the proof of our main theorem.

Let us now prove the theorem 1.5 of Attouch and Cazernicki.

Proof. The dynamical system (1.6) is the particular case of the general system (GAVD$_{\gamma, \varepsilon}$) corresponding to $\theta = 0$ and $U(x) = \frac{1}{2} \left\Vert x \right\Vert^2$. It follows from Theorem 1.2 that any solution $x(.)$ of the dynamical system (1.6) satisfies the assumptions of the assertion (Q$_{1}$) of Theorem 1.4. Therefore, we deduce that $x(t)$ converges strongly to the unique minimizer $p^{\ast}$ of the function of $U$. To conclude we notice that, in this case, $p^{\ast}$ is the projection of zero on $S_{\Phi}$.

Remark 4.1. Without the regularizing term $\epsilon(t) \nabla U (x(t))$ and under appropriate assumptions on the damping term, the trajectories of the dynamical system (GAVD$_{\gamma, \varepsilon}$) weakly converge to a non-specified minimizer of the objective functional $\Phi$ (See Theorem 1.3 with $\epsilon (t) = 0)$. As it is shown in the main theorem (Theorem 1.4), if $\epsilon (t)$ vanishes slowly at infinity, then te regularizing term $\epsilon(t) \nabla U (x(t))$ forces the trajectories of the differential system (GAVD$_{\gamma, \varepsilon}$) to converges strongly to a particular minimizer of $\Phi$. In this sense, The added term $\epsilon(t) \nabla U (x(t))$ may be considered as a stabilizer factor for our dynamical system.

Acknowledgments

The authors acknowledge the Deanship of Scientific Research at King Faisal University for the financial support under the annual research project (Grant No. 170065). The authors are also indebted to the referee for his careful reading and insightful comments.

Conflict of interest

The authors declare that they have no conflict of interest.