  Second-order continuous-time dissipative dynamical systems with viscous and
Hessian driven damping have inspired effective first-order algorithms for
solving convex optimization problems. While preserving the fast convergence
properties of the Nesterov-type acceleration, the Hessian driven damping makes
it possible to significantly attenuate the oscillations. To study the stability
of these algorithms with respect to perturbations, we analyze the behaviour of
the corresponding continuous systems when the gradient computation is subject
to exogenous additive errors. We provide a quantitative analysis of the
asymptotic behaviour of two types of systems, those with implicit and explicit
Hessian driven damping. We consider convex, strongly convex, and non-smooth
objective functions defined on a real Hilbert space and show that, depending on
the formulation, different integrability conditions on the perturbations are
sufficient to maintain the convergence rates of the systems. We highlight the
differences between the implicit and explicit Hessian damping, and in
particular point out that the assumptions on the objective and perturbations
needed in the implicit case are more stringent than in the explicit case.
