We introduce non-autonomous continuous dynamical systems which are linked to Newton and Levenberg-Marquardt methods. They aim at solving inclusions governed by maximal monotone operators in Hilbert spaces. Relying on Minty representation of maximal monotone operators as lipschitzian manifolds, we show that these dynamics can be formulated as first-order in time differential systems, which are relevant to Cauchy-Lipschitz theorem. By using Lyapunov analysis, we prove that their trajectories asymptotically weakly converge to equilibria. Discrete time version of these results provides new insight on Newton's method for solving monotone inclusions.
Submitted in Jan 27, 2010 to SIAM Journal on Control and Optimization