The main objective of this paper is to provide new explicit criteria to characterize weak lower semi-continuous Lyapunov pairs or functions associated to first-order differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator. The dual criteria we give are expressed by the means of the proximal subdifferential of the nominal functions while primal conditions are in terms of the Dini directional derivative. Further, we propose a unifying review of many other criteria given in the literature. Our approach is based on advanced tools of variational analysis and generalized differentiation.
XLIM, 123 avenue A. Thomas, 87060 LIMOGES CEDEX, May 2010