Abstract
This thesis is devoted to finding zeroes of structured maximal monotone operators, by using discrete and continuous dissipative dynamical systems. The solutions are obtained as the limits of trajectories when the time t tends towards infinity. We pay special attention to the dynamics that are obtained by Levenberg-Marquardt regularization of Newton's method. We also revisit the approaches based on some related dynamical systems.In a Hilbert framework, we are interested in finding zeroes of a structured maximal monotone operator M = A + B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. We introduce discrete and continuous dynamical systems which are linked to Newton's method. They involve separately B and the resolvents of A, and are designed to splitting methods. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy-Lipschitz theorem. We focus on the particular case where A is the subdifferential of a convex lower semicontinuous proper function, and B is the gradient of a convex, continuously differentiable function. We study the asymptotic behavior of trajectories. When the regularization parameter does not tend to zero too rapidly, and by using Lyapunov asymptotic analysis, we show the convergence of trajectories. Besides, we show the Lipschitz continuous dependence of the solution with respect to the regularization term. Then we extend our study by considering various classes of dynamical systems which aim at solving inclusions governed by structured monotone operators M = ∂Φ + B, where ∂Φ is the subdifferential of a convex lower semicontinuous function, and B is a monotone cocoercive operator. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems. The time discretization of these dynamics gives various forward-backward splittingmethods (some new). Finally, we focus on the study of the asymptotic behavior of trajectories of the regularized Newton dynamics, in which we introduce an additional vanishing Tikhonov-like viscosity term. We thus obtain the asymptotic selection of the solution of minimal norm.