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

• Published in: AIMS mathematics Vol. 6; no. 5; pp. 4901 - 4914
• Main Authors: May, Ramzi, Mnasri, Chokri, Elloumi, Mounir
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2021
• Subjects: asymptotic behavior; convex optimization; ordinary differential equations; regularization method
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2021287

Let $\mathcal{H}$\ be a real Hilbert space. We investigate the long time behavior of the trajectories $x(.)$ of the vanishing damped nonlinear dynamical system with regularizing term% \begin{equation} x^{\prime\prime}(t)+\gamma(t)x^{\prime}(t)+\nabla\Phi(x(t))+\varepsilon (t)\nabla U(x(t))=0, \tag{GAVD$_{\gamma,\varepsilon}$}% \end{equation} where $\Phi,U:\mathcal{H}\rightarrow\mathbb{R}$ are two convex continuously differentiable functions, $\varepsilon(.)$ is a decreasing function satisfying $\displaystyle\lim_{t\rightarrow+\infty}\varepsilon(t)=0,$ and $\gamma(.)$ is a nonnegative function which behaves, for $t$ large enough, like $\displaystyle\frac{K}{t^{\theta}}$ where $K>0$ and $0\leq\theta\leq1.$ The main contribution of this paper is the following control result: If $\displaystyle\int_{0}^{+\infty}\frac{\varepsilon(t)}% {\gamma(t)}dt=+\infty,$ $U$ \ is strongly convex and its unique minimizer $x^{\ast}$ is also a minimizer of $\Phi$ then every trajectory $x(.)$ of (GAVD$_{\gamma,\varepsilon}$) converges strongly to $x^{\ast}$ and the rate of convergence to $0$ of its energy function $$W(t)=\frac{1}{2}\left\Vert x^{\prime}(t)\right\Vert ^{2}+\Phi(x(t))-\Phi ^{\ast}+\varepsilon(t)(U(x(t))-U^{\ast})$$ is of order to $\circ(1/t^{1+\theta})$. Moreover, we prove a new result concerning the weak convergence of the trajectories of (GAVD$_{\gamma,\varepsilon}$) to a common minimizer of $\Phi$ and $U$ (if one exists) under a simple condition on the speed of decay of the regularizing factor $\varepsilon(t)$ to $0$.