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Second-order continuous-time dissipative dynamical systems with viscous andHessian driven damping have inspired effective first-order algorithms forsolving convex optimization problems. While preserving the fast convergenceproperties of the Nesterov-type acceleration, the Hessian driven damping makesit possible to significantly attenuate the oscillations. To study the stabilityof these algorithms with respect to perturbations, we analyze the behaviour ofthe corresponding continuous systems when the gradient computation is subjectto exogenous additive errors. We provide a quantitative analysis of theasymptotic behaviour of two types of systems, those with implicit and explicitHessian driven damping. We consider convex, strongly convex, and non-smoothobjective functions defined on a real Hilbert space and show that, depending onthe formulation, different integrability conditions on the perturbations aresufficient to maintain the convergence rates of the systems. We highlight thedifferences between the implicit and explicit Hessian damping, and inparticular point out that the assumptions on the objective and perturbationsneeded in the implicit case are more stringent than in the explicit case.

On the effect of perturbations in first-order optimization methods with inertia and Hessian driven damping