The heavy ball with friction method, I. The continuous dynamical system: Global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system

Let H be a real Hilbert space and Phi : H --> R a continuously differentiable function, whose gradient is Lipschitz continuous on bounded sets. We study the nonlinear dissipative dynamical system: x(t) + lambda--x(t) + del Phi(x(t)) = 0, lambda > 0, plus Cauchy data, mainly in view of the unconstrained minimization of the function Phi. New results concerning the convergence of a solution to a critical point are given in various situations, including when Phi is convex (possibly with multiple minima) or is a Morse function (the critical point being then generically a local minimum); a counterexample shows that, without peculiar assumptions, a trajectory may not converge. By following the trajectories, we obtain a method for exploring local minima of Phi. A singular perturbation analysis links our results with those concerning gradient systems.