The Heavy ball method regularized by Tikhonov term. Simultaneous convergence of values and trajectories

Let $ f: {\mathcal H} \rightarrow \mathbb{R} be a convex differentiable function whose solution set {{\rm{argmin}}}\; f is nonempty. To attain of the problem \min_{\mathcal H}f $, we consider second order dynamic system \;\ddot{x}(t) + \alpha \, \dot{x}(t) \beta (t) \nabla f(x(t)) c x(t) = 0 where positive such that \lim_{t\rightarrow +\infty}\beta(t) +\infty $. By imposing adequate hypothesis on first and derivatives simultaneously prove value objective in generated trajectory converges O}\big(\frac{1}{\beta(t)}\big) to global minimum function, strongly norm element \Vert \dot{x}(t)\Vert zero \mathcal{O} \big( \sqrt{\frac{\dot{\beta}(t)}{\beta (t)}}+ e^{-\mu t} \big) \mu<\frac{\alpha}2 We then present two choices illustrate these results. On basis Moreau regularization technique, extend results non-smooth functions with extended real values.