We consider the dynamical system \begin{ equation*}\left\{ \begin{ array}{ ll} v(t)\in\partial\phi(x(t))\\ \lambda\dot x(t) + \dot v(t) + v(t) + \nabla \psi(x(t))=0, \end{ array}\right.\end{ equation*} where $\phi:\R^n\to\R\cup\{ +\infty\}$ is a proper, convex and lower semicontinuous function, $\psi:\R^n\to\R$ is a (possibly nonconvex) smooth function and $\lambda>0$ is a parameter which controls the velocity. We show that the set of limit points of the trajectory $x$ is contained in the set of critical points of the objective function $\phi+\psi$, which is here seen as the set of the zeros of its limiting subdifferential. If the objective function satisfies the Kurdyka-\L{ }ojasiewicz property, then we can prove convergence of the whole trajectory $x$ to a critical point. Furthermore, convergence rates for the orbits are obtained in terms of the \L{ }ojasiewicz exponent of the objective function, provided the latter satisfies the \L{ }ojasiewicz property.
Categories: math.OC math.DS