We consider a class of inertial second order dynamical system with Tikhonov regularization, which can be applied to solving the minimization of a smooth convex function. Based on the appropriate choices of the parameters in the dynamical system, we first show that the function value along the trajectories converges to the optimal value, and prove that the convergence rate can be faster than \(o(1/t^2)\). Moreover, by constructing proper energy function, we prove that the trajectories strongly converges to a minimizer of the objective function of minimum norm. Finally, some numerical experiments have been conducted to illustrate the theoretical results.

中文翻译：

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具有Tikhonov正则化的一类惯性动力系统的收敛性

我们考虑一类具有Tikhonov正则化的惯性二阶动力系统，该系统可用于求解光滑凸函数的最小化。基于所述参数中的动力系统的适当的选择，我们首先表明，沿轨迹收敛到最优值的函数值，并证明了收敛速度可以比更快\（O（1 / T ^ 2）\ ）。此外，通过构造适当的能量函数，我们证明了该轨迹强烈收敛于最小范数的目标函数的极小值。最后，进行了一些数值实验以说明理论结果。