Strong convergence behavior of a second order dynamical system with Hessian-Driven and Vanishing Damping

In relation to the minimization of a convex differentiable function, we examine the convergence behavior of solutions to a second order dynamical system that contains a Hessian term, time rescaling and a Tikhonov regularisation parameter. The system also contains a viscous damping coefficient term scaled by the square root of the Tikhonov regularisation parameter. The introduction of a Tikhonov regularisation term ensures the strong convergence of the generated trajectory towards the minimum norm solution, rather than weak convergence to an arbitrary minimizer. In addition, the time scale parameter has been used to accelerate the convergence rate of the objective values along its trajectories, and the Hessian term to attenuate any transverse oscillations that might arise. So, on the basis of the Lyapunov analysis and for appropriate conditions on the time scale parameter, we simultaneously demonstrate that the trajectories converge strongly toward the minimum norm minimizer, and that the objective function values converge rapidly to the global minimum, as well as a convergence towards zero of the velocity and the gradient. Moreover we show some integral estimates.