Abstract
Let H be a real Hilbert space. We investigate the long time behavior of the trajectories x(.) of the vanishing damped nonlinear dynamical system with regularizing term
x ''(t) + gamma(t)x0(t) + del phi(x(t)) + epsilon(t) del U(x(t)) = 0, (GAVD gamma,epsilon)
where phi, U : H -> R are two convex continuously differentiable functions, epsilon(.) is a decreasing function satisfying lim(t ->+infinity) epsilon(t) = 0, and gamma(.) is a nonnegative function which behaves, ofr t large enough, like K/t(theta) where K > 0 and 0 <= theta <= 1. The main contribution of this paper is the following control result: If integral(+infinity)(0)epsilon(t)/gamma(t)dt = +infinity, U is strongly convex and its unique minimizer x* is also a minimizer of phi then epsilon(t) 0 every trajectory x(.) of (GAVD gamma,epsilon) converges strongly to x* and the rate of convergence to 0 of its energy function
W(t) = 1/2 parallel to x'(t)parallel to(2) + Phi(x(t)) - Phi* + epsilon(t)(U(x(t)) - U*)
is of order to o(1/t(1+theta)). Moreover, we prove a new result concerning the weak convergence of the trajectories of (GAVD gamma,epsilon) to a common minimizer of Phi and U (if one exists) under a simple condition on the speed of decay of the regularizing factor epsilon(t) to 0.