Abstract
We propose and study the convergence properties of the trajectories generated by a damped inertial dynamic which is driven by the sum of potential and nonpotential operators. Precisely, we seek to reach asymptotically the zeros of sums of potential term (the gradient of a continuously differentiable convex function) and nonpotential monotone and cocoercive operator. As an original feature, in addition to viscous friction, the dynamic involves implicit Newton-type damping. This contrasts with the authors' previous study where explicit Newton-type damping was considered, which, for the potential term, corresponds to Hessian-driven damping. We show the weak convergence, as time goes to infinity, of the generated trajectories towards the zeros of the sum of the potential and nonpotential operators. Our results are based on Lyapunov analysis and appropriate setting of damping parameters. The introduction of geometric dampings allows to control and attenuate the oscillations known for the viscous damping of inertial methods. Rewriting the second-order evolution equation as a system involving only first order derivative in time and space allows us to extend the convergence analysis to nonsmooth convex potentials. Our study concerns the autonomous case with positive fixed parameters. These results open the door to their extension to the nonautonomous case and to the design of new first-order accelerated algorithms in optimization taking into account the specific properties of potential and nonpotential terms. The proofs and techniques are original due to the presence of the nonpotential term.