In this talk, we will discuss the asymptotic behavior of the solutions of differential inclusions governed by maximally monotone operators. In the case where the LaSalle’s invariance principle is inconclusive, we provide a refined version of the invariance principle theorem. This result derives from the problem of locating the ω-limit set of a bounded solution of the dynamic. In addition, we propose an extension of LaSalle’s invariance principle, which allows us to give a sharper location of the ω-limit set. The provided results are given in terms of nonsmooth Lyapunov pair-type functions. We will conclude this presentation by applying our results to an important second-order gradient-like dissipative dynamical system with Hessian-driven damping called (DIN) by Alvarez et and his collaborators.