Approaching nonsmooth nonconvex optimization problems through first order dynamical systems with hidden acceleration and Hessian driven damping terms

In this paper we carry out an asymptotic analysis of the proximal-gradient dynamical system \begin{equation*}\left{ \begin{array}{ll} \dot x(t) +x(t) = \prox_{\gamma f}\big[x(t)-\gamma\nabla\Phi(x(t))-ax(t)-by(t)\big],\ \dot y(t)+ax(t)+by(t)=0 \end{array}\right.\end{equation*} where $f$ is a proper, convex and lower semicontinuous function, $\mathrm{\Phi }$ a possibly nonconvex smooth function and $\gamma ,a$ and $b$ are positive real numbers. We show that the generated trajectories approach the set of critical points of $f+\mathrm{\Phi }$, here understood as zeros of its limiting subdifferential, under the premise that a regularization of this sum function satisfies the Kurdyka-\L{}ojasiewicz property. We also establish convergence rates for the trajectories, formulated in terms of the \L{}ojasiewicz exponent of the considered regularization function.