Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization

Abstract

We consider the Tikhonov-like dynamics -(u) over dot(t) is an element of A(u(t)) + epsilon(t)u(t) where A is a maximal monotone operator on a Hilbert space and the parameter function epsilon(t) tends to 0 as t -> infinity with integral(infinity)(0)epsilon(t) dt = infinity. When A is the subdifferential of a closed proper convex function f, we establish strong convergence of u(t) towards the least-norm minimizer of f. In the general case we prove strong convergence towards the least-norm point in A(-1)(0) provided that the function epsilon(t) has bounded variation, and provide a counterexample when this property fails.