  In a Hilbert space setting H, for convex optimization, we analyze the fast
convergence properties as t tends to infinity of the trajectories generated by
a third-order in time evolution system. The function f to minimize is supposed
to be convex, continuously differentiable, with a nonempty set of minimizers.
It enters into the dynamic through its gradient. Based on this new dynamical
system, we improve the results obtained by [Attouch, Chbani, Riahi: Fast convex
optimization via a third-order in time evolution equation, Optimization 2020].
As a main result, when the damping parameter $\alpha$ satisfies $\alpha > 3$,
we show that the convergence of the values at the order 1/t3 as t goes to
infinity, as well as the convergence of the trajectories. We complement these
results by introducing into the dynamic an Hessian driven damping term, which
reduces the oscillations. In the case of a strongly convex function f, we show
an autonomous evolution system of the third order in time with an exponential
rate of convergence. All these results have natural extensions to the case of a
convex lower semicontinuous function with extended real values. Just replace f
with its Moreau envelope.
