On a class of second-order dynamical systems involving viscous and Hessian-driven damping

In the framework of real Hilbert space, we consider the problem of minimizing a convex and twice continuously differentiable function with Lipschitz continuous gradient on bounded sets. Our approach is based on a second-order dynamical system involving both viscous and Hessian-driven dampings, as well as a time scaling term. This dynamics is closely related to the inertial systems with Hessian-driven damping, introduced by Attouch and his co-authors in [17,18]. The first damping term vanishes asymptotically, while the latter fulfills certain growth conditions. The time scaling coefficient in our case, however, is chosen in a specific way in terms of damping parameters to enhance the convergence properties. We first study the existence and uniqueness of the trajectory. By assuming that the gradient tends to zero, we show the generated trajectory converges strongly to a specific element in the solution set. Our analysis will give a precise expression for this destination, which is indeed constructed from the initialization. We then derive the convergence rate of the velocities, gradient norm, and function value, which are influenced by the growth condition of the parameter. Some numerical experiments are conducted to illustrate the theoretical results. As a final remark, we show that there is actually a connection between our considered equation and a class of first-order dynamical systems with Tikhonov regularization, hence explaining the strong convergence of the trajectory as well as fast rates following recent studies.