Second-Order Viability Problems with Perturbation in Hilbert Spaces

In this paper, we study the existence of viable solutions to the differential inclusion

$ \ddot{x}(t) \in f\left( {t,x(t),\dot{x}(t)} \right) + F\left( {x(t),\dot{x}(t)} \right), $

where f is a Carathéodory single-valued map and F is an upper semi-continuous multifunction with compact values contained in the Clarke subdifferential ? _c V of an uniformly regular function V.