We begin by considering second order dynamical systems of the from $\ddotx(t) + \gamma(t)\dot x(t) + \lambda(t)B(x(t))=0$, where $B: {\calH}\rightarrow{\cal H}$ is a cocoercive operator defined on a real Hilbert space${\cal H}$, $\lambda:[0,+\infty)\rightarrow [0,+\infty)$ is a relaxationfunction and $\gamma:[0,+\infty)\rightarrow [0,+\infty)$ a damping function,both depending on time. For the generated trajectories, we show existence anduniqueness of the generated trajectories as well as their weak asymptoticconvergence to a zero of the operator $B$. The framework allows to address fromsimilar perspectives second order dynamical systems associated with the problemof finding zeros of the sum of a maximally monotone operator and a cocoerciveone. This captures as particular case the minimization of the sum of anonsmooth convex function with a smooth convex one. Furthermore, we prove thatwhen $B$ is the gradient of a smooth convex function the value of the latterconverges along the ergodic trajectory to its minimal value with a rate of${\cal O}(1/t)$.
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机译：我们首先考虑$ \ ddotx（t）+ \ gamma（t）\ dot x（t）+ \ lambda（t）B（x（t））= 0 $的二阶动力系统，其中$ B： {\ calH} \ rightarrow {\ cal H} $是在实希尔伯特空间$ {\ cal H} $，$ \ lambda：[0，+ \ infty）\ rightarrow [0，+ \ infty）上定义的矫顽算子$是松弛函数，$ \ gamma：[0，+ \ infty）\ rightarrow [0，+ \ infty）$是阻尼函数，两者都取决于时间。对于生成的轨迹，我们显示了生成的轨迹的存在性和唯一性，以及它们到算子$ B $零的弱渐近收敛性。该框架允许从相似的角度解决与寻找最大单调算子和椰油子酮之和为零的问题有关的二阶动力学系统。在特定情况下，这捕获了具有平滑凸函数的非光滑凸函数之和的最小化。此外，我们证明当$ B $是光滑凸函数的梯度时，后者的值沿遍历轨迹收敛到其最小值，且速率为$ {\ cal O}（1 / t）$。
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