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

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, \;\;\;\;\;\;\;\;\;({\rm{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 $ \lim\limits_{t\rightarrow+\infty}\varepsilon(t) = 0, $ and $ \gamma(.) $ is a nonnegative function which behaves, for $ t $ large enough, like $ \frac{K}{t^{\theta}} $ where $ K > 0 $ and $ 0\leq\theta\leq1. $ The main contribution of this paper is the following control result: If $ \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 $.